Gauss frame map of an immersion into Euclidean space

Supply the canonical smooth tangent-bundle structures and smooth global differential, established under ${\mathrm{A}\mathrm{C}}_{\omega }$ by Assuming countable choice, the tangent bundle has a canonical smooth 2n-manifold structure and Assuming countable choice, the global differential of a smooth map is smooth. Once these are supplied, the following constructions use no further choice.

Let $f:M^m\to {\mathbb{R}}^n$ be a smooth immersion with $n\ge m$. Canonically trivializing the target tangent bundle makes its differential a smooth section $x⟼d{f}_x\text{of}\mathcal{M}=\mathrm{Mono}(TM,{\epsilon }^n)⟶M,$ whose fibre consists of injective linear maps $T_{x}M\to {\mathbb{R}}^n$, with the subspace smooth structure in the space of linear maps. This is the unnormalized Gauss data of $f$.

For a smooth local tangent frame $(s_1,\dots ,s_m)$, the local Gauss frame map is the full-rank matrix $g_U(x)=(d{f}_x{s}_1(x)\mid \cdots \mid d{f}_x{s}_m(x))\mathrm{.}$ A change of frame by $A(x)\in {\mathrm{G}\mathrm{L}}_m(\mathbb{R})$ changes this matrix to $g_U(x)A(x)$. These frame changes define the monomorphism bundle, and the smoothness of $df$ gives a smooth section in every such trivialization.

If a smooth tangent metric is supplied, write $E=V(TM,{\epsilon }^n)$ for the separate bundle of isometric linear injections into the Euclidean target. Its fibre in an orthonormal tangent frame is the Stiefel manifold $V_m({\mathbb{R}}^n)$. The normalized Gauss section is the fibrewise polar part $Q_f=df(d{f}^{\ast }df{)}^{-1\mathrm{/}2}\in \mathrm{\Gamma }(E)\mathrm{.}$ Positivity of $d{f}^{\ast }df$, smoothness of its positive square root, and independence of orthonormal tangent frames are proved in Euclidean formal immersions are homotopy equivalent to Stiefel-bundle sections ↗. For an orthogonal change $U$ of tangent frame the normalized matrix changes to $Q_{f}U$. Ordinary Gram–Schmidt in an arbitrary frame does not supply this equivariant formula; polar normalization is the convention here.

A global tangent frame identifies $\mathcal{M}$ with the product bundle with fibre $\mathrm{Mono}({\mathbb{R}}^m,{\mathbb{R}}^n)$. Orthonormalizing that frame in the supplied metric identifies $E$ with the product bundle with fibre $V_m({\mathbb{R}}^n)$, so the normalized section becomes a global map into that Stiefel manifold. Without a global frame it remains a section of $E$. Polar decomposition retains a positive-definite factor in the monomorphism fibre, and the cited proposition proves the resulting deformation retraction onto the Stiefel fibre. For $m=0$ both fibres are points; for $n=m$ they are respectively ${\mathrm{G}\mathrm{L}}_m(\mathbb{R})$ and $O(m)$.

No tangent metric is part of the unnormalized datum; normalization uses the supplied metric. No orientation, properness or normal framing is required. The normal bundle is the quotient ${\epsilon }^n\mathrm{/}df(TM)$ and is not part of these Gauss data. The model assertions in this definition are justified by the cited proposition, whose proof uses the raw full-rank matrix and frame-change definitions without assuming those assertions.