 Shape operator in a normal direction

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 $⟨\xi ,Y⟩=0$ shows that $A_{\xi }X=-(D_X\xi {)}^{\top }$. In higher codimension there is a different operator for each normal direction.