% Reorganized and reconstructed exposition. % Build: latexmk -pdf -interaction=nonstopmode -halt-on-error main.tex \documentclass[11pt]{amsart} \IfFileExists{lmodern.sty}{\usepackage[T1]{fontenc}\usepackage{lmodern}}{} \usepackage[utf8]{inputenc} \usepackage{amsmath,amssymb,amsthm,mathtools,mathrsfs} \usepackage[margin=1in]{geometry} \usepackage[expansion=false]{microtype} \usepackage{enumitem} \usepackage{xcolor} \usepackage[colorlinks=true,linkcolor=blue!60!black,citecolor=blue!60!black,urlcolor=blue!60!black]{hyperref} \usepackage[capitalise,nameinlink]{cleveref} \setlength{\emergencystretch}{3em} \setcounter{tocdepth}{1} \hypersetup{pdftitle={Small-image isometric embeddings of parallelizable and stably parallelizable Riemannian manifolds}} % ---------- notation ---------- \newcommand{\R}{\mathbb{R}} \newcommand{\C}{\mathbb{C}} \newcommand{\Z}{\mathbb{Z}} \newcommand{\cL}{\mathcal{L}} \newcommand{\cR}{\mathcal{R}} \newcommand{\cP}{\mathcal{P}} \newcommand{\cX}{\mathcal{X}} \newcommand{\cY}{\mathcal{Y}} \newcommand{\dd}{\mathrm{d}} \newcommand{\ii}{\mathrm{i}} \newcommand{\ip}[2]{\langle #1,#2\rangle} \newcommand{\abs}[1]{\lvert #1\rvert} \newcommand{\norm}[1]{\lVert #1\rVert} \newcommand{\sym}{\odot} \newcommand{\av}{\operatorname{av}} \newcommand{\Cb}{C_b} \newcommand{\triv}[1]{\underline{\R}^{#1}} \newcommand{\T}{\mathsf T} \DeclareMathOperator{\Sym}{Sym} \DeclareMathOperator{\Hom}{Hom} \DeclareMathOperator{\im}{im} \DeclareMathOperator{\rank}{rank} \DeclareMathOperator{\spanop}{span} \DeclareMathOperator{\Id}{Id} \DeclareMathOperator{\SO}{SO} \DeclareMathOperator{\diag}{diag} \DeclareMathOperator{\Cross}{Cross} \DeclareMathOperator{\Mat}{Mat} \DeclareMathOperator{\Gr}{Gr} % ---------- theorem environments ---------- \theoremstyle{plain} \newtheorem{thm}{Theorem}[section] \newtheorem{lem}[thm]{Lemma} \newtheorem{prop}[thm]{Proposition} \newtheorem{cor}[thm]{Corollary} \newtheorem{mainthm}{Theorem} \renewcommand{\themainthm}{\Alph{mainthm}} \newtheorem{maincor}[mainthm]{Corollary} \theoremstyle{definition} \newtheorem{defn}[thm]{Definition} \newtheorem{setup}[thm]{Setup} \newtheorem{notation}[thm]{Notation} \theoremstyle{remark} \newtheorem{rem}[thm]{Remark} \numberwithin{equation}{section} \crefname{mainthm}{Theorem}{Theorems} \crefname{maincor}{Corollary}{Corollaries} \crefname{setup}{Setup}{Setups} \crefname{notation}{Notation}{Notations} \crefname{thm}{Theorem}{Theorems} \crefname{lem}{Lemma}{Lemmas} \crefname{prop}{Proposition}{Propositions} \crefname{cor}{Corollary}{Corollaries} \crefname{defn}{Definition}{Definitions} \crefname{rem}{Remark}{Remarks} \title[Isometric embeddings]{Isometric embeddings of parallelizable and stably parallelizable Riemannian manifolds} \author{Wenchang Yang} \address{Tsinghua University, Beijing, China} \author{Yi Wang} \address{Tsinghua University, Beijing, China} \date{} \keywords{Isometric embedding, parallelizable manifold, second fundamental form, Hopf action, metric correction, h-principle} \subjclass[2020]{53C42, 53C24, 57R25} \begin{document} \begin{abstract} Let $s_n=n(n+1)/2$. We prove that every smooth parallelizable Riemannian $n$-manifold, $n\ge 3$, admits a smooth isometric embedding into $\mathbb{R}^{s_n+1}$ whose image lies in an arbitrarily small ball. For closed stably parallelizable manifolds, each of whose connected components has vanishing Euler characteristic, we obtain small-image isometric embeddings into $\mathbb{R}^{s_n+5}$. The proof combines explicit partial isometries along commuting Hopf flows, an iterative scheme for exact metric correction using oscillatory perturbations constructed from the second fundamental form, and a transversality argument for metric-preserving perturbations. \end{abstract} \maketitle \tableofcontents % ---- begin sec1-intro.tex ---- \section{Introduction}\label{sec:intro} \subsection{Background} A smooth map $F\colon M\to\R^N$ induces the symmetric tensor $F^*\delta$, where $\delta$ is the Euclidean metric; in local coordinates \begin{equation}\label{eq:intro-metric} (F^*\delta)_{ij}=\ip{\partial_iF}{\partial_jF}. \end{equation} The isometric embedding problem asks for $F$ with $F^*\delta=g$ for a prescribed Riemannian metric $g$. Throughout we write \[ s_n=\frac{n(n+1)}2,\qquad q_n=s_{n-1}=\frac{n(n-1)}2 . \] The system \eqref{eq:intro-metric} consists of $s_n$ equations, so $N=s_n$ is the determined dimension. By the Cartan--Janet theorem \cite{Janet26,Cartan27}, real-analytic metrics are locally induced by analytic embeddings into $\R^{s_n}$. Globally, Nash \cite[Theorem~2]{Nash56} showed that every closed Riemannian $n$-manifold embeds isometrically into an arbitrarily small ball of $\R^{n(3n+11)/2}$. Nash's argument perturbs \emph{free} maps, i.e.\ maps whose first and second derivatives are linearly independent; this already requires $N\ge s_n+n$. G\"unther \cite{Gunther1991} (see also \cite{Yang1998Gunther}) recast the perturbation step as a contraction and obtained the dimension \[ N=\max\Bigl\{\frac{n(n+5)}2,\frac{n(n+3)}2+5\Bigr\}=s_n+\max\{2n,\,n+5\}. \] In this paper we show that, for manifolds with trivial tangent bundle, one extra dimension beyond the determined case suffices. \subsection{Main results} Manifolds are smooth, Hausdorff and second countable, and have no boundary unless stated otherwise; a closed manifold is compact without boundary. Write $\triv j$ for the trivial real vector bundle of rank $j$ and $B_r(p)$ for the open Euclidean ball. An immersion $F\colon M\to\R^N$ is \emph{phase regular} if at every point $x$ there is a tangent hyperplane $P\subset T_xM$ such that the normal second fundamental form of $F$ is injective on $\Sym^2P$; see \cref{defn:phase}. \begin{mainthm}\label{thm:A} Let $M$ be a parallelizable $n$-manifold, $n\ge 3$, possibly noncompact, disconnected, or with smooth boundary, and let $g$ be a smooth Riemannian metric on $M$, smooth up to the boundary. For every $p\in\R^{s_n+1}$ and $r>0$ there is a smooth phase-regular embedding \[ F\colon M\longrightarrow\R^{s_n+1},\qquad F^*\delta=g,\qquad F(M)\subset B_r(p). \] If $M$ is noncompact and has no boundary, $F$ can moreover be chosen so that $F(x)\ne p$ for all $x$, $F(x)\to p$ as $x$ leaves every compact set, and $F$ is a proper embedding into $\R^{s_n+1}\setminus\{p\}$. \end{mainthm} For instance, every closed orientable $3$-manifold is parallelizable (Stiefel \cite{Stiefel1935}; see also \cite{Kirby1989}), so every closed orientable Riemannian $3$-manifold embeds isometrically into an arbitrarily small ball of $\R^7$. Products $S^1\times N$ with $N$ stably parallelizable, and all Lie groups, are further examples. For tangent bundles that are only stably trivial we obtain: \begin{mainthm}\label{thm:B} Let $M$ be a closed $n$-manifold, $n\ge3$, with $TM\oplus\triv k\cong\triv{n+k}$ for some $k\ge0$, and assume that every component of $M$ has Euler characteristic zero. Then every smooth Riemannian metric on $M$ is induced by a smooth phase-regular embedding into an arbitrarily small ball of $\R^{s_n+5}$. \end{mainthm} For example, the spheres $S^{2m+1}$, $m\ge1$, and the products $S^{2m+1}\times N$ with $N$ closed and stably parallelizable satisfy the hypotheses of \cref{thm:B}. On manifolds all of whose components are noncompact, stable triviality of the tangent bundle already implies triviality (\cref{prop:open-cancellation}), so \cref{thm:A} applies: \begin{maincor}\label{cor:C} Let $M$ be an $n$-manifold without boundary, $n\ge3$, all of whose components are noncompact, and suppose $TM\oplus\triv k\cong\triv{n+k}$ for some $k\ge 0$. Then the conclusions of \cref{thm:A}, including the properness statement, hold for every smooth Riemannian metric on $M$. \end{maincor} No completeness, finite topology, or bounded geometry is assumed in the noncompact case. \subsection{Phase regularity} Let $F$ be an immersion and $x\in M$. The normal second fundamental form is a linear map $B_F(x)\colon\Sym^2T_xM\to\nu_x$ into the normal space, of dimension $N-n$. A nonzero covector $\xi$ is \emph{good} if $B_F(x)$ is injective on $\Sym^2\ker\xi$, a space of dimension $q_n$. A good covector can exist only if $N\ge s_n$, and when it exists, the space of \emph{admissible normals} \[ E_\xi=\nu_x\ominus B_F(x)\bigl(\Sym^2\ker\xi\bigr) \] has dimension $N-s_n$. In the dimension $N=s_n+1$ of \cref{thm:A}, $E_\xi$ is a line. Freeness, by contrast, is the injectivity of $B_F$ on all of $\Sym^2T_xM$, which needs $N\ge s_n+n$. Admissible normals are what make metric correction possible. Let $\psi$ be a local function with $\dd\psi$ good, let $\nu\in E_{\dd\psi}$ be a unit normal field, and add the oscillation $U=\tau\delta b\,\nu\cos(\psi/\tau)$ with $0<\delta\le\tau\ll1$. The quadratic part of the change of metric has mean $\tfrac12\delta^2b^2\,\dd\psi^2$, which is the increment we want. The linear part, $-2\ip{B_F}{U}$ to leading order, vanishes on $\ker\dd\psi\times\ker\dd\psi$ precisely because $\nu$ is admissible. The components that do involve the direction $\dd\psi$ are absorbed by tangential corrections, because conjugation by $e^{\ii\psi/\tau}$ turns the operator $u\mapsto\dd\psi\sym u$ into a dominant term of size $\tau^{-1}$; a finite Neumann series cancels them to any prescribed order in $\tau$ divided by the scale of the data. Iterating such explicit increments, with smoothing, produces exact solutions. The method needs $E_\xi\ne0$, hence $N\ge s_n+1$. \subsection{Strategy} The proof of \cref{thm:A} for closed $M$ has four steps. \smallskip\noindent\emph{Step 1: model partial isometries} (\cref{sec:model}). On a product $X=\prod_{i=1}^nS^{2C-1}$ of Hopf spheres with commuting period-one Hopf fields $Y_1,\dots,Y_n$, we construct maps $S_k\colon X\to\R^{s_n+1}$, $k\to\infty$, with \[ \ip{Y_iS_k}{Y_jS_k}=\delta_{ij}+O(k^{-\infty}),\qquad \norm{S_k}_{C^j}\lesssim k^{\max(j-1,0)}, \] whose normal second form along $\Sym^2\spanop(Y_1,\dots,Y_{n-1})$ has smallest singular value $\gtrsim k^{-1/2}$. They are obtained from explicit curves on Hopf spheres, matrix-valued primitives with vanishing mean, formal power-series corrections, a normal collar, and a final wrapping along the last factor. \smallskip\noindent\emph{Step 2: exact metric correction} (\cref{sec:correction}). If an immersion into $\R^N$, $N\ge s_n+1$, has uniformly elliptic induced metric, $C^{119}$ norm $O(\epsilon^{-A})$ and phase margin $\gtrsim\epsilon^\kappa$, then for small $\epsilon$ every metric within $O(\epsilon^{K})$ of its induced metric in $C^{117}$, with $K=K(A,\kappa,b)$, is realized exactly by a $C^2$-perturbation of size $\epsilon^b$. All estimates are uniform over atlases of bounded overlap, with no dependence on the number of charts. \smallskip\noindent\emph{Step 3: local phases} (\cref{sec:immersion}). Given an orthonormal coframe $\alpha_1,\dots,\alpha_n$ of $(M,g)$, we build maps $z_\ell\colon M\to S^{2C-1}$ which near each point of a fine net look like $e^{2\pi\ii T\lambda(x)}$ with $\dd\lambda\approx\alpha_\ell$, glued by a colouring argument. Then $S_k(z_1,\dots,z_n)$ has induced metric $T^2g$ up to a polynomially small error; Step~2 makes it exact, and division by $T$ gives a phase-regular isometric immersion into a small ball. The coframe need not be closed. \smallskip\noindent\emph{Step 4: separation of points} (\cref{sec:embedding}). Exact-metric perturbations with prescribed first variations of values and second forms (\emph{probes}) are produced by the iteration of Step~2. A dimension estimate for spaces of quadratic forms shows that at least $n+1$ admissible normal value directions exist, except along a rank stratum and an exceptional locus of second forms, which are removed by adding their defining equations. Sard's theorem and a Baire argument then give an embedding with the same metric. \smallskip For \cref{thm:B} (\cref{sec:stable}), a nonvanishing vector field splits $TM=H\oplus\triv1$ with $H\oplus\Sym^2H$ trivial; the model map of Step~1 is replaced by a partially free map produced from this bundle data by an h-principle for open ample relations and by G\"unther-type local metric additions, which cost the four extra dimensions. For noncompact manifolds (\cref{sec:noncompact}), a conformal change $h=a^{-2}g$ to bounded geometry (M\"uller--Nardmann) and the weighted derivative $\dd+\dd\log a$ make every estimate uniform; the decay $a\to0$ gives properness. \subsection{Organization} \Cref{sec:prelim} fixes notation for second forms, partial geometry, Hopf actions and uniform atlases. \Cref{sec:model,sec:correction,sec:immersion,sec:embedding} carry out Steps 1--4 and prove \cref{thm:A} for closed manifolds. \Cref{sec:stable} proves \cref{thm:B}, and \cref{sec:noncompact} treats noncompact manifolds and boundary, completing \cref{thm:A} and \cref{cor:C}. % ---- end sec1-intro.tex ---- % ---- begin sec2-prelim.tex ---- \section{Preliminaries}\label{sec:prelim} \subsection{Second forms and phase regularity} For a map $F\colon M\to\R^N$ we write $\gamma(F)=F^*\delta$. If $F$ is an immersion, $\Pi_F$ denotes the orthogonal projection of $\R^N$ onto the normal space $\nu_F=\dd F(TM)^\perp$; in coordinates $\Pi_F=I-\dd F\,\gamma(F)^{-1}\dd F^{\T}$. \begin{defn}[Second form, phase regularity]\label{defn:phase} The \emph{normal second form} of an immersion $F$ is $B_F(v,w)=\Pi_F\bigl(\nabla^2F(v,w)\bigr)$, where $\nabla$ is any torsion-free connection on $TM$; changing $\nabla$ adds a tangent vector to $\nabla^2F$, so $B_F$ is well defined, and in coordinates $B_{ij}=\Pi_F\partial_i\partial_jF$. We regard $B_F(x)$ as a linear map $\Sym^2T_xM\to\nu_x$. \begin{enumerate}[label=(\roman*)] \item A nonzero $\xi\in T^*_xM$ is \emph{good} for $F$ if $B_F(x)|_{\Sym^2\ker\xi}$ is injective. The immersion is \emph{phase regular} if every point has a good covector. \item For a good $\xi$, the space of \emph{admissible normals} is $E_\xi=E_\xi(F,x)=\nu_x\cap B_F(x)(\Sym^2\ker\xi)^\perp$; it has dimension $N-s_n$. \item Given an auxiliary norm on $TM$, the \emph{phase margin} is \[ \mu(F,x)=\max_{\abs\xi=1}\sigma_{\min}\bigl(B_F(x)|_{\Sym^2\ker\xi}\bigr), \] where $\sigma_{\min}$ is the smallest singular value. \end{enumerate} \end{defn} If $\gamma(F)=h$, the Gauss formula identifies $B_F$ with the full Hessian of $F$ for the Levi-Civita connection of $h$: \begin{equation}\label{eq:gauss} B_{ij}=\partial_i\partial_jF-\Gamma(h)^k_{ij}\partial_kF . \end{equation} \begin{lem}[Stability of second forms]\label{lem:projector} Let $F$ be an immersion of a coordinate ball with $cI\le\gamma(F)$ and $\norm{F}_{C^2}\le C$. There are $\rho,C'>0$, depending only on $c,C,N$, such that $\norm{V}_{C^1}\le\rho$ implies that $F+V$ is an immersion and \[ \abs{B_{F+V}-B_F}\le C'\bigl(\norm V_{C^2}+\norm V_{C^1}\norm F_{C^2}\bigr). \] In particular good covectors, phase regularity, and lower bounds on the phase margin persist under $C^2$-small perturbations, with explicit loss. \end{lem} \begin{proof} The formula $\Pi=I-\dd F\gamma^{-1}\dd F^\T$ shows $\abs{\Pi_{F+V}-\Pi_F}\le C'\norm{V}_{C^1}$ once $\gamma(F+V)\ge cI/2$. Write $B_{F+V}-B_F=\Pi_{F+V}\partial^2V+(\Pi_{F+V}-\Pi_F)\partial^2F$. \end{proof} \begin{rem}[Dimensions]\label{rem:dimension} A good covector forces $N-n\ge q_n$, that is $N\ge s_n$. The constructions of this paper add oscillations along admissible normals, so they need $N\ge s_n+1$. In the dimension $N=s_n+1$, a good covector $\xi$ determines the admissible normal line $E_\xi$, and $B_F(x)$ has rank $q_n$ or $q_n+1$. \end{rem} \subsection{Partial geometry}\label{sec:partial} Let $D\subset TX$ be an integrable distribution of rank $m$. A \emph{flat structure} on $D$ is a covering of $X$ by open sets carrying frames $(Y_1,\dots,Y_m)$ of $D$ by commuting vector fields, such that on overlaps the transition matrices are constant along the leaves of $D$. A partial metric $h_D$ is \emph{flat} if these frames can be chosen $h_D$-orthonormal. \begin{defn}[Partial jets]\label{defn:partial} Let $D$ carry a flat structure and let $H\colon X\to\R^N$ be smooth. \begin{enumerate}[label=(\roman*)] \item The \emph{partial metric} is $\gamma_D(H)=(\dd H|_D)^*\delta$, so $\gamma_D(H)_{\alpha\beta}=\ip{Y_\alpha H}{Y_\beta H}$. \item The \emph{partial Hessian} $\nabla^2_DH\in\Gamma(\Sym^2D^*\otimes\R^N)$ is $\nabla^2_DH(Y_\alpha,Y_\beta)=Y_\alpha Y_\beta H$; this is independent of the flat frame since transition matrices are constant along leaves. \item The \emph{partial jet} is the bundle map \[ J_D(H)\colon D\oplus\Sym^2D\to X\times\R^N,\qquad (v,A)\mapsto \dd H(v)+\nabla^2_DH(A). \] $H$ is \emph{$D$-free} if $J_D(H)$ is injective. Its adjoint $T_H\colon\R^N\to D^*\oplus\Sym^2D^*$ is the \emph{pairing map}; given a partial metric, $C_H=J_D(H)(J_D(H)^*J_D(H))^{-1}$ is the minimum-norm right inverse of $T_H$. \item If $\gamma_D(H)>0$, the \emph{partial normal second form} is $B^D_H=\Pi^D_H\nabla^2_DH$, where $\Pi^D_H$ is the orthogonal projection off $\dd H(D)$. \end{enumerate} \end{defn} Freeness does not depend on the choice of torsion-free partial connection: another choice adds a $\dd H(D)$-valued term to the Hessian. If $\gamma_D(H)$ is constant in a flat orthonormal frame, then $\ip{Y_\alpha Y_\beta H}{Y_\gamma H}=0$: the coefficients are symmetric in $(\alpha,\beta)$ and, by differentiating the constant Gram matrix, antisymmetric in $(\beta,\gamma)$, hence zero. The main example is the following. \begin{defn}[Coefficient distributions]\label{defn:coefficient} Let $Z=\prod_{\ell=1}^pS^{2C_\ell-1}$ carry the Hopf actions of \cref{sec:hopf}, with commuting generators $X_1,\dots,X_p$. For a manifold $B$ and a subbundle $W\subset B\times\R^p$ of rank $m$, the \emph{coefficient distribution} on $B\times Z$ is \[ D_W=\Bigl\{Y_w=\sum_{\ell=1}^pw_\ell X_\ell\ :\ w\in W_b\Bigr\}, \qquad h_W(Y_w,Y_{w'})=\ip{w}{w'} . \] \end{defn} For a local orthonormal frame $(w_\alpha)$ of $W$, the fields $Y_{w_\alpha}$ commute, because their coefficients depend only on $b$ and the $X_\ell$ commute and are tangent to $Z$. Their transition matrices depend only on $b$. Hence $D_W$ is integrable, it carries a canonical flat structure, and the \emph{coefficient metric} $h_W$ is flat. When $W=B\times\R^p$ we recover the span of all Hopf generators. \subsection{Hopf actions, averages and primitives}\label{sec:hopf} On $S^{2C-1}\subset\C^C$ we use the period-one circle action $R_tz=e^{2\pi\ii t}z$ with generator $X$, and on products the corresponding commuting actions. For a smooth function $h$ (scalar, vector or matrix valued) put \[ \av(h)(z)=\int_0^1h(R_tz)\,\dd t,\qquad I_Xh(z)=\int_0^1t\,h(R_tz)\,\dd t . \] \begin{lem}\label{lem:primitive} $\av(h)$ is $R$-invariant, and $X\av(h)=0$. If $\av(h)=0$, then $XI_Xh=h$ and $\av(I_Xh)=0$, and $I_Xh$ is the unique primitive of $h$ along $X$ with zero mean. Consequently, if $\Phi$ commutes with the action and $\av(h)=0$, then $I_X(h\circ\Phi)=(I_Xh)\circ\Phi$; in particular, if $h\circ R_{1/2}=\pm h$ then $I_Xh\circ R_{1/2}=\pm I_Xh$. These operations are smooth, continuous in every $C^j$ norm, and commute with derivatives in parameters on which the action does not depend. \end{lem} \begin{proof} Along an orbit, $XI_Xh=\int_0^1t\,\partial_t[h(R_tz)]\,\dd t =h(z)-\av(h)(z)$ by parts, and $\av(I_Xh)=\int_0^1t\,\av(h)\,\dd t=0$. Two primitives differ by an invariant function, which equals its own mean. The equivariance statements follow from uniqueness. \end{proof} We repeatedly use the antipodal map $z\mapsto-z=R_{1/2}z$; a function is \emph{even} or \emph{odd} if it is invariant or anti-invariant under it. The mean of an odd function vanishes. \subsection{Uniform atlases and weighted norms}\label{sec:atlas} \begin{setup}[Uniform atlases]\label{setup:uniform} A \emph{uniform atlas} on an $n$-manifold is a finite or countable, locally finite family of charts $\phi_i\colon\widehat U_i\to B_3(0)\subset\R^n$ whose cores $\phi_i^{-1}(B_1(0))$ cover the manifold, such that each $\widehat U_i$ meets at most $\Lambda$ others and all transition maps satisfy uniform bounds at every order. The $C^j$ norm of a map or tensor is the supremum over charts of its coordinate $C^j$ norm on $B_3(0)$; ellipticity $cI\le\gamma\le CI$ and singular values refer to coordinate norms. When the atlas is countable we only consider maps and tensors all of whose $C^j$ norms are finite. A compact parameter space is included by using product charts and full derivative norms in source and parameter; only source derivatives enter metrics and second forms. \end{setup} All constants below depend on the atlas only through $n$, $N$, $\Lambda$ and the transition bounds, never on the number of charts. For $00$ be smooth on $S^{2C-1}$, put $L=\av(a)$, $h=I_X(a/L-1)$ and $\Phi(z)=R_{h(z)}z$. Then $\Phi$ is a diffeomorphism mapping each orbit onto itself. If $p$ is smooth with $\abs{Xp}=a$, then $\widetilde p=(p\circ\Phi^{-1})/L$ satisfies $\abs{X\widetilde p}=1$, and at corresponding points $\spanop(X\widetilde p,X^2\widetilde p)=\spanop(Xp,X^2p)$. If $a$ is even, $\Phi$ commutes with $z\mapsto-z$. \end{lem} \begin{proof} On the orbit $t\mapsto R_tz_0$ the map $\Phi$ is $t\mapsto\phi(t)=t+h(R_tz_0)$, with $\phi'=1+Xh=a/L>0$ and $\phi(t+1)=\phi(t)+1$; so it is a diffeomorphism of the orbit. Since $\dd\Phi(X)=(a/L)X$ and $\dd\Phi$ induces the differential of $R_{h(z)}$ transversally to the orbits, $\dd\Phi$ is invertible, and $\Phi$ is a bijective local diffeomorphism. Differentiating $\widetilde p(R_{\phi(t)}z_0)=p(R_tz_0)/L$ gives $\phi'\,X\widetilde p=Xp/L$, hence $\abs{X\widetilde p}=1$, and a second derivative gives the statement about spans. If $a$ is even, so is $h$ (\cref{lem:primitive}), and $\Phi(-z)=R_{h(z)}(-z)=-\Phi(z)$. \end{proof} \begin{lem}[Curves on Hopf spheres]\label{lem:curves} Let $C\ge1$. \begin{enumerate}[label=(\alph*)] \item There is a smooth $f\colon S^{2C-1}\to\R^2$ with \[ \abs{Xf}=1,\qquad f(-z)=\diag(1,-1)f(z),\qquad \av\bigl(f_1Xf_2-f_2Xf_1\bigr)=0 . \] \item There is a smooth $c\colon S^{2C-1}\to\R^3$ with $\abs{Xc}=1$ such that $Xc$ and $X^2c$ are linearly independent everywhere. Then $b=Xc\times X^2c/\abs{Xc\times X^2c}$ is a smooth unit field perpendicular to $Xc$ and $X^2c$. \end{enumerate} \end{lem} \begin{proof} (a) Let $k_j=2j-1$ and $u(z)=\sum_{j=1}^C\operatorname{Re}(z_j^{k_j})$. Along an orbit $u$ is a trigonometric polynomial with distinct frequencies $k_j$, so \[ A=\av\bigl((Xu)^2\bigr)=2\pi^2\sum_{j=1}^Ck_j^2\abs{z_j}^{2k_j}>0 , \] and $A$ is invariant. Put $v=u/\sqrt A$ and $q=Xv$, so $\av(q^2)=1$, and \[ p=\bigl(I_X(1-q^2),\,v\bigr),\qquad Xp=(1-q^2,\,q),\qquad \abs{Xp}^2=\bigl(q^2-\tfrac12\bigr)^2+\tfrac34 . \] Since the $k_j$ are odd, $v$ and $q$ are odd, so $p_1$ is even and $p_2$ is odd. Apply \cref{lem:reparam} with $a=\abs{Xp}$, which is even, and put $f=\widetilde p$. Then $\abs{Xf}=1$; the antipodal map commutes with $\Phi$, so $f(-z)=\diag(1,-1)f(z)$. The integrand $f_1Xf_2-f_2Xf_1$ is odd, so its mean vanishes. (b) Let $w=Xf$ and $c_0=(f,\abs f^2/2)$. Then \[ Xc_0=\bigl(w,\ip fw\bigr),\qquad X^2c_0=\bigl(Xw,\,1+\ip f{Xw}\bigr), \] using $\abs w=1$. If $Xw\ne0$, the planar parts $w\perp Xw$ are independent; if $Xw=0$, the second vector is $(0,1)$ while $w\neq0$. Hence $Xc_0,X^2c_0$ are independent everywhere. \Cref{lem:reparam}, applied with $a=\abs{Xc_0}$, gives $c$. \end{proof} \begin{lem}[Partial immersions]\label{lem:partial-immersion} For $X_m=\prod_{i=1}^mS^{2C_i-1}$ with Hopf generators $Y_1,\dots,Y_m$, there is a smooth $A_m\colon X_m\to\R^{m+1}$ such that $Y_1A_m,\dots,Y_mA_m$ are linearly independent everywhere. \end{lem} \begin{proof} For $m=1$ take $f$ from \cref{lem:curves}(a). Given $A_m$, let $\nu_m$ be the unit normal in $\R^{m+1}$ to $Y_1A_m,\dots,Y_mA_m$ (the normalized generalized cross product). With $(a,b)=f$ on the new factor, set \[ A_{m+1}(x,z)=\bigl(A_m(x)+\epsilon a(z)\nu_m(x),\ \epsilon b(z)\bigr). \] The old columns are $(Y_iA_m+\epsilon aY_i\nu_m,0)$ and the new one is $\epsilon\bigl((Xa)\nu_m,Xb\bigr)$, of length $\epsilon$. They are orthogonal, since $\ip{Y_iA_m}{\nu_m}=\ip{Y_i\nu_m}{\nu_m}=0$. By compactness the old columns remain independent for small $\epsilon>0$. \end{proof} \subsection{Matrix primitives} The next proposition provides the displacement used in the inductive step of \cref{thm:free-families}. Its essential property is that the mixed metric terms it creates have zero mean along the new orbit. \begin{lem}[Uniform oscillatory decay]\label{lem:decay} Let $\eta(z)=\sum_{j=1}^C\operatorname{Re}(z_j^{2j-1})$ on $S^{2C-1}$. For every smooth function $a$ there is $C_a$ with \[ \norm{\av(a\,e^{\ii T\eta})}_{C^0}\le C_aT^{-1/(2C)}\qquad(T\ge1). \] \end{lem} \begin{proof} Along an orbit, $\partial_t^j\eta(R_tz)=\sum_{l=1}^C\operatorname{Re}\bigl((2\pi\ii k_l)^jz_l^{k_l}e^{2\pi\ii k_lt}\bigr)$ with $k_l=2l-1$. If these vanished for $j=1,\dots,2C$ at some $t$, then, writing the real parts as sums over the $2C$ distinct nonzero nodes $\pm2\pi\ii k_l$, the invertibility of the Vandermonde matrix would force all $z_l=0$. By compactness there is $c>0$ with $\max_{1\le j\le2C}\abs{X^j\eta}\ge c$ on the sphere. Cover $(\R/\Z)\times S^{2C-1}$ by finitely many sets $I_\alpha\times U_\alpha$, $I_\alpha$ an interval, on which a fixed derivative $\partial_t^{j_\alpha}\eta(R_tz)$ has absolute value at least $c/2$, and take a subordinate smooth partition of unity $\rho_\alpha(t,z)$. By van der Corput's lemma with amplitude \cite[Chapter~VIII, Section~1.2, Corollary]{Stein1993}, for $j_\alpha\ge2$, \[ \Bigl|\int_{I_\alpha}\rho_\alpha(t,z)\,a(R_tz)\,e^{\ii T\eta(R_tz)}\,\dd t\Bigr| \le C_{j_\alpha}(cT)^{-1/j_\alpha}\int_{I_\alpha}\abs{\partial_t(\rho_\alpha a)}\,\dd t , \] the amplitude vanishing at the endpoints. For $j_\alpha=1$, one integration by parts using $\abs{\partial_t\eta}\ge c/2$ and the bound on $\partial_t^2\eta$ gives $O(T^{-1})$. The constants are uniform in $z$, and $j_\alpha\le2C$. \end{proof} \begin{prop}[Matrix primitives]\label{prop:matrix} For all $C\ge1$ and $d\ge2$ there is a smooth $P\colon S^{2C-1}\to\Mat_{d\times d}(\R)$ such that \begin{equation}\label{eq:matrix-moments} Q=XP\ \text{is invertible everywhere},\qquad \av(P^\T Q)=0 . \end{equation} \end{prop} \begin{proof} Block-diagonal matrices inherit both properties from their blocks, and every $d\ge2$ is a sum of $2$'s and at most one $3$. Let $f$ be as in \cref{lem:curves}(a), $v=Xf=(s,t)$ and $u=J_2v=(-t,s)$. \emph{$d=2$.} Put $P_2=f_1I+f_2J_2$, so $Q_2=sI+tJ_2\in\SO(2)$ and $P_2^\T Q_2=\ip f{Xf}\,I+(f_1Xf_2-f_2Xf_1)\,J_2$. The first coefficient is $\frac12X\abs f^2$ and the second has zero mean by \cref{lem:curves}(a). \emph{$d=3$.