Hessian Matrix - Generalizations To Riemannian Manifolds

Generalizations To Riemannian Manifolds

Let be a Riemannian manifold and its Levi-Civita connection. Let be a smooth function. We may define the Hessian tensor

by ,

where we have taken advantage of the first covariant derivative of a function being the same as ordinary derivative. Choosing local coordinates we obtain the local expression for the Hessian as

where are the Christoffel symbols of the connection. Other equivalent forms for the Hessian are given by

and .