Shape operator

Assume ${\mathrm{A}\mathrm{C}}_{\omega }$. For a normal field $\nu \in \mathrm{\Gamma }(\nu M)$ along an embedded Riemannian submanifold and a tangent field $X\in \mathrm{\Gamma }(TM)$, define the shape operator in the normal direction $\nu$ by

$S_{\nu }X:=-({\bar{\mathrm{\nabla }}}_X\nu {)}^{\mathrm{\top }}\mathrm{.}$

The ambient derivative of a normal field along $M$ is well defined by the local calculation recorded in Normal connection, and the tangent projection is the smooth projection of Tangential and normal projections along a Riemannian submanifold. The minus sign is part of the convention.

The value at $p$ depends only on $X_p$ and ${\nu }_p$. Function-linearity in the direction gives $S_{\nu }(fX)=f{S}_{\nu }X$, while the section Leibniz rule gives

$S_{f\nu }X=-(X(f)\nu +f{\bar{\mathrm{\nabla }}}_X\nu {)}^{\mathrm{\top }}=f{S}_{\nu }X,$

because $X(f)\nu$ is normal. Real linearity follows from the same connection laws. Consequently the assignment is a smooth fibrewise bilinear map

$\nu M{\times }_{M}TM⟶TM,({\nu }_p,X_p)⟼S_{{\nu }_p}{X}_p,$

or equivalently a smooth bundle map $\nu M\to \mathrm{End}(TM)$. Its self-adjointness is proved in the next theorem and is not assumed here.

The hypothesis ${\mathrm{A}\mathrm{C}}_{\omega }$ is inherited exactly through the smooth normal-bundle and projection construction; the formula introduces no further choice. If $M$ is empty, if $TM$ has rank zero, or if $\nu M$ has rank zero, the bilinear map is uniquely zero. Rank-one and boundary cases use the same formula, and degenerate metrics are outside the Riemannian hypothesis.