} Regard $f,v,u$ as vectors in the first two coordinates of $\R^3$, let $\eta$ be as in \cref{lem:decay}, and for $T\ge1$ set \[ Q_0=\bigl[\,v,\ e_3\cos(T\eta)-u\sin(T\eta),\ -e_3\sin(T\eta)-u\cos(T\eta)\,\bigr]. \] Its columns are orthonormal and $\det Q_0=\det[v,u,e_3]=1$, so $Q_0\in\SO(3)$. Let $M_T=\av(Q_0)$. Its first column is $\av(Xf)=0$ and, by \cref{lem:decay}, $\norm{M_T}_{C^0}\to0$ as $T\to\infty$. Put $Q_1=Q_0-M_T$ and $P_1=I_XQ_1$, so $XP_1=Q_1$ and the first column of $P_1$ is $p_1=f-\av(f)$. Let $R=\diag(1,-1,1)$ and $D=\diag(1,1,-1)$. Under $z\mapsto-z$ we have $v\mapsto Rv$, $u\mapsto-Ru$ and $\eta\mapsto-\eta$, hence $Q_0(-z)=RQ_0(z)D$. Since $M_T$ is invariant, $M_T=RM_TD$, so $Q_1(-z)=RQ_1(z)D$ and, by \cref{lem:primitive}, $P_1(-z)=RP_1(z)D$. The matrix $\Omega=\av(P_1^\T Q_1)$ is skew, because $\av X(P_1^\T P_1)=0$, and satisfies $\Omega=D\Omega D$. Therefore only $\omega=\Omega_{12}=-\Omega_{21}$ can be nonzero. Since $p_1\perp e_3$ and $\av(p_1)=0$, \[ \omega=\av\ip{p_1}{e_3\cos(T\eta)-u\sin(T\eta)-M_Te_2} =-\av\bigl(\ip{p_1}u\sin(T\eta)\bigr), \] which tends to $0$ uniformly by \cref{lem:decay}. Let $P$ be $P_1$ with its second column replaced by $p_2+\omega v$. As $\omega$ is invariant, $XP=Q_1+\omega\,(Xv)e_2^\T$. Again $\av(P^\T XP)$ is skew and $D$-symmetric, and its $(1,2)$ entry is \[ \omega+\omega\,\av\ip{p_1}{Xv}=\omega-\omega\,\av\ip{Xp_1}v =\omega-\omega\,\av\abs v^2=0 . \] Hence $\av(P^\T XP)=0$. Finally $\norm{XP-Q_0}_{C^0}\le\norm{M_T}_{C^0}+\norm\omega_{C^0}\norm{Xv}_{C^0}<1$ for $T$ large, so $XP$ is invertible. \end{proof} % ---- begin sec3b-model.tex ---- \subsection{Formal corrections} \begin{lem}[Borel lemma with parameters]\label{lem:borel} Let $P$ be a compact manifold, possibly with boundary or corners, and let $a_j\in C^\infty(P,\R^N)$, $j\ge0$. There is $a\in C^\infty((-1,1)\times P,\R^N)$ whose Taylor series at $\epsilon=0$ is $\sum_j\epsilon^ja_j$. If $\Lambda$ is a continuous linear map from $C^\infty(P,\R^N)$ to a Fr\'echet space (for instance, restriction of some derivatives to a submanifold) with $\Lambda a_j=0$ for all $j$, then $a$ can be chosen with $\Lambda a(\epsilon,\cdot)=0$ for every $\epsilon$. \end{lem} \begin{proof} Fix $\chi\in C^\infty_c(-1,1)$ equal to $1$ near $0$. For $j\ge2$ choose $\lambda_j\ge1$ so large that all derivatives of order at most $j-1$ of $\chi(\lambda_j\epsilon)\epsilon^ja_j(x)$ are bounded by $2^{-j}$; this is possible because each such derivative carries a factor $\epsilon^{i}$ with $i\ge1$ on the support $\abs\epsilon\le\lambda_j^{-1}$. Then $a=a_0+\epsilon a_1+\sum_{j\ge2}\chi(\lambda_j\epsilon)\epsilon^ja_j$ converges in $C^\infty$ and has the required Taylor series. The cutoffs do not depend on $x$, so $\Lambda$ passes through the sum. \end{proof} For a family of maps $P$ and a displacement $U$, in a flat frame of $D$, \begin{equation}\label{eq:basic-identity} \gamma_D(P+U)_{\alpha\beta}=\gamma_D(P)_{\alpha\beta} +Y_\alpha\ip{Y_\beta P}U+Y_\beta\ip{Y_\alpha P}U -2\ip{Y_\alpha Y_\beta P}U+\gamma_D(U)_{\alpha\beta} . \end{equation} In particular, if $\ip{\dd_DP}U=0$, the linear part of the change of partial metric is $-2\ip{\nabla^2_DP}U$. \begin{lem}[Formal correction of partial metrics]\label{lem:formal} Let $X$ be compact, $D$ an integrable distribution with a flat structure, and $h_D$ a partial metric. Let $P_\epsilon$ be a smooth family with \[ \gamma_D(P_\epsilon)-h_D=\epsilon^{2L+1}S_\epsilon,\qquad L\ge1, \] and suppose the pairing map $T_{P_\epsilon}$ has, for $\epsilon\ne0$, a right inverse $C_\epsilon$ such that $\widetilde C_\epsilon=\epsilon^LC_\epsilon$ extends smoothly through $\epsilon=0$. Then there is a smooth family $\widehat U_\epsilon$ such that, with $U_\epsilon=\epsilon^{L+1}\widehat U_\epsilon$, the error $\gamma_D(P_\epsilon+U_\epsilon)-h_D$ is flat at $\epsilon=0$. \end{lem} \begin{proof} By continuity $T_{P_\epsilon}\widetilde C_\epsilon=\epsilon^L\Id$ for all $\epsilon$. Write $U=\epsilon^Lv$ and consider \begin{equation}\label{eq:formal-fixed} v=\widetilde C_\epsilon\bigl(0,\tfrac12(\epsilon S_\epsilon+\gamma_D(v))\bigr). \end{equation} Seek $v=\sum_{j\ge1}\epsilon^jv_j$. Since $\gamma_D(v)$ is quadratic and $v=O(\epsilon)$, the coefficient of $\epsilon^j$ on the right depends only on $v_1,\dots,v_{j-1}$, through finitely many derivatives and products. Hence all $v_j$ are determined and smooth, and \cref{lem:borel} gives a smooth $v=\epsilon\widehat v$ satisfying \eqref{eq:formal-fixed} up to a flat error. Applying $T_{P_\epsilon}$, we obtain modulo flat terms $\ip{\dd_DP}{U}=0$ and $2\ip{\nabla^2_DP}{U}=\epsilon^{2L+1}S+\gamma_D(U)$. Inserting this in \eqref{eq:basic-identity} shows that $\gamma_D(P+U)-h_D$ is flat. \end{proof} \subsection{Free families with flat metric error} Let $X_a=\prod_{i=1}^aS^{2C_i-1}$ with Hopf generators $Y_1,\dots,Y_a$ and $D$ their span. The partial jet of $H\colon X_a\to\R^N$ has the $a+s_a$ \emph{columns} $Y_iH$ and $Y_iY_jH$ ($i\le j$). Give the columns $Y_iH$ and $Y_iY_iH$ weight $0$, and $Y_iY_jH$ with $i0$ and smooth families $H_\epsilon\colon X_a\to\R^{s_a+a+1}$, $\abs\epsilon<\epsilon_a$, such that: \begin{enumerate}[label=(\roman*)] \item $\gamma_D(H_\epsilon)-I_a$ is flat at $\epsilon=0$; \item each column divided by $\epsilon^{\text{weight}}$ extends smoothly to $\epsilon=0$, and these \emph{normalized columns} are linearly independent for all $\abs\epsilon<\epsilon_a$. \end{enumerate} Consequently, for $\epsilon\ne0$, $H_\epsilon$ is $D$-free; the unit normal $\nu_\epsilon$ to the normalized columns is smooth through $0$ and perpendicular to the partial jet; and the minimum-norm right inverse $C_{H,\epsilon}$ of the pairing map is such that $\epsilon^{a-1}C_{H,\epsilon}$ extends smoothly through $0$. \end{thm} \begin{proof} The consequences follow from (ii): the number of columns is $a+s_a=N-1$, and if $M_\epsilon$ is the matrix of normalized columns and $\Delta_\epsilon$ the diagonal matrix of the powers $\epsilon^{\text{weight}}$, then $T_{H_\epsilon}=\Delta_\epsilon M_\epsilon^\T$ and $C_{H,\epsilon}=M_\epsilon(M_\epsilon^\T M_\epsilon)^{-1}\Delta_\epsilon^{-1}$; the largest weight is $a-1$. For $a=1$ take $H=c$ from \cref{lem:curves}(b), independent of $\epsilon$, with $\nu=b$. Assume the statement for $a=m$ and put $L=m$. Let $c,b$ be the curve and binormal of \cref{lem:curves}(b) on the new factor $S^{2C_{m+1}-1}$, with generator $X=Y_{m+1}$. In $\R^N=\R^{s_m+m+1}\times\R^3\times\R^{m-1}$, $N=s_{m+1}+m+2$, set \[ F_{0,\epsilon}=(H_\epsilon,c,0),\qquad T_\epsilon=\bigl[(\nu_\epsilon,0,0),(0,b,0),(0,0,e_1),\dots,(0,0,e_{m-1})\bigr], \] an orthonormal $N\times(m+1)$ frame. Choose $A=A_m\colon X_m\to\R^{m+1}$ from \cref{lem:partial-immersion}, $P$ and $Q=XP$ from \cref{prop:matrix} with $d=m+1$, and define the displacement \[ Z_\epsilon=T_\epsilon\,P\,A . \] Each column of $T$ depends on one factor and the columns lie in orthogonal blocks; hence \begin{equation}\label{eq:frame-identities} T^\T T=I,\quad T^\T Y_iT=0,\quad T^\T XT=0,\quad (Y_iT)^\T XT=0,\quad Y_iXT=0 . \end{equation} The columns of $T$ are perpendicular to all first and second columns of $F_0$, and the mixed columns $Y_iXF_0$ vanish. By \eqref{eq:basic-identity} the linearization of $\gamma_D$ at $F_0$ kills $Z$, and \eqref{eq:frame-identities} gives \begin{equation}\label{eq:mixed} \ip{Y_iZ}{XZ}=\ip{PY_iA}{QA},\qquad T^\T Y_iXZ=QY_iA . \end{equation} The first expression has zero mean along $X$ by \eqref{eq:matrix-moments}, because $A$ does not depend on the new factor; the vectors $QY_iA$ are linearly independent. \emph{Second-order cancellation.} Put $e_{ij}=\ip{Y_iZ}{Y_jZ}$, $e=\abs{XZ}^2$ and $w_i=-I_X\ip{Y_iZ}{XZ}$. For $\epsilon\ne0$ use $C_{H,\epsilon}$ to choose $W_H$ with \[ \ip{Y_iH}{W_H}=w_i,\qquad 2\ip{Y_iY_jH}{W_H}=Y_iw_j+Y_jw_i+e_{ij}, \] and set $W=\bigl(W_H,\ \tfrac{e}{2\abs{X^2c}^2}X^2c,\ 0\bigr)$. Since $\ip{Xc}{X^2c}=0$, a direct computation with \eqref{eq:basic-identity} shows that the linearization of $\gamma_D$ at $F_0$ maps $W$ to $-\gamma_D(Z)$: the $(ij)$, $(iX)$ and $(XX)$ entries are $-e_{ij}$, $Xw_i=-\ip{Y_iZ}{XZ}$ and $-e$. By the inductive hypothesis $\widetilde W_\epsilon=\epsilon^{m-1}W_\epsilon$ is smooth through $0$. The family \[ P_\epsilon=F_{0,\epsilon}+\epsilon^LZ_\epsilon+\epsilon^{L+1}\widetilde W_\epsilon \] therefore satisfies \[ \gamma_D(P_\epsilon)-I_{m+1}=\diag\bigl(\gamma_D(H_\epsilon)-I_m,0\bigr) +\epsilon^{2L+1}S_3+\epsilon^{2L+2}S_4 \] with $S_3,S_4$ smooth, and the first term is flat. \emph{Independence.} Keep the old weights and give the new mixed columns $Y_iXP_\epsilon$ weight $L$. At $\epsilon=0$ the normalized columns of $P_\epsilon$ are the old normalized columns (in the first block), $Xc$ and $X^2c$ (in the second block), and $Y_iXZ_0$. The orthogonal complement of the first two groups is spanned by the columns of $T_0$, and by \eqref{eq:mixed} the projection of $Y_iXZ_0$ onto it is $T_0QY_iA$. Hence all $N-1$ normalized columns are independent at $\epsilon=0$, so on an interval, and as in the first paragraph the pairing map of $P_\epsilon$ has a right inverse $C_\epsilon$ with $\epsilon^LC_\epsilon$ smooth. \Cref{lem:formal} (the flat error is divisible by $\epsilon^{2L+1}$) gives $U_\epsilon=O(\epsilon^{L+1})$ with flat metric error. Since all weights are at most $L$, adding $U_\epsilon$ changes the normalized columns by $O(\epsilon)$. Thus $H_{m+1,\epsilon}=P_\epsilon+U_\epsilon$ satisfies (i) and (ii) on a smaller interval. \end{proof} \subsection{Normal collars and wrapping} \begin{prop}[Wrapping]\label{prop:wrapping} Let $X$ be compact, $D$ an integrable distribution of rank $m\ge1$ with a flat structure and a flat partial metric $h_D$. Let $H_\epsilon\colon X\to\R^q$ and $\nu_\epsilon$ be smooth through $\epsilon=0$, with $\gamma_D(H_\epsilon)-h_D$ flat at $0$, and assume that for $\epsilon\ne0$ the map $H_\epsilon$ is $D$-free, $\nu_\epsilon$ is a unit vector perpendicular to its partial jet, and $\epsilon^KC_{H,\epsilon}$ extends smoothly through $0$ for some integer $K\ge0$. On $\widehat X=X\times S^{2C-1}$ let $\widehat D=D\oplus\R X_{\rm new}$ and $\widehat h=h_D+(X_{\rm new}^*)^2$. Then for all large integers $k$ there are maps $S_k\colon\widehat X\to\R^{q+1}$ such that, for all $j$ and $R$, \begin{align} \norm{S_k}_{C^j}&\le C_jk^{\max(j-1,0)},\label{eq:wrap-jets}\\ \norm{\gamma_{\widehat D}(S_k)-\widehat h}_{C^j}&\le C_{j,R}\,k^{-R},\label{eq:wrap-error} \end{align} and the partial normal second form of $S_k$ along $\widehat D$, restricted to $\Sym^2D$, has smallest singular value at least $c\,k^{-K/(2K+1)}\ge ck^{-1/2}$. \end{prop} \begin{proof} Let $M=2K+1$, let $(a,b)=f$ be the planar curve of \cref{lem:curves}(a) on $S^{2C-1}$, and fix $\bar a$ with $\abs a\le\bar a$. Work in flat orthonormal frames $Y_\alpha$ of $D$. \emph{Step 1: a normal collar.} On $X\times[-\bar a,\bar a]$ we seek \[ G_\epsilon(x,s)=H_\epsilon(x)+\epsilon^Ms\,\nu_\epsilon(x)+\epsilon^{2M-K}W(\epsilon,x,s), \qquad W|_{s=0}=\partial_sW|_{s=0}=0 . \] In the variable $t=\epsilon^Ms$ we have $G_t=\nu_\epsilon+\epsilon^{M-K}W_s$ and $G_{tt}=\epsilon^{-K}W_{ss}$. Let $T_Gv=(\ip v{G_t},\ip v{\dd_DG},\ip v{\nabla^2_DG})$ and $R_H(r,y)=r\nu_\epsilon+C_{H,\epsilon}y$. Since $C_{H,\epsilon}$ takes values in the span of the partial jet, $R_H$ is a right inverse of $T_G$ at $(G,G_t)=(H_\epsilon,\nu_\epsilon)$. The differences of the entries of $T_G$ from these values are $O(\epsilon^{M-K})$ and $O(\epsilon^M)$, while $R_H=O(\epsilon^{-K})$; hence $A_G=T_GR_H=I+\epsilon^{M-2K}\Theta=I+\epsilon\Theta$, with $\Theta$ smooth in $\epsilon$, $x$, $s$ and the $2$-jet of $W$. Consider \begin{equation}\label{eq:collar} W_{ss}=\epsilon^KR_HA_G^{-1}\bigl(0,0,\ip{\dd_DG_t}{\dd_DG_t}\bigr). \end{equation} Every occurrence of $W$ on the right carries a positive power of $\epsilon$, so the coefficient of $\epsilon^j$ determines $\partial_s^2W_j$ from $W_0,\dots,W_{j-1}$; integrating twice with zero initial data gives smooth coefficients, and \cref{lem:borel} realizes them by a smooth $W$ with both initial conditions. Then \eqref{eq:collar} holds up to a flat error, so the physical residual $R=G_{tt}-R_HA_G^{-1}(0,0,\ip{\dd_DG_t}{\dd_DG_t})$ is flat. Let $E_{tt}=\abs{G_t}^2-1$, $E_{\alpha t}=\ip{Y_\alpha G}{G_t}$, $E_{\alpha\beta}=\ip{Y_\alpha G}{Y_\beta G}-h_{\alpha\beta}$ and $\theta_\alpha=\ip R{Y_\alpha G}$. The three components of $T_GG_{tt}=(0,0,\ip{\dd_DG_t}{\dd_DG_t})+T_GR$ give \[ \partial_tE_{tt}=2\ip{G_t}R,\qquad \partial_tE_{\alpha t}=\tfrac12Y_\alpha E_{tt}+\theta_\alpha,\qquad \partial_t^2E_{\alpha\beta}=Y_\alpha\theta_\beta+Y_\beta\theta_\alpha -2\ip R{Y_\alpha Y_\beta G}. \] At $s=0$ we have $E_{tt}=E_{\alpha t}=0$, $\partial_tE_{\alpha\beta}=-2\ip{\nu_\epsilon}{Y_\alpha Y_\beta H_\epsilon}=0$, and $E_{\alpha\beta}$ is flat. Since $\partial_s=\epsilon^M\partial_t$ and $s$ ranges over a fixed interval, integration shows that all three errors are flat, with all derivatives in $x$ and $s$. \emph{Step 2: power maps.} Let $P_k(z)=(z_\ell^k)_\ell\big/\bigl(\sum_\ell\abs{z_\ell}^{2k}\bigr)^{1/2}$. Then $P_k(R_tz)=R_{kt}P_k(z)$, so $X(\phi\circ P_k)=k(X\phi)\circ P_k$. Moreover $\norm{P_k}_{C^j}\le C_jk^j$: at a point where $m_z=\max_\ell\abs{z_\ell}\ge C^{-1/2}$, the numerator has norm at least $m_z^k$ and its derivatives of order $j$ are at most $C_jk^jm_z^{k-j}$. \emph{Step 3: wrapping.} Let $\epsilon=k^{-1/M}$, so $\epsilon^M=k^{-1}$, and \[ S_k(x,z)=\bigl(G_\epsilon(x,a(P_kz)),\ k^{-1}b(P_kz)\bigr). \] Every derivative of $G_\epsilon$ in $s$ carries the factor $\epsilon^M=k^{-1}$, while $j$ derivatives of $a\circ P_k$ or $b\circ P_k$ cost $k^j$; this gives \eqref{eq:wrap-jets}. With $\varrho=(Xa)\circ P_k$ and $\varsigma=(Xb)\circ P_k$ we have $\varrho^2+\varsigma^2=1$, $Y_\alpha S_k=(Y_\alpha G,0)$ and $X_{\rm new}S_k=(\varrho G_t,\varsigma)$, so the entries of $\gamma_{\widehat D}(S_k)-\widehat h$ are $E_{\alpha\beta}$, $\varrho E_{\alpha t}$ and $\varrho^2E_{tt}$ evaluated at $s=a(P_kz)$. Flatness in $\epsilon$ absorbs every power of $k$ produced by the chain rule, proving \eqref{eq:wrap-error}. \emph{Margin.