Tangential and normal projections along a Riemannian submanifold

Assume ${\mathrm{A}\mathrm{C}}_{\omega }$. Let $M^n\subseteq {\bar{M}}^m$ be an embedded submanifold of a Riemannian manifold $(\bar{M},\bar{g})$, and equip $M$ with the induced metric. Inside the restricted ambient tangent bundle set

$\nu M:=(TM{)}^{\perp }={\coprod }_{p\in M}\{\xi \in T_p\bar{M}:{\bar{g}}_p(\xi ,u)=0\text{ for every }u\in T_{p}M\}\mathrm{.}$

This is a smooth vector subbundle by Orthogonal complements of subbundles are smooth subbundles, and the ambient metric identifies it with the quotient normal bundle of Normal and conormal bundles of an embedded submanifold by Assuming countable choice, an ambient metric identifies the two normal bundles. The latter supplier assumes ${\mathrm{A}\mathrm{C}}_{\omega }$ in order to construct the smooth restricted ambient tangent bundle; that is the exact choice use inherited here. The fibrewise orthogonal decompositions assemble as

$T\bar{M}{\mathrm{\mid }}_M=TM\oplus \nu M\mathrm{.}$

For a smooth vector field $V$ along $M$, meaning a smooth section of $T\bar{M}{\mathrm{\mid }}_M$, define its tangential component and normal component by the unique decomposition

$V=V^{\mathrm{\top }}+V^{\perp },V^{\mathrm{\top }}\in \mathrm{\Gamma }(TM),V^{\perp }\in \mathrm{\Gamma }(\nu M)\mathrm{.}$

Equivalently, $V^{\mathrm{\top }}={\pi }^{\mathrm{\top }}V$ and $V^{\perp }={\pi }^{\perp }V$, where ${\pi }^{\mathrm{\top }}$ and ${\pi }^{\perp }$ are the two orthogonal bundle projections. They are smooth: in a local smooth orthonormal frame $(e_1,\dots ,e_n,e_{n+1},\dots ,e_m)$ adapted so that the first $n$ vectors span $TM$, one has

${\pi }^{\mathrm{\top }}(V)={\sum }_{i=1}^n\bar{g}(V,e_i)e_i,{\pi }^{\perp }(V)={\sum }_{\alpha =n+1}^m\bar{g}(V,e_{\alpha })e_{\alpha },$

whose coefficient functions are smooth. Empty sums cover zero-dimensional tangent or normal fibres, including the codimension-zero case. These are canonical metric projections and require no choice of a global frame.