Euclidean formal immersions are homotopy equivalent to Stiefel-bundle sections

Use the supplied smooth tangent-bundle structures of [F12] and the supplied metric; [F10] supplies one under ${\mathrm{A}\mathrm{C}}_{\omega }$ if needed. In any local tangent frame the condition that a matrix have rank $m$ is open, so [F1] and [F4] give the smooth monomorphism bundle $\mathcal{M}$. Orthonormal tangent frames give the associated bundle with fibre $V_m({\mathbb{R}}^n)$ in [F6] and [F7], which is exactly its isometric-injection subbundle $E$. The map $B\mapsto (x\mapsto B_x)$ is a bijection onto $\mathrm{\Gamma }(\mathcal{M})$ by [F3]. It is a homeomorphism: section coordinates are precisely the matrix coefficients on each compact base chart piece. A compact piece of $TM$ has compact base projection and bounded vector coordinates, so controlling coefficient jets there controls every jet of the fibre-linear total-space map. Conversely evaluating that map along the finitely many local basis-vector sections over a compact base piece recovers each coefficient and all its derivatives. These two estimates show that the weak total-space and coefficient topologies agree with the section topology in [F2].

For an injective $B$ put $R=(B^{\ast }B{)}^{1\mathrm{/}2}$ and $Q=B{R}^{-1}$. By [F11] these operations are smooth, and $Q^{\ast }Q=I$. Conversely $(Q,R)$ with $Q$ isometric and $R$ positive gives $B=QR$, uniquely, so this is the asserted local polar diffeomorphism. Under a change of orthonormal tangent frame $U\in O(m)$, $B$ changes to $BU$, $R$ to $U^{\ast }RU$, and $Q$ to $QU$; hence it descends globally without a global frame. The path $B_t=B((1-t)I+t{R}^{-1})$ remains injective because its second factor is positive definite, has $B_0=B$ and $B_1=Q$, and fixes every isometric $B$. The operations and this path are continuous for weak $C^{\mathrm{\infty }}$ section topologies: on finitely many compact chart pieces the positive spectra stay bounded away from zero, and the smooth matrix operations and all chain-rule derivatives vary continuously there. Thus it is an equivariant strong deformation retraction of section spaces. Rank zero gives the unique empty matrix and the same formulas.

The canonical identification $T{\mathbb{R}}^n\cong {\mathbb{R}}^n\times {\mathbb{R}}^n$ sends a formal immersion $(f,F)$ to $(f,B)$ with $B_x:T_{x}M\to {\mathbb{R}}^n$ the same fibre map. This is a bijection onto $C^{\mathrm{\infty }}(M,{\mathbb{R}}^n)\times \mathrm{\Gamma }(\mathcal{M})$ and a homeomorphism for the identical local matrix/derivative neighbourhoods, as in step 1.1. The contraction $(f,B)\mapsto ((1-t)f,B)$ deforms the first factor to zero; combining it with step 2.1 on the second factor gives the homotopy equivalence to $\mathrm{\Gamma }(E)$. Its homotopy inverse sends an isometric section $Q$ to $(0,Q)$. For an immersion the smooth formal pair is $(f,df)$ by [F2] and [F12], so the normalized section is the polar part of $df$, as stated. No compactness of $M$ is needed because every basic neighbourhood controls only finitely many compact chart pieces.

If $TM$ has a smooth global frame by [F9], orthonormalizing it in the supplied metric gives a global orthonormal frame and identifies $E$ with $M\times V_m({\mathbb{R}}^n)$. For $M=S^m$ with its standard metric, [F8] gives $T_x{S}^m=x^{\perp }$; its projection $I-x{x}^{\ast }$ varies smoothly, so the Gauss map to the Grassmannian is smooth in its graph charts. The tautological-plane pullback of [F6] therefore is $T{S}^m$, and its isometric-injection bundle pulls back to $E$. On $S^1$ the standard unit angular field is a global orthonormal frame, giving $E\cong S^1\times V_1({\mathbb{R}}^2)=S^1\times S^1$. When $m=0$ the Stiefel and monomorphism fibres are points, and when $n=m$ the normalized fibre is $O(m)$ while the monomorphism fibre retains its positive-definite polar factor. These are included in the same construction.