} Compare the first columns $(Y_\alpha G,0)$, $(\varrho G_t,\varsigma)$ and the second columns $(Y_\alpha Y_\beta G,0)$ with $(Y_\alpha H_\epsilon,0)$, $(\varrho\nu_\epsilon,\varsigma)$ and $(Y_\alpha Y_\beta H_\epsilon,0)$; the differences are $O(\epsilon^{M-K})$ and $O(\epsilon^M)$. The comparison second columns are perpendicular to $(\varrho\nu_\epsilon,\varsigma)$, so their projection off the comparison first columns is $(\Pi^D_{H_\epsilon}\nabla^2_DH_\epsilon,0)$. For unit $\Xi\in\Sym^2D$, $\abs{\Pi^D_H\nabla^2_DH(\Xi)}=\min_v\abs{J_D(H)(-v,\Xi)}\ge\sigma_{\min}(J_D(H)) =\norm{C_{H,\epsilon}}^{-1}\ge c\epsilon^K$. The comparison Gram matrices are uniformly positive and all second jets are bounded, so the projector formula changes the normal second form by $O(\epsilon^{M-K})=O(\epsilon^{K+1})$. Hence the margin is at least $\tfrac c2\epsilon^K=\tfrac c2k^{-K/(2K+1)}$. \end{proof} \begin{cor}[Model maps]\label{cor:model} Let $n\ge2$, $C_1,\dots,C_n\ge1$, and let $Y_1,\dots,Y_n$ be the Hopf generators of $X_n=\prod_{i=1}^nS^{2C_i-1}$. For all large $k$ there are maps $S_k\colon X_n\to\R^{s_n+1}$ satisfying \eqref{eq:wrap-jets}, \eqref{eq:wrap-error} with $\widehat h=I_n$, whose partial normal second form has smallest singular value at least $ck^{-1/2}$ on $\Sym^2\spanop(Y_1,\dots,Y_{n-1})$. \end{cor} \begin{proof} \Cref{thm:free-families} with $a=n-1$ gives families into $\R^{s_{n-1}+n}=\R^{s_n}$ with $K=n-2$; apply \cref{prop:wrapping}. \end{proof} % ---- end sec3b-model.tex ---- % ---- end sec3-model.tex ---- % ---- begin sec4-correction.tex ---- \section{Exact metric correction}\label{sec:correction} This section proves \cref{prop:repair}: an immersion with polynomially controlled jets and phase margin, whose metric error is polynomially small of sufficiently high order, can be corrected to an exact isometric immersion. The basic step (\cref{lem:increment}) adds a prescribed positive tensor to the metric by explicit oscillations along admissible normals; an iteration with smoothing (\cref{lem:iteration}) produces exact solutions. Throughout, norms refer to a uniform atlas as in \cref{setup:uniform}, and $C$ denotes constants depending only on the indicated data. \subsection{Smoothing} \begin{lem}[Smoothing operators]\label{lem:smoothing} For every integer $r\ge1$ there are linear operators $S_\ell$, $0<\ell\le\ell_0$, acting on maps and on tensors through their chart components, such that for all $m$: \begin{enumerate}[label=(\roman*)] \item $\abs{S_\ell u}_{m,\ell}\le C_{m,r}\norm u_{C^0}$ and $\abs{S_\ell G}^{(2)}_{m,\ell}\le C_{m,r}\norm G_{C^2}$; \item if $\tau\le\ell\le t\le1$ and $\abs u_{r,t}\le\Lambda_0$, then $\abs{u-S_\ell u}_{m,\tau}\le C_{m,r}\bigl[\Lambda_0(\ell/t)^r+(\tau/t)^r\abs u_{m,t}\bigr]$, and the same holds for $\abs{\cdot}^{(2)}$ in place of $\abs\cdot$. \end{enumerate} \end{lem} \begin{proof} Let $\kappa$ be an even bump supported in the unit ball and $\kappa_\lambda(x)=\lambda^{-n}\kappa(x/\lambda)$. Choose $r+1$ distinct $\lambda_a\in[\frac12,1]$ and solve the Vandermonde system $\sum_ac_a=1$, $\sum_ac_a\lambda_a^p=0$ ($1\le p\le r$); then $\rho=\sum_ac_a\kappa_{\lambda_a}$ has $\int\rho=1$ and vanishing moments of orders $1,\dots,r$. Let $\{\zeta_i\}$ be a partition of unity subordinate to the charts, with $\operatorname{supp}\zeta_i\subset B_2$ and uniform bounds, and put $S_\ell u=\sum_i\zeta_i\,(\rho_\ell*u_i)$, where $u_i$ is the chart representative and $\rho_\ell(x)=\ell^{-n}\rho(x/\ell)$. Differentiating the kernel gives (i). For (ii), the vanishing moments and Taylor's formula give $\norm{u-S_\ell u}_{C^j}\le C\ell^{r-j}\norm u_{C^r}\le C\Lambda_0\ell^{r-j}t^{-r}$ for $j\le r$, while for $j>r$ we use $\norm{u-S_\ell u}_{C^j}\le C\norm u_{C^j}$ and $\tau^j\le(\tau/t)^rt^j$. Only suprema and the bounded overlap of the charts enter. \end{proof} \subsection{Oscillatory modes} For an immersion $F$ let $L_F(V)=\ip{\dd F}{\dd V}+\ip{\dd V}{\dd F}$ be the linearized metric, so that \begin{equation}\label{eq:quadratic-identity} \gamma(F+V)=\gamma(F)+L_F(V)+\gamma(V). \end{equation} For a covector field $\xi=\dd\psi$ that is good for $F$, let $\beta_\psi\colon\nu_F\to\Sym^2(\ker\dd\psi)^*$, $v\mapsto\ip{B_F}v|_{\ker\dd\psi}$; it is onto, its kernel is $E_{\dd\psi}$, and we write $J_\psi=\beta_\psi^*(\beta_\psi\beta_\psi^*)^{-1}$. \begin{lem}[Oscillatory modes]\label{lem:modes} Let $F$ be an immersion of $B_3$ with $cI\le\gamma(F)$, let $\psi$ be a function with $\abs{\dd\psi}\ge c$ such that $\dd\psi$ is good for $F$ with margin at least $c$, and let $0<\tau\le\ell\le1$. Assume $\abs F^{(2)}_{m,\ell}+\abs\psi_{m+2,\ell}\le K_m$ for all $m$. For each integer $p\ge1$ there is a linear differential operator $\mathcal Z_{p,\tau}$, whose coefficients are polynomials in $\tau$ with coefficients depending smoothly on finitely many derivatives of $F$ and $\psi$ (and on a fixed partition of unity), such that for a complex symmetric tensor $h$ and a complex section $v_f$ of $E_{\dd\psi}$, both supported in a set $S\subset B_2$, the field $Z=\mathcal Z_{p,\tau}(h,v_f)$ is supported in $S$ and \[ L_F\bigl(Ze^{\ii\psi/\tau}\bigr)=(h+\mathfrak e)\,e^{\ii\psi/\tau}, \] where, with $\abs{(h,v_f)}_{m}=\abs h_{m,\ell}+\abs{v_f}_{m,\ell}$, \begin{gather*} \abs Z_{m,\ell}\le C_m\abs{(h,v_f)}_{m+p},\qquad \abs{\mathfrak e}_{m,\ell}\le C_m\Bigl(\frac\tau\ell\Bigr)^p\abs{(h,v_f)}_{m+p},\\ \bigl|Z-v_f+\tfrac12J_\psi(h|_{\ker\dd\psi})\bigr|_{m,\ell}\le C_m\frac\tau\ell\,\abs{(h,v_f)}_{m+p}. \end{gather*} The constants depend on $m$, $p$, $c$ and finitely many $K_j$. \end{lem} \begin{proof} By linearity and a fixed partition of unity we may assume that $S$ lies in a ball of fixed radius carrying coordinates with $x^1=\psi$; their bounds depend only on $c$ and the $K_j$. Write $g=\gamma(F)$, $Z=\dd F(g^{-1}u)+v$ with $u_i=\ip{\partial_iF}Z$ and $v$ normal. Then \[ L_F(Z)_{ij}=\partial_iu_j+\partial_ju_i-2\Gamma^k_{ij}u_k-2\ip{B_{ij}}v, \] and conjugation by $e^{\ii\psi/\tau}$ adds $(\ii/\tau)(\delta_{i1}u_j+\delta_{j1}u_i)$. Indices $a,b$ run from $2$ to $n$. Put $T(u)_{ab}=\frac12(\partial_au_b+\partial_bu_a)-\Gamma^k_{ab}u_k$ and $v=v_f+J_\psi\bigl(T(u)-\frac12h_H\bigr)$, $h_H=(h_{ab})$; then the $(ab)$ equations hold exactly. The remaining equations become $(\ii/\tau+\cL)u=k$, where \begin{equation}\label{eq:mode-system} \begin{aligned} (\cL u)_1&=\partial_1u_1-\Gamma^k_{11}u_k-\ip{B_{11}}{J_\psi T(u)},& k_1&=\tfrac12h_{11}+\ip{B_{11}}{v_f}-\tfrac12\ip{B_{11}}{J_\psi h_H},\\ (\cL u)_a&=\partial_1u_a+\partial_au_1-2\Gamma^k_{1a}u_k-2\ip{B_{1a}}{J_\psi T(u)},& k_a&=h_{1a}+2\ip{B_{1a}}{v_f}-\ip{B_{1a}}{J_\psi h_H}. \end{aligned} \end{equation} A residual $e$ in this system produces the metric residual $2e_1$ in the $(11)$ entry, $e_a$ in the $(1a)$ entries, and $0$ in the $(ab)$ entries. Set \[ u=\frac\tau\ii\sum_{l=0}^{p-1}\Bigl(-\frac\tau\ii\cL\Bigr)^lk , \] whose residual is $(-1)^{p-1}(\tau/\ii)^p\cL^pk$. Each application of the first-order operator $\cL$ costs one derivative, i.e.\ a factor $\ell^{-1}$ in the slow norms; this gives all three estimates. \end{proof} \subsection{Phase data} The increments need phases that are good, together with all the phases created by quadratic interactions. \begin{defn}[Phase data]\label{defn:phase-data} Let $F_0$ be an immersion and $h_0$ a positive tensor. \emph{Phase data} for $(F_0,h_0)$ with constant $c_*>0$ consist of: \begin{enumerate}[label=(\alph*)] \item functions $\chi_i$, $i\in I$, with $\sum_i\chi_i^2=1$ and uniform bounds, each supported in a ball $B_2$ of some chart, and a colouring of $I$ by at most $\Lambda'$ colours such that equally coloured supports are disjoint; \item for each $i$, functions $\phi_{i,1},\dots,\phi_{i,s_n}$ with uniform bounds whose differentials are constant in the chart of $\chi_i$ and form a \emph{rank-one basis}: every symmetric matrix is uniquely $A=\sum_r\varpi_{i,r}(A)\,\dd\phi_{i,r}^2$, with linear functionals $\varpi_{i,r}$ satisfying $\varpi_{i,r}(h_0)\ge c_*$ on $\operatorname{supp}\chi_i$; \item for every \emph{label} $\lambda=(i,r)$, and every pair of distinct labels $\lambda,\lambda'$ whose supports meet, the covectors $\dd\phi_\lambda$, $2\dd\phi_\lambda$ and $\dd\phi_\lambda\pm\dd\phi_{\lambda'}$ have length at least $c_*$ and are good for $F_0$ with margin at least $c_*$ on the corresponding supports; \item for each label a constant \emph{seed} vector $a_\lambda\in\R^N$ whose orthogonal projection onto $E_{\dd\phi_\lambda}(F_0)$ has length at least $c_*$ on $\operatorname{supp}\chi_i$. \end{enumerate} \end{defn} By \cref{lem:projector}, conditions (c) and (d) persist, with $c_*/2$, for all immersions $F$ with $\norm{F-F_0}_{C^2}\le\rho_0$, where $\rho_0>0$ depends on $c_*$ and $\norm{F_0}_{C^2}$; for such $F$ we use the unit normals $n_\lambda(F)$ obtained by projecting $a_\lambda$ onto $E_{\dd\phi_\lambda}(F)$ and normalizing. Likewise (b) persists, with $c_*/2$, for tensors $A$ with $\norm{A-h_0}_{C^0}\le\eta_0$. \begin{lem}[Finite positive increments]\label{lem:increment} Fix phase data for $(F_0,h_0)$ and $\rho_0,\eta_0$ as above, and an integer $q\ge1$. Let $F$ and $H$ satisfy $\norm{F-F_0}_{C^2}\le\rho_0$, $\norm{H-h_0}_{C^0}\le\eta_0/2$ and, for some $0<\ell\le1$, $\abs F^{(2)}_{m,\ell}+\abs H_{m,\ell}\le K_m$ for all $m$. If $0<\delta\le\tau\le\ell$ and $\tau/\ell\le\theta$, where $\theta>0$ depends on $q$, the phase data and finitely many $K_j$, then an explicit finite differential construction gives $U$ with, for every $m$, \begin{equation}\label{eq:increment} \abs U_{m,\tau}\le C_m\,\delta\tau,\qquad \bigl|\gamma(F+U)-\gamma(F)-\delta^2H\bigr|_{m,\tau} \le C_m\Bigl[\delta\Bigl(\frac\tau\ell\Bigr)^q+\frac{\delta^3}\tau\Bigr]. \end{equation} The constants depend on $m$, $q$, the phase data and finitely many $K_j$. Each summand of $U$ is supported in the support of one $\chi_i$, and $U$ depends smoothly on additional parameters on which $F$ and $H$ depend smoothly, with the same estimates in full norms. \end{lem} \begin{proof} Put $p=q+2$ and $b_\lambda(A)=\chi_i\varpi_{i,r}(A)^{1/2}$ for $\lambda=(i,r)$, so that $A=\sum_\lambda b_\lambda(A)^2\dd\phi_\lambda^2$ whenever $\norm{A-h_0}_{C^0}\le\eta_0$. \emph{Primary oscillations.} For such $A$, let $Z_\lambda=\mathcal Z_{p,\tau}\bigl(0,\sqrt2\,\delta\tau\,b_\lambda(A)n_\lambda(F)\bigr)$ be given by \cref{lem:modes} with phase $\phi_\lambda$, and $U_1(A)=\operatorname{Re}\sum_\lambda Z_\lambda e^{\ii\phi_\lambda/\tau}$. Then $\abs{U_1}_{m,\tau}\le C\delta\tau$ and $\abs{L_F(U_1)}_{m,\tau}\le C\delta\tau(\tau/\ell)^p$. The product $\gamma(U_1)=\dd U_1\cdot\dd U_1$ is a finite sum of terms with phases $0$, $\pm2\phi_\lambda$ and $\pm\phi_\lambda\pm\phi_{\lambda'}$ for interacting labels. The phase-zero terms pair each label with its conjugate; their sum is \[ \delta^2\bigl(A+\mathcal T(A)\bigr),\qquad \abs{\mathcal T(A)}_{m,\ell}\le C\tfrac\tau\ell , \] since the leading part of $Z_\lambda$ is $\sqrt2\delta\tau b_\lambda n_\lambda$, $\abs{n_\lambda}=1$, and every other contribution involves a slow derivative or a correction from \cref{lem:modes}. The map $\mathcal T$ is smooth, and for some fixed loss $l_0$, $\abs{\mathcal T(A)-\mathcal T(\bar A)}_{m,\ell}\le C(\tau/\ell)\abs{A-\bar A}_{m+l_0,\ell}$. Let $A_0=H$ and $A_{l+1}=H-\mathcal T(A_l)$ for $l0$, depending on $k_0$, $\rho_1$, $\Lambda_1$, the phase data and finitely many norms of $h_0$, with the following property. Suppose $01$. Every stage ends, so $k\to\infty$; for each $m$, all but finitely many increments satisfy $\norm{U_j}_{C^m}\le\rho_1t_j/8$, and $\sum_jt_j<1$ when $t_*$ is small. Thus $G_j$ converges in every $C^m$, the total $C^2$ displacement is at most $\rho_1$, and $\gamma(G_\infty)=\lim(h-\delta_j^2H_j)=h$ since the $H_j$ are bounded. For compact parameters we smooth in source and parameter variables and use full norms; only source derivatives enter the metric, and all thresholds are suprema over the parameter space. For countable atlases only suprema and bounded overlap were used. \end{proof} \begin{rem}[The choice of scales]\label{rem:scales} The reserve $(\delta')^2S_\ell h_0$ kept in the defect is what keeps the normalized defect $H'$ close to the positive tensor $h_0$, so that every step adds a \emph{positive} tensor. The exponents in \eqref{eq:scales} make both small ratios equal, $\tau/\ell=\ell/t=t^{1/8}$, so that the accuracy $q=r=13(k+1)$ of the increments and of the smoothing gains $t^{13(k+1)/8}$, which beats the normalization loss $\delta/\tau^{2(k+1)}=t^{-(3k+5)/2}$ with the margin $t^{(k-7)/8}$; the quadratic error $\delta^3/\tau$ is harmless once $k\ge8$. Increasing the stage $k$ is needed for convergence in high norms, since $\norm{U}_{C^m}\lesssim t^{k+\frac54(1-m)}$. \end{rem} \subsection{Polynomial entrance estimates} \begin{prop}[Polynomial metric repair]\label{prop:repair} Fix $n\ge2$, $N\ge s_n+1$, the data of a uniform atlas on an $n$-manifold, ellipticity constants $00$. There is an integer $K_{\rm rep}=K_{\rm rep}(A,\kappa,b)$ with the following property. Let $0<\epsilon<1$ and let $F_\epsilon$ be immersions into $\R^N$ and $h_\epsilon$ positive tensors with \[ cI\le\gamma(F_\epsilon)\le CI,\qquad \norm{F_\epsilon}_{C^{119}}\le C_{\rm ref}\,\epsilon^{-A},\qquad \mu(F_\epsilon,x)\ge c_{\rm ph}\,\epsilon^\kappa\ \ \forall x, \] \[ \norm{h_\epsilon-\gamma(F_\epsilon)}_{C^{117}}\le C_{\rm err}\,\epsilon^{K_{\rm rep}} . \] Then for all sufficiently small $\epsilon$ there is a smooth $F^*_\epsilon$ with \[ \gamma(F^*_\epsilon)=h_\epsilon,\qquad \norm{F^*_\epsilon-F_\epsilon}_{C^2}\le\epsilon^b,\qquad \mu(F^*_\epsilon,x)\ge\tfrac12c_{\rm ph}\,\epsilon^\kappa . \] $K_{\rm rep}$ depends only on $A,\kappa,b$ and the atlas data; the prefactors affect only how small $\epsilon$ must be, and higher norms of $F_\epsilon,h_\epsilon$ affect only the length of the iteration. The statement holds for countable uniform atlases and, with full norms, for families depending on a compact parameter. No regularity in $\epsilon$ is claimed. \end{prop} We say that a quantity is \emph{polynomial} if it is bounded by $C\epsilon^{-p}$, or below by $c\,\epsilon^{p}$, where $p$ depends only on $A,\kappa$ and the atlas data. \begin{proof} \emph{Good root directions.