In all cases, a Hessian is a symmetric bilinear form on a tangent space, encoding second-order information about a twice-differentiable function. (Compare the differential of a once-differentiable function, which is a 1-form on the tangent space.)
For the purpose of this page, a tangent vector “is” a local derivation on germs of functions. If you don’t like this model, take it that the natural isomorphism is unwritten, but trivial to insert; we do not actually name any derivations, but only germs.
If an -bilinear form on germs of at a point of a twice-differentiable manifold is -linear in one argument and symmetric, then it descends to a bilinear form on the tangent space .
Given a manifold and function , the (ordered) second derivative of , for germs and of tangent fields, is simply
At a critical point , (i.e., ) one checks that
so that for a real function germ ,
and hence the symmetric bilinear form
descends to the tangent space at , and is pronounced as the Hessian of at the critical point .
When carries a torsion-free connection , (e.g., the Levi-Civita connection of a Riemannian structure), one may define a global Hessian, defined for germs by
which again is symmetric, and hence descends to the tangent space. Note, however, this obviously depends on the particular connection used.
Last revised on July 18, 2012 at 10:40:49. See the history of this page for a list of all contributions to it.