Shape operator in a normal direction
= Shape operator in a normal direction
{title2=$\langle A_\xi X,Y\rangle=\langle II(X,Y),\xi\rangle$}
For a normal vector $\xi$, this identity defines the corresponding <shape operator> using the vector-valued <second fundamental form>. It is self-adjoint because of the <symmetry of the second fundamental form>. Differentiating $\langle\xi,Y\rangle=0$ shows that $A_\xi X=-(D_X\xi)^\top$. In higher codimension there is a different operator for each normal direction.