} For a second form $B$ at a point let $D_B(\xi)$ be the sum of squares of the $q_n\times q_n$ minors of $B\circ\Sym^2(\abs\xi^2I-\xi\xi^\T)$. It is a nonnegative homogeneous polynomial of degree $8q_n$, positive exactly at good covectors, and by Cauchy--Binet $D_B(\xi)$ is the product of the squared singular values of $B|_{\Sym^2\ker\xi}$ for unit $\xi$. Hence the phase margin gives $\norm{D_B}\ge c\,\epsilon^{2q_n\kappa}$ for the coefficient norm. On the zero-sum hyperplane of $\R^{n+1}$, the $s_n$ squares of $(e_i-e_j)/\sqrt{n+1}$, $iA+\kappa+1$ (enlarged for the finitely many geometric conditions above) preserves half the phase margin. Let $\rho_1=\epsilon^{R+b+2}$ and $\Lambda_1=C\epsilon^{-(2A+1)}$. Every smallness condition on $t$ at stage $8$ has the form $C_i\epsilon^{-a_i}t^{f_i}\le c_i\epsilon^{d_i}$ with $f_i>0$; choose $\varsigma$ with $\varsigma f_i>a_i+d_i+1$ for all $i$, $\varsigma>2A+2$ and $16\varsigma-A>R+b+3$. Put $t_0=\epsilon^\varsigma$, $h_0=\gamma(F_\epsilon)$ and \[ \delta_0=t_0^8,\qquad G_0=\sqrt{1-\delta_0^2}\,F_\epsilon,\qquad H_0=h_0+\delta_0^{-2}\bigl(h_\epsilon-h_0\bigr), \] so that $\gamma(G_0)+\delta_0^2H_0=h_\epsilon$. For $1\le j\le117$, $t_0^j\norm{F_\epsilon}_{C^{j+2}}\le C\epsilon^{\varsigma j-A}$ and $t_0^j\norm{h_0}_{C^j}\le C\epsilon^{\varsigma j-2A}$. If $P_0$ bounds the remaining polynomial losses, choose $K_{\rm rep}>16\varsigma+P_0+2$; then $\abs{H_0-h_0}_{117,t_0}=O(\epsilon^{K_{\rm rep}-16\varsigma-P_0})$, so $\mathcal W_{117}(t_0;G_0,H_0)\le\Lambda_1$ and $H_0$ lies in the required neighbourhood. The rescaling costs $\norm{G_0-F_\epsilon}_{C^2}=O(\epsilon^{16\varsigma-A})=o(\rho_1)$. \Cref{lem:iteration} gives the exact solution within the remaining budget, and \cref{lem:projector} gives the phase margin. Only suprema, bounded overlap and a fixed number of labels entered, so nothing depends on the number of charts; for compact parameters we refine product charts and freeze source covectors only. \end{proof} % ---- end sec4b-correction.tex ---- % ---- end sec4-correction.tex ---- % ---- begin sec5-immersion.tex ---- \section{Isometric immersions into small balls}\label{sec:immersion} \subsection{Composition with local phases} Let $(M,g)$ be a closed Riemannian $n$-manifold and $L\colon TM\to M\times\R^p$ an isometric bundle monomorphism with component one-forms $\alpha_1,\dots,\alpha_p$, so that $g=\sum_\ell\alpha_\ell^2$. For an integer $C\ge1$ consider $M\times(S^{2C-1})^p$ with the coefficient distribution $D=D_{L(TM)}$ of \cref{defn:coefficient} and its coefficient metric $h_{L(TM)}$. For a subbundle $E\subset TM$ we write $D_{L(E)}\subset D$ for the corresponding subdistribution. \begin{prop}[Local phases]\label{prop:local-phases} Let $n\ge2$, $N\ge s_n+1$, and let $E\subset TM$ be a hyperplane subbundle. Assume that for every $C\ge1$ and all large $k$ there are maps $S_k\colon M\times(S^{2C-1})^p\to\R^N$ such that, for all $j,R$, \begin{enumerate}[label=(\alph*)] \item $\norm{S_k}_{C^j}\le C_jk^{\max(j-1,0)}$ for derivatives in all variables; \item $\norm{\gamma_D(S_k)-h_{L(TM)}}_{C^j}\le C_{j,R}k^{-R}$; \item the partial normal second form of $S_k$ along $D$ has smallest singular value at least $ck^{-1/2}$ on $\Sym^2D_{L(E)}$. \end{enumerate} Then for every $r>0$ there is a smooth isometric immersion $F\colon(M,g)\to B_r(0)\subset\R^N$ for which every nonzero covector annihilating $E$ is good. In particular $F$ is phase regular. \end{prop} \begin{proof} \emph{Nets and colours.} For $K\ge1$ choose a maximal $K^{-1}$-separated set $\{p_i\}$ in $(M,g)$. For $K$ large, volume comparison in normal coordinates bounds the number of balls $B(p_i,4K^{-1})$ meeting a given one, uniformly in $K$. Take bumps supported in $B(p_i,2K^{-1})$, positive on $B(p_i,K^{-1})$, and normalize them by the square root of their sum of squares to get $\sum_i\chi_i^2=1$ with $\abs{\partial^j\chi_i}\le C_jK^j$. Colour the supports with a fixed number $C$ of colours so that equally coloured supports are disjoint. We fix this $C$, and with it the maps $S_k$, before choosing $k$, $K$ and $T$. \emph{Phase maps.} In normal coordinates at $p_i$ let $\varphi_{\ell i}$ be the linear function with $\dd\varphi_{\ell i}(p_i)=\alpha_\ell(p_i)$. For $T\ge K$ define $z_\ell\colon M\to\C^C$ by \begin{equation}\label{eq:phase-maps} (z_\ell)_c(x)=\sum_{\mathrm{colour}(i)=c}\chi_i(x)\,e^{2\pi\ii T\varphi_{\ell i}(x)}, \qquad H(x)=S_k\bigl(x,z_1(x),\dots,z_p(x)\bigr). \end{equation} Each summand extends by zero, and disjointness within a colour gives $\abs{z_\ell}^2=\sum_i\chi_i^2=1$; thus $z_\ell$ is a smooth map into $S^{2C-1}$. The phases are local functions; the forms $\alpha_\ell$ need not be closed. \emph{Rescaled charts.} Use normal charts of radius $3T^{-1}$, with coordinates $y\in B_3$ equal to $T$ times normal coordinates. They form a uniform atlas, uniformly in $T$, and $T^2g$ is uniformly elliptic in $y$. Put $e_{K,T}=K^{-1}+K/T$ and $C_{\ell a}=T\alpha_\ell(\partial_{y_a})$. In these charts every derivative of $z_\ell$ is bounded: derivatives of cutoffs cost $K/T\le1$, and the phase rates $T\dd\varphi_{\ell i}(\partial_{y_a})$ are bounded. Base derivatives of $S_k$ cost $T^{-1}$. By the chain rule and (a), \begin{equation}\label{eq:initial-jets} \norm H_{C^j}\le C_jk^{\max(j-1,0)} . \end{equation} \emph{First columns and metric error.} With $X_\ell$ the Hopf generators, the comparison first columns are $\Xi_a=\sum_\ell C_{\ell a}X_\ell S_k$, evaluated at the same arguments. Replacing $\dd\varphi_{\ell i}$ by $\alpha_\ell$ costs $O(K^{-1})$ on $\operatorname{supp}\chi_i$, differentiating cutoffs costs $K/T$, and base derivatives cost $T^{-1}$; hence $\partial_{y_a}H-\Xi_a=O(e_{K,T})$, and $j$ further derivatives cost at most $C_jk^j$. Since $(C_{\ell a})_\ell=L(T\partial_{y_a})$ and $L$ is isometric, the Gram matrix of the $\Xi_a$ is $T^2g$ up to the partial metric error of $S_k$. Hence, for all $j,R$, \begin{equation}\label{eq:initial-error} \norm{\gamma(H)-T^2g}_{C^j}\le C_jk^je_{K,T}+C_{j,R}k^{-R}. \end{equation} \emph{Second form.} The comparison second columns $\sum_{\ell,r}C_{\ell a}C_{rb}X_\ell X_rS_k$ differ from $\partial_{y_a}\partial_{y_b}H$ by $O(ke_{K,T})$: derivatives of the coefficients and base derivatives cost $T^{-1}$, and differentiating a first-column error uses at most the $O(k)$ second derivatives of $S_k$. For $v,w\in T_xM$ the comparison second form is $T^2\nabla^2_DS_k(Lv,Lw)$, and its projection off $\spanop(\Xi_a)=\dd S_k(D)$ is the partial normal second form pulled back by the isometry $TL\colon(TM,T^2g)\to L(TM)$. By (c) it has margin $\ge ck^{-1/2}$ on $\Sym^2E$. The first-jet projectors differ by $O(e_{K,T})$ and the second jet is $O(k)$, so the second form of $H$ has margin at least $\frac c2k^{-1/2}$ on $\Sym^2E$ provided $k^{3/2}e_{K,T}\to0$. \emph{Correction.} Apply \cref{prop:repair} in the rescaled atlas with $\epsilon=k^{-1}$, $A=118$, $\kappa=1/2$ and $b=2$, and let $K_0=K_{\rm rep}$. Choose \[ a>K_0+119,\qquad K=k^a,\qquad T=k^{2a}. \] Then $e_{K,T}=2k^{-a}$, so \eqref{eq:initial-error} at order $117$ is $o(k^{-K_0})$ (taking $R>K_0$), and $k^{3/2}e_{K,T}\to0$. All exponents are chosen before $k$, and the constants of the fixed families $S_k$ only affect how large $k$ must be. For large $k$ we obtain $H_*$ with \[ \gamma(H_*)=T^2g,\qquad \norm{H_*-H}_{C^2}\le k^{-2} \] in the rescaled atlas. By \cref{lem:projector} this changes the second form by $O(k^{-1})=o(k^{-1/2})$, so the covectors annihilating $E$ remain good. Finally $F=H_*/T$ satisfies $\gamma(F)=g$ and $\sup\abs F\le(\norm{S_k}_{C^0}+1)/T\to0$. \end{proof} \subsection{Parallelizable sources} \begin{cor}\label{cor:parallelizable-immersion} Every smooth metric on a closed parallelizable $n$-manifold, $n\ge2$, is induced by a phase-regular immersion into an arbitrarily small ball of $\R^{s_n+1}$. \end{cor} \begin{proof} Orthonormalize a global frame and let $L=(\alpha_1,\dots,\alpha_n)$ be the dual coframe, an isometric isomorphism $TM\to M\times\R^n$. Then $D_{L(TM)}$ is spanned by all Hopf generators of $(S^{2C-1})^n$, and the maps of \cref{cor:model} (independent of $x$) satisfy (a)--(c) of \cref{prop:local-phases} with $E=\ker\alpha_n$. \end{proof} % ---- end sec5-immersion.tex ---- % ---- begin sec6-embedding.tex ---- \section{Exact-metric perturbations and embeddings}\label{sec:embedding} In this section $M$ is closed. We show that a phase-regular isometric immersion can be perturbed, keeping the metric, into an embedding (\cref{thm:embedding}). The perturbations are produced by the iteration of \cref{sec:correction}; their first variations are computed in \cref{prop:probes}, and the linear algebra of \cref{sec:algebra} shows that they span enough directions to separate points. \subsection{Probes} \begin{defn}[Normal oscillations]\label{defn:oscillation} Let $F_0$ be a phase-regular immersion. A \emph{normal oscillation} is a section $A=\chi\,\nu\otimes\dd\psi^2$ of $\Sym^2T^*M\otimes\nu_{F_0}$, where $\chi$ is a cutoff supported in a chart, $\dd\psi$ is nonzero and good near $\operatorname{supp}\chi$, and $\nu$ is a normal field with $\nu\in E_{\dd\psi}$ there; $\nu$ may vanish. \end{defn} \begin{prop}[Probes]\label{prop:probes} Let $F_0\colon M\to\R^N$ be a smooth phase-regular immersion of a closed $n$-manifold, $n\ge2$, $N\ge s_n+1$, and $h_0=\gamma(F_0)$. Fix normal oscillations $A_l=\chi_l\nu_l\otimes\dd\psi_l^2$, $1\le l\le L$ (repetitions allowed), and integers $r\ge2$, $k\ge\max(8,2r)$. There are weights $w_l>0$, distinct for distinct list entries, a radius $\rho_p>0$, and for every sufficiently small $t>0$ a jointly smooth family $F_{t,u}$, $u=(u_{l,1},u_{l,2})_l$, $\abs u<\rho_p$, with $\gamma(F_{t,u})=h_0$, such that the following holds. Put \[ \tau=t^{5/4},\qquad \sigma=t^{k-5/4},\qquad a=t^{k+5/4},\qquad c_l=\cos(w_l\psi_l/\tau),\quad s_l=\sin(w_l\psi_l/\tau). \] After a smooth orthogonal identification of $\nu_{F_{t,u}}$ with $\nu_{F_0}$, the identified second form $\widetilde B$ satisfies \begin{equation}\label{eq:probes} \begin{aligned} \partial_{u_{l,1}}\widetilde B&=\sigma A_lc_l+R_{B,l1}, & \partial_{u_{l,1}}F_{t,u}&=-aw_l^{-2}\chi_l\nu_lc_l+R_{V,l1},\\ \partial_{u_{l,2}}\widetilde B&=\sigma A_ls_l+R_{B,l2}, & \partial_{u_{l,2}}F_{t,u}&=-aw_l^{-2}\chi_l\nu_ls_l+R_{V,l2}, \end{aligned} \end{equation} and, uniformly for $\abs u<\rho_p$, with functions $\eta_B,\eta_V\to0$ as $t\to0$, \begin{equation}\label{eq:probe-errors} \begin{gathered} \norm{F_{t,u}-F_0}_{C^r}\le C\,t^{k+\frac54(1-r)},\qquad \norm{\partial_uF_{t,u}}_{C^1}\le Ct^k,\qquad \norm{F_{t,u}}_{C^3}\le C,\\ \abs{R_{B,li}}\le\sigma\bigl(\eta_B(t)+C\abs u\bigr),\qquad \abs{R_{V,li}}\le a\bigl(\eta_V(t)+C\abs u\bigr). \end{gathered} \end{equation} \end{prop} \begin{proof} \emph{Phase data.} Construct phase data for $(F_0,h_0)$ as in the proof of \cref{prop:repair} (at one fixed $\epsilon$), and add one label for each list entry, with phase $w_l\psi_l$ and support $\operatorname{supp}\chi_l$. The weights $w_l$ are inserted into the weight hierarchy, so that all doubled and interacting phases, including those between probe and background labels, are good. All choices are made before $t$. Extend the parameter ball to a torus by a smooth map that equals the coordinates on a smaller ball and takes values in a small ball, so that \cref{lem:iteration} applies with a compact parameter space. \emph{First step.} Let $\delta=t^k$, $\ell=t^{9/8}$, $\delta_1=\tau^k$, and $G_0=\sqrt{1-\delta^2}F_0$, which has the same normal spaces as $F_0$. Run the construction of \cref{lem:increment} with $q=13(k+1)$, where the probe labels receive the prescribed free amplitudes \begin{equation}\label{eq:probe-amplitude} v_{f,l}=-\delta\tau\,\chi_l\,w_l^{-2}\nu_l\,(u_{l,1}-\ii u_{l,2}), \end{equation} and the background labels receive $\sqrt2\delta\tau b_\lambda(A)n_\lambda$ as before. The real part of the leading probe mode is the value term in \eqref{eq:probes}. Its mean contribution to the metric is $\delta^2H_{\rm probe}(u)$, $H_{\rm probe}=\sum_l\chi_l^2\abs{\nu_l}^2\abs{u_l}^2\dd\psi_l^2/(2w_l^2)$ to leading order. The background tensor is $A=A_q$, obtained from $A_0=\bigl(1-(\delta_1/\delta)^2\bigr)h_0-H_{\rm probe}(u)$ by the $q$ substitutions \[ A_{j+1}=\bigl(1-(\delta_1/\delta)^2\bigr)h_0-H_{\rm probe}(u)-\mathcal T(A_j,u), \] where $\mathcal T$ now includes the corrections of the probe modes. A small parameter ball keeps the $A_j$ in the positive neighbourhood. Distinct labels have no phase-zero cross terms, so all phase-zero terms are sesquilinear in the amplitude of a single label; hence the probe contributions, and inductively the $A_j$, are even in $u$. Their second $u$-derivatives are bounded independently of $t$, because only source variables are differentiated and the square roots have uniform reserves; so $\partial_uA_j=O(\abs u)$. Then correct the nonzero-phase quadratic terms as in \cref{lem:increment}, and let $U_0$ be the resulting increment, $G_1=G_0+U_0$ and $H_1=\delta_1^{-2}(h_0-\gamma(G_1))$. The normalized errors are $O\bigl(t^{(k+13)/8}+t^{(2k-5)/4}\bigr)$, so $G_1,H_1$ satisfy the entrance conditions of \cref{lem:iteration} at scale $t_1=\tau$ and stage $k$, with the weighted bound $\mathcal W_{r_k}(t_1;G_1,H_1)\le\Lambda_1$. \emph{First variations.