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
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 .
which again is symmetric, and hence descends to the tangent space. Note, however, this obviously depends on the particular connection used.