} Two derivatives of \eqref{eq:probe-amplitude} times $e^{\ii w_l\psi_l/\tau}$ give $\sigma A_l$ times $c_l$ or $s_l$. Slow derivatives and the corrections of \cref{lem:modes} contribute $O(\sigma\tau/\ell)$ to the second-derivative columns and $O(a\tau/\ell)$ to the values; the $u$-dependence of the background contributes $O(\sigma\abs u)$ and $O(a\abs u)$; the secondary increment contributes $O(\sigma^2)$ in $C^2$ and $O(\delta^2)=O(a\sigma)$ in $C^0$. \emph{Tail.} Let $C_*\ge C_m(k)$ for $m\in\{1,2,3,r\}$, where $C_m(k)$ are the constants of \eqref{eq:increment-sizes} in full norms. At the entrance of every later stage $k'>k$ we impose $C_m(k')\,t_j^{k'-k}\le C_*$ at these orders, as \cref{lem:iteration} permits. Then every later increment satisfies $\norm{U_j}_{C^m}\le C_*\,t_j^{\,k+\frac54(1-m)}$ for these $m$, and since $t_{j+1}=t_j^{5/4}$ the sums are dominated by their first terms. Summing from $t_1=t^{5/4}$, the tail satisfies \begin{align*} \norm{\mathrm{Tail}}_{C^2_xC^1_u}&\le2C_*t^{\frac{5k}4-\frac{25}8}=o(\sigma),\\ \norm{\mathrm{Tail}}_{C^1_xC^1_u}&\le2C_*t^{\frac{5k}4-\frac{25}{16}}=o(t^k),\\ \norm{\partial_u\mathrm{Tail}}_{C^0}&\le2C_*t^{\frac{5k}4}=o(a), \end{align*} with gaps $(2k-15)/8$, $(4k-25)/16$ and $(k-5)/4$, positive for $k\ge8$. At order $r$ the same sum is $o(t^{k+\frac54(1-r)})$ because $k\ge2r$, and at order $3$ the first increment is $O(t^{k-5/2})$. This gives the first line of \eqref{eq:probe-errors}. \emph{Second forms.} Since $\gamma(F_{t,u})=h_0$, \eqref{eq:gauss} gives $B=\partial^2F-\Gamma(h_0)\partial F$; the $u$-derivative of the connection term is $O(t^k)=o(\sigma)$. If $\Pi_{t,u}$ is the normal projection of $F_{t,u}$ and $\iota$ the inclusion of $\nu_{F_0}$, then $\Pi_{t,u}\iota(\iota^*\Pi_{t,u}\iota)^{-1/2}$ is an orthogonal isomorphism $\nu_{F_0}\to\nu_{F_{t,u}}$ whose distance to $\iota$ and whose $u$-derivative are $O(t^k)$. The resulting errors are $o(\sigma)$, which completes the proof of \eqref{eq:probes} and \eqref{eq:probe-errors}. \end{proof} \subsection{Admissible normal directions}\label{sec:algebra} \begin{lem}[Spaces of quadratic forms]\label{lem:quadratic-span} Let $V$ be a Euclidean space, $\cP\subset\Sym^2V$ a linear subspace, and $d=\max_\xi\dim\cP\xi$, where $\cP\xi=\{Q\xi:Q\in\cP\}$. Then $\dim\cP\le d(d+1)/2$, and if $d\ge2$ equality holds only if $\cP=\Sym^2U$ for a $d$-dimensional subspace $U$. \end{lem} \begin{proof} The case $d=0$ is trivial. Choose $\xi_0$ with $\dim\cP\xi_0=d$ and $\xi_0\not\perp\cP\xi_0$; such points form a nonempty Zariski-open set, since a nonzero quadratic form cannot vanish on a nonempty open set. Put $U_0=\cP\xi_0$ and $E=U_0\cap\xi_0^\perp$, so $\dim E=d-1$. Choose $Q_1,\dots,Q_d\in\cP$ with $Q_i\xi_0$ independent and let $A(\xi)$ have columns $Q_i\xi$. Near $\xi_0$ every $Q\in\cP$ satisfies $Q\xi=A(\xi)c_Q(\xi)$ with $c_Q=(A^*A)^{-1}A^*Q\xi$ smooth. If $Q\xi_0=0$, then $c_Q(\xi_0)=0$, and differentiating gives $Qv=A(\xi_0)\,\dd c_Q(\xi_0)v\in U_0$ for all $v$. Thus $\im Q\subset U_0\cap\xi_0^\perp=E$, i.e.\ $Q\in\Sym^2E$. Hence $\dim\cP\le d+\dim\Sym^2E=d(d+1)/2$. Suppose equality and $d\ge2$. Then $\Sym^2E\subset\cP$ is the kernel of evaluation at $\xi_0$. Let $\pi$ be the projection onto $E^\perp$ and $\cL=\pi\cP\subset\Hom(V,E^\perp)$; then $\dim\cL=d$. If $\xi$ has nonzero component in $E$, then $(\Sym^2E)\xi=E$, so $\dim\cL\xi\le1$; the corresponding $2\times2$ minors are polynomials in $\xi$, so $\dim\cL\xi\le1$ for all $\xi$. If some $A\in\cL$ had rank at least two, pick $x,y$ with $Ax,Ay$ independent. For $A'\in\cL$, pointwise dependence at $x,y,x+y$ gives $A'x=cAx$ and $A'y=cAy$ with one scalar $c$. If $Az\ne0$, then $Az$ is independent of $Ax$ or of $Ay$, and comparing at $x+z$ or $y+z$ gives $A'z=cAz$; if $Az=0$, both comparisons give $A'z\in\R Ax\cap\R Ay=0$. So $A'=cA$, contradicting $\dim\cL\ge2$. Hence every nonzero member of $\cL$ has rank one, and two members with different image lines would give independent values at a vector outside both kernels, contradicting $\dim\cL\xi\le1$. So all images lie in one line $\R f$. Every $Q\in\cP$ then has image in $U=E\oplus\R f$, so $Q\in\Sym^2U$, and $\cP=\Sym^2U$ by dimension. \end{proof} \begin{prop}[Admissible value directions]\label{prop:admissible-span} Let $\dim V=n$, let $B\colon\Sym^2V\to W$ be phase regular (some $\xi\in V^*$ has $B$ injective on $\Sym^2\ker\xi$), $K=\ker B$, $d=\dim K$, and $E_\xi=B(\Sym^2\ker\xi)^\perp$ for good $\xi$. Let \[ \cR(K)=\{P\in\Sym^2V:\ P\xi\in K\xi\text{ for every good }\xi\}. \] Then $\dim\cR(K)\le d(d+1)/2$, $(\spanop_\xi E_\xi)^\perp=B(\cR(K))$, and \begin{equation}\label{eq:value-dimension} \dim\spanop_\xi E_\xi=\dim W-\dim\cR(K)+d\ \ge\ \dim W-\frac{d(d-1)}2 . \end{equation} If moreover $n\ge3$, $B$ is onto and $\dim W=q_n+1$, then $\dim\spanop_\xi E_\xi\ge n+1$ unless \begin{equation}\label{eq:exceptional} K\subset\Sym^2U\quad\text{for a hyperplane }U\subset V, \end{equation} in which case $U$ is unique, $\cR(K)=\Sym^2U$, and the span has dimension $n$. \end{prop} \begin{proof} Goodness of $\xi$ means that $Q\mapsto Q\xi$ is injective on $K$. For a basis $Q_1,\dots,Q_d$ of $K$, the condition $P\in\cR(K)$ is the polynomial identity $Q_1\xi\wedge\dots\wedge Q_d\xi\wedge P\xi=0$ on the open cone of good covectors, so $\cR(K)$ is a linear space whose evaluation rank equals $d$ on that cone and is at most $d$ everywhere. \Cref{lem:quadratic-span} gives the dimension bound. A vector orthogonal to all $E_\xi$ lies in every $B(\Sym^2\ker\xi)\subset\im B$; write it as $B(P)$. For good $\xi$ it equals $B(P_\xi)$ with $P_\xi\xi=0$, so $(P-P_\xi)\in K$ and $P\xi\in K\xi$; conversely $P\in\cR(K)$ gives $B(P)\in B(\Sym^2\ker\xi)$ for each good $\xi$. This proves the second identity, and \eqref{eq:value-dimension} follows from $K\subset\cR(K)$. In the onto case $d=s_n-q_n-1=n-1\ge2$ and the lower bound in \eqref{eq:value-dimension} is $n$. Equality means $\dim\cR(K)=d(d+1)/2$, i.e.\ $\cR(K)=\Sym^2U$ for a hyperplane $U$ by \cref{lem:quadratic-span}, and then $K\subset\Sym^2U$. Conversely, if $K\subset\Sym^2U$ then $K\xi=U$ for good $\xi$, so $\Sym^2U\subset\cR(K)$ and equality holds; $U=K\xi$ is unique. \end{proof} % ---- begin sec6b-embedding.tex ---- \begin{prop}[The exceptional locus]\label{prop:exceptional} Let $n\ge3$, $\dim W=q_n+1$, and let $\mathcal O\subset\Hom(\Sym^2V,W)$ be the open set of onto, phase-regular maps. The set $\mathcal E\subset\mathcal O$ defined by \eqref{eq:exceptional} is a smooth, relatively closed submanifold. At every $B\in\mathcal E$ there are a good $\xi$ and $\nu\in E_\xi$ such that the variation $\dot B=\nu\otimes\xi^2$, i.e.\ $P\mapsto P(\xi,\xi)\nu$, is not tangent to $\mathcal E$. \end{prop} \begin{proof} For $B\in\mathcal E$ write $K=\ker B\subset\Sym^2U$, $U=e^\perp$, $\abs e=1$. For good $\xi$ the map $Q\mapsto Q\xi=Q(\xi-\ip\xi ee)$ is injective on $K$ with image $U$; in particular no nonzero $\zeta\in U$ satisfies $K\zeta=0$, for otherwise $K\xi$ would lie in a subspace of dimension $n-2$. \emph{Submanifold.} The map $B\mapsto\ker B$ is a submersion from onto maps to $\Gr_{n-1}(\Sym^2V)$, so it suffices to treat the set of kernels. These are parametrized by pairs $(U,K)$ with $K\in\Gr_{n-1}(\Sym^2U)$, a Grassmann bundle over the space of hyperplanes. An infinitesimal rotation $u\mapsto\ip u\zeta e$ ($\zeta\in U$) of $U$ moves $Q\in K$ by $2e\sym(Q\zeta)$, which lies in the complement $e\sym U$ of $\Sym^2U$; it vanishes for all $Q\in K$ only if $\zeta=0$. Variations within $\Sym^2U$ are the usual Grassmannian tangent vectors. Hence the parametrization is an immersion. It is injective, since $U=K\xi$ for any good $\xi$, and $U$ depends continuously on $K$; so it is an embedding. Limits of exceptional kernels in $\mathcal O$ are exceptional by compactness of the space of hyperplanes. \emph{Transversality.} Let $A=B|_{K^\perp}$ and $G=(A^*A)^{-1}$. For a variation $\dot B$, the kernel, written as a graph over $K$ in $K^\perp$, has derivative $\dot K=-A^{-1}\dot B|_K$. If the variation is tangent to $\mathcal E$, differentiating $Q_s(e_s,e_s)=0$ for $Q_s\in K_s$ and using $Qe=0$ gives $\dot K(Q)(e,e)=0$ for all $Q\in K$. Take a good covector, project it along $e$ to get a good $z\in U$, and choose $Q_0\in K$ with $Q_0z=z$, so $Q_0(z,z)=\abs z^2>0$. Each $\xi_t=z+te$ is good. Let $\nu_t=(A^*)^{-1}(e\sym\xi_t)$; since $e\sym\xi_t\perp K$ we have $B^*\nu_t=e\sym\xi_t$, which vanishes on $\Sym^2\ker\xi_t$, so $\nu_t\in E_{\xi_t}$. For $\dot B=\nu_t\otimes\xi_t^2$ we get $\dot K(Q)=-Q(z,z)\,G(e\sym\xi_t)$ and \[ \dot K(Q_0)(e,e)=-\abs z^2\bigl(\ip{G(e\sym z)}{e\sym e}+t\ip{G(e\sym e)}{e\sym e}\bigr). \] The coefficient of $t$ is nonzero because $G$ is positive and $e\sym e\in K^\perp$. For all but one value of $t$ the variation is not tangent to $\mathcal E$. \end{proof} \begin{lem}[Parametric avoidance]\label{lem:avoidance} Let $P\subset\R^p$ be open, $Y$ a second-countable manifold of dimension $m$, and $\Phi\colon P\times Y\to\R^b$ smooth. If the derivative of $\Phi$ has rank greater than $m$ at every point of a set $Z\subset\Phi^{-1}(0)$, then the projection of $Z$ to $P$ has measure zero. \end{lem} \begin{proof} Near each point of $Z$, $m+1$ suitable components of $\Phi$ cut out a submanifold of dimension $p-1$ containing $Z$ locally; its projection to $P$ is null by Sard's theorem. Countably many such neighbourhoods cover $Z$. \end{proof} \subsection{Separation of points} For a compact $P\subset(M\times M)\setminus\Delta$, a map \emph{separates} $P$ if it takes different values at the two points of every pair in $P$. \begin{lem}[Separation on compact pair sets]\label{lem:separation} Let $n\ge3$, $N=s_n+c$ with $c\ge1$, and let $F\colon M\to\R^N$ be a smooth phase-regular immersion of a closed manifold. For every compact $P\subset(M\times M)\setminus\Delta$, $r\ge2$ and $\epsilon>0$ there is a smooth phase-regular immersion $G$ with $\gamma(G)=\gamma(F)$, $\norm{G-F}_{C^r}<\epsilon$, which separates $P$. \end{lem} \begin{proof} Replace $P$ by $P\cup P^\T$ and let $C_0$ be the union of its two projections. Put $q=q_n$; the normal spaces have dimension $q+c$, and phase regularity gives $\rank B\ge q$. Second forms are written in local orthonormal frames of $TM$ and of $\nu_F$, transported to nearby maps by the identification of \cref{prop:probes}. \emph{Removal principle.} Let $G$ be a smooth phase-regular immersion with $\gamma(G)=\gamma(F)$, and let $Q\subset P$ be compact. Let $\mathcal U$ be an open set of second forms (invariant under the frame changes) containing $B_G(x)$ for all first points $x$ of pairs in $Q$, and let $\Sigma\subset\mathcal U$ be a submanifold, closed in $\mathcal U$, with local defining maps $\Phi_\Sigma$ (or $\Sigma=\mathcal U$, with no equations). Suppose that at every $(x,y)\in Q$ with $G(x)=G(y)$ and $B_G(x)\in\Sigma$ there are normal oscillations $A_l=\chi_l\nu_l\otimes\dd\psi_l^2$, supported near $x$ and away from $y$, such that the vectors \begin{equation}\label{eq:limit-columns} \bigl(\dd\Phi_\Sigma(A_l(x)),0\bigr),\qquad \bigl(0,\chi_l\nu_l(x)\bigr),\qquad \bigl(0,\dd G(v)\bigr)\ \ (v\in T_xM) \end{equation} span a space of dimension greater than $2n$. Then arbitrarily $C^r$-small exact-metric perturbations of $G$ have no such pairs in $Q$. Indeed, since $\Sigma$ is closed in $\mathcal U$, the set of such pairs is compact, and by compactness finitely many oscillations work on a neighbourhood $\mathcal N$ of it. Apply \cref{prop:probes} to the list in which every oscillation appears twice, with weights $w_{l,1}\ne w_{l,2}$. At a point $(x,y)$, combine each cosine and sine column with the coefficients $(c_l,s_l)$ of its own weight: the normalized second-form and value columns become $(A_l,-w_{l,j}^{-2}\chi_l\nu_l)$ up to $o(1)+O(\abs u)$, uniformly, and the response at $y$ is controlled by the global error. Since $\bigl(\begin{smallmatrix}1&1\\-w_{l,1}^{-2}&-w_{l,2}^{-2}\end{smallmatrix}\bigr)$ is invertible, bounded combinations isolate both kinds of columns. Divide the $\Phi_\Sigma$-rows by $\sigma$, the value rows by $a$, and multiply the derivatives in $x$ by $a$; their $\Phi_\Sigma$-entries become $(a/\sigma)\partial_x\Phi_\Sigma(B)=O(\tau^2)$ by the $C^3$ bound, and their value entries converge to $\dd G$. Hence for $\abs u$ and then $t$ small the derivative of $(x,y,u)\mapsto\bigl(\Phi_\Sigma(B_{t,u}(x)),\,G_{t,u}(x)-G_{t,u}(y)\bigr)$ has rank greater than $2n$ on $\mathcal N$, and $C^2$-convergence excludes such pairs outside $\mathcal N$. \Cref{lem:avoidance} gives a parameter $u$ for which none remain. By \eqref{eq:probe-errors} the perturbation is $C^r$-small, and it can be chosen to preserve any previously established positive separation margin on a compact set of pairs, and the phase regularity. \emph{Rank $q$.} Let $\mathcal U$ be the open set of matrices of rank at least $q$, which contains all phase-regular second forms, and $\Sigma_q\subset\mathcal U$ the closed submanifold of matrices of rank $q$, of codimension $nc$. At $B\in\Sigma_q$ with kernel $K$ ($\dim K=n$) and cokernel $L=(\im B)^\perp$ ($\dim L=c$), the normal space of $\Sigma_q$ is $\Hom(K,L)$, via $\dot B\mapsto\pi_L\dot B|_K$. Every good $\xi$ has $E_\xi=L$, and the variations $\nu\otimes\xi^2$ project to $Q\mapsto Q(\xi,\xi)\nu$; as $\xi$ ranges over the open cone of good covectors and $\nu$ over $L$, these span $\Hom(K,L)$, since a quadratic form vanishing on an open cone is zero. The span in \eqref{eq:limit-columns} has dimension $nc+c+n>2n$. The set of pairs with $\rank B(x)=q$ is compact, because lower rank is excluded. The removal principle gives $G_1$ without coincidences at rank-$q$ points. Choose an open $O\supset\{x\in C_0:\rank B_{G_1}(x)=q\}$ such that $G_1$ separates every pair of $P$ meeting $\overline O$ with a positive margin, which we protect from now on. On $C_0\setminus O$ the rank is at least $q+1$ with a uniform margin. \emph{The case $c\ge2$.} On $C_0\setminus O$ we have $d=\dim\ker B\le n-1$, and \cref{prop:admissible-span} gives at least $q+c-\frac{(n-1)(n-2)}2=n-1+c\ge n+1$ admissible value directions; with the $n$ tangent directions the span in \eqref{eq:limit-columns} (with $\Sigma=\mathcal U$ and $Q$ the pairs of $P$ with first point in $C_0\setminus O$) has dimension at least $2n+1$. One more application of the removal principle separates $P$. \emph{The case $c=1$.} On $C_0\setminus O$ the second form is onto. At exceptional points, \cref{prop:admissible-span} gives $n$ admissible value directions and \cref{prop:exceptional} an admissible variation transverse to $\Sigma=\mathcal E$, which is closed in $\mathcal U=\mathcal O$; for the pairs of $P$ with first point in $C_0\setminus O$ the span in \eqref{eq:limit-columns} has dimension at least $1+n+n$. The removal principle eliminates the exceptional coincidences; protect a neighbourhood $O_{\mathcal E}$ of the exceptional points as before. On $C_0\setminus(O\cup O_{\mathcal E})$, \cref{prop:admissible-span} gives $n+1$ admissible value directions, and a final application of the removal principle separates $P$. \end{proof} \begin{rem}\label{rem:separation-axioms} The proof uses the probe families only through: membership in a class of phase-regular immersions with the same metric that is closed under the construction; the leading columns in \eqref{eq:probes}, possibly multiplied by one fixed smooth positive function; $a/\sigma\to0$; the error bounds \eqref{eq:probe-errors} on a compact neighbourhood of $C_0$, with global control of the responses away from the supports; and smallness in a prescribed norm. It therefore applies to any class with such families. \end{rem} \begin{lem}[Baire separation]\label{lem:baire} Let $M$ be a second-countable manifold, $\cX$ a Baire space, and $v\mapsto F_v\colon M\to\R^N$ a family of maps such that $v\mapsto F_v|_K$ is continuous in the uniform norm for every compact $K\subset M$. If for every compact $P\subset(M\times M)\setminus\Delta$ the parameters whose maps separate $P$ are dense, then the parameters giving injective maps are dense. \end{lem} \begin{proof} Let $K_j$ be a compact exhaustion, $d$ a metric for the topology, and $P_j=\{(x,y)\in K_j^2:d(x,y)\ge1/j\}$. Separating $P_j$ is an open condition (the minimum of $\abs{F_v(x)-F_v(y)}$ over $P_j$ is positive, or $P_j=\emptyset$) and dense by hypothesis. The intersection over $j$ is dense, and its maps are injective. \end{proof} \subsection{Embeddings with the same metric} \begin{thm}\label{thm:embedding} Let $n\ge3$, $N\ge s_n+1$, and let $F\colon M\to\R^N$ be a smooth phase-regular immersion of a closed $n$-manifold. For every $\epsilon>0$ there is a smooth phase-regular embedding $G$ with $\gamma(G)=\gamma(F)$ and $\norm{G-F}_{C^2}<\epsilon$. \end{thm} \begin{proof} $C^\infty(M,\R^N)$ is a Fr\'echet space and $\gamma$ is continuous, so $\cY=\{G:\gamma(G)=\gamma(F)\}$ is closed, hence completely metrizable. By \cref{lem:projector} there is $0<\rho<\epsilon$ such that every map in the nonempty open set $\cX=\{G\in\cY:\norm{G-F}_{C^2}<\rho\}$ is phase regular; $\cX$ is a Baire space. Every basic neighbourhood in $\cX$ contains a small ball for some $C^r$ norm, $r\ge2$, within which \cref{lem:separation} separates any compact off-diagonal pair set. By \cref{lem:baire} some $G\in\cX$ is injective; being an injective immersion of a compact manifold, it is an embedding. \end{proof} \begin{proof}[Proof of \cref{thm:A} for closed manifolds] By \cref{cor:parallelizable-immersion} there is a phase-regular isometric immersion into $B_{r/2}(p)$. \Cref{thm:embedding} with $N=s_n+1$ gives an isometric embedding within $C^0$-distance $r/2$, hence into $B_r(p)$. Connectedness is not used. \end{proof} % ---- end sec6b-embedding.tex ---- % ---- end sec6-embedding.tex ---- % ---- begin sec7-stable.tex ---- \section{Stably parallelizable closed manifolds}\label{sec:stable} For \cref{thm:B} the explicit model maps of \cref{sec:model} are replaced by a partially free map produced from bundle data. We use the notation of \cref{sec:partial}; all distributions are integrable and carry flat structures. A \emph{foliation box} is a chart $\Omega\cong B^m\times B^d$ in which the flat frame is $\partial_{\theta_1},\dots,\partial_{\theta_m}$ for the leaf coordinates $\theta$; a \emph{linear leaf coordinate} is $\phi=\sum_\alpha c_\alpha\theta_\alpha$ with constant $c\ne0$. \subsection{Bundle-theoretic input} \begin{lem}[Orthogonal transport and cancellation]\label{lem:transport} \begin{enumerate}[label=(\alph*)] \item A smooth homotopy $P_t$ of orthogonal projections in a trivial bundle is covered by a smooth homotopy of orthogonal automorphisms $O_t$ with $O_0=\Id$ and $P_tO_t=O_tP_0$. \item If a manifold has the homotopy type of a CW complex of dimension $d$, every stably trivial bundle of rank greater than $d$ over it is trivial. \end{enumerate} \end{lem} \begin{proof} (a) Solve $\partial_tO_t=[\partial_tP_t,P_t]O_t$, $O_0=\Id$. The coefficient is skew, so $O_t$ is orthogonal. Differentiating $P_t^2=P_t$ gives $\partial_tP_t=[[\partial_tP_t,P_t],P_t]$, so $P_tO_t$ and $O_tP_0$ solve the same linear equation. No compactness is needed. (b) It suffices to cancel one trivial line. If $E\oplus\triv1\cong\triv{r+1}$ with $r=\rank E>d$, the image of the last line gives $u\colon M\to S^r$ with $E\cong u^\perp$. By cellular approximation \cite[Theorem~4.8]{Hatcher2002}, $u$ is nullhomotopic, since $S^r$ has no cells in dimensions $1,\dots,d$. Make the homotopy smooth relative to its ends (approximate in the ambient space and normalize) and apply (a) to $\Id-u_tu_t^*$. \end{proof} \begin{prop}[The hyperplane bundle]\label{prop:hyperplane-bundle} Under the hypotheses of \cref{thm:B} there is a unit vector field $e$ such that, with $H=e^\perp$, \[ TM=H\oplus\R e,\qquad H\oplus\Sym^2H\cong\triv{s_n-1}. \] \end{prop} \begin{proof} Stable triviality orients every component $P$, and $\chi(P)=0$, so the Euler class vanishes \cite[Corollary~11.12]{MilnorStasheff1974}; it is the only obstruction to a section of the unit sphere bundle of $TP$ over the $n$-dimensional base \cite[Proposition~3.22]{HatcherVB2017}. Symmetric squares preserve stable triviality: if $E\oplus\triv a$ is trivial, then $\Sym^2(E\oplus\triv a)\cong\Sym^2E\oplus aE\oplus\triv{s_a}$ is trivial, and adding $\triv{a^2}$ trivializes $aE$. Hence $\Sym^2TM\cong\Sym^2H\oplus H\oplus\triv1$ is stably trivial, so $H\oplus\Sym^2H$ is stably trivial of rank $s_n-1>n$, and trivial by \cref{lem:transport}(b). \end{proof} \subsection{An h-principle for partially free maps} \begin{lem}[Relative realization of partial jets]\label{lem:hprinciple} Let $D$ be an integrable distribution of rank $m$ on a compact manifold $X$, and $Q\ge m+s_m+1$. Let $J\colon D\oplus\Sym^2D\to X\times\R^Q$ be a monomorphism and $f\colon X\to\R^Q$ a smooth map with $J=J_D(f)$ near a closed set $K\supset\partial X$. Then there are a smooth $D$-free map $F$, equal to $f$ near $K$ and arbitrarily $C^0$-close to $f$, and a smooth homotopy of monomorphisms from $J$ to $J_D(F)$, fixed near $K$. If the first component of $J$ is $\dd_Df$, then $F$ can be chosen arbitrarily $C^1$-close to $f$. \end{lem} \begin{proof} \emph{Full jets.} There is a torsion-free connection $\nabla$ on $TX$ with $\nabla_{Y_\alpha}Y_\beta=0$ for all flat frames: locally take the coordinate connection of a foliation box, and patch with a partition of unity. Both conditions are affine and, because transition matrices of flat frames are constant along leaves, independent of the frame. Then $J_D(f)$ is the restriction of the full $2$-jet $(\dd f,\nabla^2f)$ to $D$ and $\Sym^2D$. Fix a complement $TX=D\oplus C$. If $J=(j_1,j_2)$, the full formal jet $(f,\ L,\ \Sigma)$ with $L=j_1\oplus\dd f|_C$ and $\Sigma=j_2$ on $\Sym^2D$, $\Sigma=\nabla^2f$ on the components involving $C$, equals $j^2f$ near $K$. \emph{Ampleness.} Let $\cR_1\subset J^1(X,\R^Q)$ be defined by injectivity of $L|_D$ and $\cR_2\subset J^2(X,\R^Q)$ by injectivity of the partial jet; both are open. Consider a principal direction given by a covector $\zeta$. If $\zeta|_D=0$, the variations $\zeta\otimes u$ and $\zeta^2\otimes u$ do not change the partial jet, so the principal slices are empty or full. Otherwise choose a basis $v_1,\dots,v_m$ of $D_x$ with $\zeta(v_i)=0$ for $i0$, free maps $P_{\epsilon,k}$, equal to $F$ outside an inner leaf support, with displacement $O(\epsilon)$ and \[ e=\gamma_D(F)+a^4\dd\theta_1^2-\gamma_D(P_{\epsilon,k}),\qquad \norm e_{C^{2,\alpha}}\le C_k\epsilon^{k-2}. \] These formulas consist of pairing systems, solved with the canonical inverse $A(A^*A)^{-1}$, and periodic integrations, solved with the zero-mean primitive, applied to a fixed periodic auxiliary map and a fixed plane oscillator; the fast substitution $z=\rho(\theta_1/\epsilon)$ does not depend on $p$. Hence all coefficients depend smoothly on $p$, and the rank and Gram margins of the auxiliary periodic configurations are uniform by compactness. The formulas remain smooth across zeros of $a$: the first correction factors as $\widetilde u_2=aR$, because its pairing right sides contain the factor $a$ and the canonical inverse is linear, so the apparent quotient $\abs{\partial_z\widetilde u_2}^2/(2a^2)$ equals $\abs{\partial_zR}^2/2$; the later systems use a fixed partition in $a^2$, dividing by $a$ only where $\abs a$ is bounded below. If $a(p,\cdot)\equiv0$, all oscillators, corrections, residuals and right sides vanish by induction, and the canonical inverses map $0$ to $0$. \emph{Exact correction.} The pairing inverse $E[P]$ is rational in the first and second partial derivatives of $P$; the free Gram determinant of $P_{\epsilon,k}$ is bounded below independently of small $\epsilon$, while leaf derivatives of order $j$ cost $C_{j,k}\epsilon^{-j}$. Hence $\norm{E[P_{\epsilon,k}]}_{C^{2,\alpha}}\le C_k\epsilon^{-b}$ with $b$ independent of $k$. Let $\chi$ be a leaf cutoff equal to $1$ on the support of $e$. By the supported correction theorem \cite[Theorem~2.12 and Sections~3.1--3.3, (3.23)--(3.28)]{Zergaenge2017}, $e$ is removed by $\chi^2v$, where $v$ is the unique small fixed point of a map $\Phi[P,\chi,e]$ on the leaf space $C^{2,\alpha}$, provided \begin{equation}\label{eq:addition-smallness} \norm{E[P]}_{2,\alpha}\norm{E[P](0,e)}_{2,\alpha}\le C_k\epsilon^{k-2-2b},\qquad \norm{\chi^2v}_{2,\alpha}\le C_k\epsilon^{k-2-b} \end{equation} are small. Choose $k$ with $k-2-2b>0$ and $k-2-b$ larger than the loss in the freeness stability estimate \cite[Satz~4.4(i)]{Zergaenge2016}, then one $\epsilon$ giving a strict contraction, uniformly in $p$, that preserves freeness and meets the $C^0$ tolerance. \emph{Parameter dependence.} Write the fixed-point equation as $v(p)=\Phi(p,v(p))$ with common contraction constant $q<1$ on a convex invariant ball. Then $\norm{v(p+h)-v(p)}\le(1-q)^{-1}\norm{\Phi(p+h,v(p))-\Phi(p,v(p))}$, and the difference quotients $w_h$ satisfy $w_h=A_hw_h+b_h$ with $A_h=\int_0^1D_v\Phi\bigl(p+h,v(p)+s(v(p+h)-v(p))\bigr)\,\dd s$, $\norm{A_h}\le q$, and $b_h\to\partial_p\Phi$. Neumann series give $\partial_pv=(I-D_v\Phi)^{-1}\partial_p\Phi$, continuously in $p$. At every higher order the top derivative of $v$ appears only through $D_v\Phi$, so induction gives smooth dependence on $p$ with values in leaf $C^{2,\alpha}$; higher leaf regularity follows inductively from the Schauder and product estimates \cite[Sections~3.2--3.3, (3.12)--(3.13) and (3.27)--(3.28)]{Zergaenge2017}, in which the highest leaf derivative again has the small coefficient. Hence $G=P_{\epsilon,k}+\chi^2v$ is jointly smooth with the asserted metric. On a plaque where $a\equiv0$ we have $P=F$ and $e=0$, so uniqueness gives $v=0$; the nested supports allow extension by $F$ outside $\Omega$. \end{proof} \subsection{Relative partial metric realization} \begin{prop}\label{prop:realization} Let $\pi\colon X\to B$ be a surjective submersion of compact manifolds, either both closed or with $\partial X=\pi^{-1}(\partial B)$, and let $D\subset\ker\dd\pi$ be an integrable distribution of rank $m\ge1$ with a flat structure. Let $Q\ge m+s_m+5$, $A\subset B$ closed with $\partial B\subset A$, and $h$ a positive partial metric on $D$. Let $F_A$ be a smooth $D$-free map defined above a neighbourhood of $A$ with $\gamma_D(F_A)=h$. The following are equivalent: \begin{enumerate}[label=(\roman*)] \item there is a monomorphism $D\oplus\Sym^2D\to X\times\R^Q$ equal to $J_D(F_A)$ above a neighbourhood of $A$; \item there is a smooth $D$-free map $F\colon X\to\R^Q$ with $\gamma_D(F)=h$, equal to $F_A$ above a neighbourhood of $A$. \end{enumerate} \end{prop} \begin{proof} (ii)$\Rightarrow$(i) is clear. For the converse assume first $\emptyset\ne A\ne B$. Choose neighbourhoods $V_0\Subset V_1\Subset V_2\Subset V_3$ of $A$ with $\overline V_3$ inside the region where the monomorphism equals $J_D(F_A)$; cut $F_A$ off by a base function equal to $1$ near $\overline V_3$ to get a global $f$, and apply \cref{lem:hprinciple} relative to $\pi^{-1}(\overline V_3)$. The result is a free $H$ with $\gamma_D(H)=h$ over $\pi^{-1}(\overline V_3)$. \emph{Positive deficit.} Choose $M_0\ge1$ with $\gamma_D(H)\le M_0h$, $0<\lambda_0<1$ with $\lambda_0^2M_0<\frac12$, and $\beta_0=(1-\lambda_0^2)^{1/4}$. Let $\beta\colon B\to[0,\beta_0]$ vanish near $\overline V_1$ and equal $\beta_0$ outside $V_2$, and put $\lambda=(1-\beta^4)^{1/2}$, both pulled back to $X$. Since $D$ is vertical, $\lambda$ is constant along leaves, so $H_s=\lambda H$ is free with $\gamma_D(H_s)=\lambda^2\gamma_D(H)$. Let \begin{equation}\label{eq:deficit} \Delta=\frac{h-\lambda^2\gamma_D(H)}{\beta^4}\ \text{ on }\{\beta\ne0\},\qquad \Delta=h\ \text{ on }\{\beta=0\}. \end{equation} Where $\beta\ne\beta_0$ we are inside $V_2$, where $\gamma_D(H)=h$, so the numerator equals $\beta^4h$; thus $\Delta$ is smooth. Where $\beta=\beta_0$ the numerator is at least $h/2$. Hence $\Delta>0$ and $h-\gamma_D(H_s)=\beta^4\Delta$. \emph{Fourth-power decomposition.} In a small foliation box choose constant leaf covectors $\vartheta_i$ with $\sum_i\vartheta_i^2$ equal to $\Delta$ at the centre. The forms $\vartheta_i^2$, $(\vartheta_i\pm\vartheta_j)^2$ span the symmetric forms; giving weight $\eta>0$ to each $(\vartheta_i\pm\vartheta_j)^2$ and $1-2\eta(m-1)>0$ to each $\vartheta_i^2$ represents the centre value, and a fixed linear right inverse of the spanning map represents nearby values with smooth positive coefficients $c_{\ell j}$ after shrinking the box. These covectors are $\dd_D\phi_{\ell j}$ for linear leaf coordinates. With finitely many bumps $\psi_\ell$ in such boxes and $S=\sum_\ell\psi_\ell^4>0$, \[ h-\gamma_D(H_s)=\sum_{\ell,j}a_{\ell j}^4(\dd_D\phi_{\ell j})^2,\qquad a_{\ell j}=\beta\psi_\ell S^{-1/4}c_{\ell j}^{1/4}. \] The amplitudes are smooth and vanish on $\pi^{-1}(\overline V_1)$. Apply \cref{lem:addition} successively to these summands, each time to the free output of the previous step. Every plaque lies in a fibre of $\pi$, so plaques over $\{\beta=0\}$ are fixed; the final map is free, has partial metric $h$, and equals $F_A$ over $V_0$. If $A=\emptyset$, the source is closed; apply \cref{lem:hprinciple} to any $f$, shrink the result by a constant so that $h-\gamma_D(H)>0$, and decompose and add as above. If $A=B$ there is nothing to prove. \end{proof} \begin{prop}[A free normal]\label{prop:free-normal} In the setting of \cref{prop:realization} with $X,B$ closed, a monomorphism $(D\oplus\Sym^2D)\oplus\triv1\to X\times\R^Q$ yields a smooth $D$-free map $F$ with $\gamma_D(F)=h$ for any prescribed positive $h$, together with a smooth unit vector field $\nu\perp\im J_D(F)$. \end{prop} \begin{proof} Let $J$ be the restriction to $D\oplus\Sym^2D$; the projection of the last column off $\im J$ is nowhere zero. \Cref{lem:hprinciple} gives a free $F_0$ and a homotopy of monomorphisms from $J$ to $J_D(F_0)$, and \cref{lem:transport}(a), applied to the complementary projections, gives a unit normal for $F_0$. To reach the metric $h$ through free maps, work on $S^1\times X\to S^1\times B$ with the pulled-back distribution. Let $\chi\colon S^1\to[0,1]$ vanish near a closed arc and equal $1$ near a point $\theta_*$, and prescribe $h_\theta=(1-\chi(\theta))\gamma_D(F_0)+\chi(\theta)h$. The constant family $F_0$ realizes $h_\theta$ near the arc, and $J_D(F_0)$ is a global formal extension. By \cref{prop:realization} there is a jointly smooth free family $F_\theta$ with $\gamma_D(F_\theta)=h_\theta$, equal to $F_0$ near the arc. Along an arc from the protected region to $\theta_*$, orthogonal transport of the complements of the partial jets carries the unit normal of $F_0$ to one of $F=F_{\theta_*}$. \end{proof} \subsection{Proof of \cref{thm:B}} Let $e$ and $H$ be as in \cref{prop:hyperplane-bundle}. Finitely many orthonormal local frames of $H$ and a partition $\sum\chi_i^2=1$ give an isometric monomorphism $I_H\colon H\to\triv a$; put $W=I_H(H)$ and $L(u+te)=(I_Hu,t)\in\R^{a+1}$, an isometric monomorphism of $TM$. Fix $C\ge1$ and let $X_H=M\times(S^{2C-1})^a\to M$ carry the coefficient distribution $D_W$ of rank $n-1$, with coefficient metric $h_W$. Then $D_W\oplus\Sym^2D_W$ is the pullback of $W\oplus\Sym^2W\cong H\oplus\Sym^2H\cong\triv{s_n-1}$, so $(D_W\oplus\Sym^2D_W)\oplus\triv1$ embeds in $\triv{s_n+4}$. Since $s_n+4=(n-1)+s_{n-1}+5$, \cref{prop:free-normal} gives a $D_W$-free map $F_H\colon X_H\to\R^{s_n+4}$ with $\gamma_{D_W}(F_H)=h_W$ and a unit free normal $\nu$. Apply \cref{prop:wrapping} to the constant family $(F_H,\nu)$, with $K=0$, and one more Hopf sphere for the line $\R e$. The resulting maps $S_k\colon M\times(S^{2C-1})^{a+1}\to\R^{s_n+5}$ satisfy (a)--(c) of \cref{prop:local-phases} for $L$ and $E=H$, because $D_W\oplus\R X_{a+1}=D_{L(TM)}$ and $h_W+1=h_{L(TM)}$; the margin is even bounded below. Hence \cref{prop:local-phases} gives a phase-regular isometric immersion into an arbitrarily small ball of $\R^{s_n+5}$, and \cref{thm:embedding} with $N=s_n+5$ turns it into an embedding, as in the proof of \cref{thm:A}. \qed % ---- end sec7b-stable.tex ---- % ---- end sec7-stable.tex ---- % ---- begin sec8-noncompact.tex ---- \section{Noncompact manifolds and boundary}\label{sec:noncompact} \subsection{Weighted maps} Let $M$ carry a countable uniform atlas, write $\Cb^j$ for the supremum of chart derivatives through order $j$ and $\Cb^\infty=\bigcap_j\Cb^j$, and let $\beta$ be a closed one-form with all $\Cb^j$ norms finite. For $V\colon M\to\R^N$ put \[ D_iV=\partial_iV+\beta_iV,\qquad \gamma_\beta(V)_{ij}=\ip{D_iV}{D_jV},\qquad B^\beta_V=\Pi\,(D_iD_jV), \] where $\Pi$ projects off $\spanop(D_iV)$ when $\gamma_\beta(V)>0$; since $\beta$ is closed, $D_iD_jV$ is symmetric. Weighted phase regularity and margins are defined with $B^\beta$ in place of $B$. \begin{prop}[Weighted correction and probes]\label{prop:weighted} \begin{enumerate}[label=(\alph*)] \item If $a>0$ satisfies $\dd\log a=\beta$ on an open set, then there \begin{equation}\label{eq:gauge} \dd(aV)=a\,DV,\qquad \gamma(aV)=a^2\gamma_\beta(V),\qquad B_{aV}=a\,B^\beta_V . \end{equation} \item \Cref{lem:increment,lem:iteration} and \cref{prop:repair} hold for $\gamma_\beta,B^\beta$ in place of $\gamma,B$, with constants depending in addition on finitely many $\Cb^j$ norms of $\beta$. \item \Cref{prop:probes} holds for $\gamma_\beta,B^\beta$, for finite lists of compactly supported normal oscillations, with all estimates in global $\Cb$ norms; for prescribed $\Cb^r$-closeness the iteration starts at a stage $k\ge\max(8,2r)$. \end{enumerate} \end{prop} \begin{proof} (a) is the product rule $\dd(aV)=a(\dd V+V\dd\log a)$ applied once and twice; the tangent spaces of $aV$ are spanned by the $D_iV$. (b) On each chart ball choose $a_i>0$ with $\dd\log a_i=\beta$ and $a_i=1$ at the centre; the bounds on $\beta$ give uniform bounds on $a_i^{\pm1}$ and all their derivatives. By (a), $V\mapsto a_iV$ transforms all weighted quantities on the chart into unweighted ones with uniformly comparable norms, ellipticity constants and margins. Alternatively, with $u_i=\ip{D_iV}Z$, $Z=DV(\gamma_\beta^{-1}u)+v$ and $D_iD_jV=\widetilde\Gamma^k_{ij}D_kV+B^\beta_{ij}$, \[ \gamma_\beta(V+Z)=\gamma_\beta(V)+L_{\beta,V}Z+\gamma_\beta(Z),\quad (L_{\beta,V}Z)_{ij}=\partial_iu_j+\partial_ju_i+2(\beta_iu_j+\beta_ju_i)-2\widetilde\Gamma^k_{ij}u_k-2\ip{B^\beta_{ij}}v, \] which differs from the unweighted operator only by terms of order zero in $u$. Hence \cref{lem:modes,lem:increment} hold verbatim. In the iteration we use the true weighted defect, \[ (\delta')^2(H'-S_\ell h_0)=\delta^2(H-S_\ell H)-E-\Cross_\beta(V-S_\ell V,U), \] with $\Cross_\beta(A,U)_{ij}=\ip{D_iA}{D_jU}+\ip{D_jA}{D_iU}$; the bounded coefficients of $\beta$ leave all exponents unchanged, and no commutation of $D$ with smoothing is needed. All constructions are local, so their locally finite sums define global maps. (c) The probe construction has the same even mean, since $\beta$ does not depend on $u$. For solutions with fixed $\gamma_\beta(V)=h$, the identity $\ip{D_iD_jV}{D_lV}=\frac12(\partial_ih_{jl}+\partial_jh_{il}-\partial_lh_{ij})+\beta_ih_{jl}+\beta_jh_{il}-\beta_lh_{ij}$ shows that $\widetilde\Gamma$ depends only on $h$ and $\beta$, so the weighted Gauss formula $B^\beta=D^2V-\widetilde\Gamma DV$ holds and the $u$-derivatives of all terms of first order in $V$ are $O(t^k)=o(\sigma)$. The remaining estimates are those of \cref{prop:probes} in global seminorms. \end{proof} \subsection{Conformal normalization} \begin{prop}\label{prop:normalization} Let $M$ be a parallelizable manifold without boundary and $g$ a smooth metric. There is $a\colon M\to(0,1)$, tending to $0$ off compact sets, such that $h=a^{-2}g$ and $\beta=\dd\log a$ satisfy: $h$ is complete with positive injectivity radius and bounded covariant derivatives of its curvature; $h$ has a global orthonormal coframe with bounded covariant derivatives of all orders; and $\beta$ has bounded covariant derivatives of all orders. \end{prop} \begin{proof} Normalize a global coframe to a $g$-orthonormal one, $\omega$, and write $h=e^{2u}g$, $a=e^{-u}$, $\theta=e^u\omega$. The scaling theorem \cite[Theorem~4.1, Definitions~2.2 and~2.9]{MN2015} controls any countable list of quantities bounded by expressions $e^{-\alpha u(x)}P_x(u,\abs{\nabla_gu},\dots,\abs{\nabla^k_gu})$ with $\alpha>0$, together with their versions on neighbouring annuli of an exhaustion; each is controlled off a compact set and bounded on it by continuity. Curvature quantities are of this form \cite[Example~2.3]{MN2015}, and placing the inverse convexity radius first gives completeness and positive injectivity radius \cite[Theorem~3.8 and the proof of Theorem~1.8]{MN2015}. For the coframe, the conformal formula $\nabla^h_XY=\nabla^g_XY+\dd u(X)Y+\dd u(Y)X-g(X,Y)\operatorname{grad}_gu$ gives $(\nabla^h_i\theta)_j=e^u\bigl((\nabla^g_i\omega)_j-u_j\omega_i+g_{ij}\ip{\dd u}\omega_g\bigr)$, and by induction \[ \abs{(\nabla^h)^r\theta}_h\le e^{-ru}P_{r,x}(j^r_gu)\ (r\ge1),\qquad \abs{(\nabla^h)^r\dd u}_h\le e^{-(r+1)u}Q_{r,x}(j^{r+1}_gu), \] so these quantities are admissible as well. Choosing $u$ above a positive proper exhaustion gives $a<1$ and $a\to0$ at infinity. \end{proof} \subsection{Bounded initial realizations} \begin{prop}\label{prop:bounded-initial} In the situation of \cref{prop:normalization}, let the uniform atlases be built from normal charts of $h$. There are arbitrarily large $T$ and maps $V_0\in\Cb^\infty(M,\R^{s_n+1})$, with respect to the atlas of $h$-radius $T^{-1}$, such that \[ \gamma_\beta(V_0)=T^2h,\qquad 2<\abs{V_0}\sup_k\norm{S_k}_{C^0}+3$, so that $H=v_0+S_k(z_1,\dots,z_n)$ has $3<\abs H0$ be smooth and bounded with $\dd\log a_1=\beta$. For every compact $P\subset(M\times M)\setminus\Delta$, $r\ge2$ and $\epsilon>0$ there is $V'\in\Cb^\infty$ with $\gamma_\beta(V')=\gamma_\beta(V)$ and $\norm{V'-V}_{\Cb^r}<\epsilon$ such that $a_1V'$ separates $P$; for small $\epsilon$ the uniform weighted phase margin persists. \end{lem} \begin{proof} Consider the class of maps $a_1V''$ with $V''\in\Cb^\infty$, $\gamma_\beta(V'')=\gamma_\beta(V)$ and a positive uniform weighted margin. By \eqref{eq:gauge}, weighted probe families (\cref{prop:weighted}(c)) for $V''$ give exact-metric families for $a_1V''$ whose leading second-form and value columns are multiplied by the same positive factor $a_1(x)$; on a compact neighbourhood of the projections of $P$, $a_1$ is bounded above and below with bounded derivatives, and responses away from the supports are bounded by the global errors times $\sup a_1$. By \cref{rem:separation-axioms} the proof of \cref{lem:separation} applies, with closeness order at least $\max(r,3)$. \end{proof} \begin{thm}\label{thm:noncompact} Let $M$ be a parallelizable $n$-manifold without boundary, $n\ge3$, and $g$ any smooth metric. For every $p\in\R^{s_n+1}$ and $R>0$ there is a smooth phase-regular isometric embedding $F\colon(M,g)\to B_R(p)$. If $M$ is noncompact, $F$ can be chosen with $F(x)\ne p$, $F(x)\to p$ as $x$ leaves compact sets, and $F$ proper into $\R^{s_n+1}\setminus\{p\}$. \end{thm} \begin{proof} We may take $p=0$ and $M\ne\emptyset$. Let $a,h,\beta$ be as in \cref{prop:normalization} and $V_0$, $T$, $C_V$ as in \cref{prop:bounded-initial}, with $T$ chosen so that $(C_V+1)/T0$ such that $\cX=\{V\in\cY:\norm{V-V_0}_{\Cb^2}<\epsilon_0\}$ consists of maps with a uniform positive weighted margin and $1<\abs V0$ from $0$ is closed in the compact set $\{a\ge T\eta/(C_V+1)\}$, so $F$ is proper into $\R^{s_n+1}\setminus\{0\}$; a proper injective immersion is an embedding. If $M$ is compact, injectivity suffices. Pairs on different components are included throughout. \end{proof} \subsection{Boundary and stable triviality} \begin{cor}\label{cor:boundary} \Cref{thm:A} holds for parallelizable manifolds with smooth boundary and metrics smooth up to the boundary; the embedding extends isometrically across an exterior collar. \end{cor} \begin{proof} In a collar of $\partial M$ inside the double, extend the frame and the metric locally and patch with a locally finite partition of unity; independence and positivity persist near each boundary point. Choose a locally finite cover $U_i$ and widths $w_i>0$ on which both hold for $-2w_i