nLab Hessian

Idea

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.

Trivial preliminary lemma

If an $\mathbb{R}$-bilinear form $G$ on germs $X,Y$ of $T M$ at a point $x$ of a twice-differentiable manifold $M$ is $C^2$-linear in one argument and symmetric, then it descends to a bilinear form on the tangent space $T_x M$.

Definitions

Given a $C^2$ manifold $M$ and $C^2$ function $\varphi:M\to \mathbb{R}$, the (ordered) second derivative of $\varphi$, for germs $X$ and $Y$ of tangent fields, is simply

$$X (Y \varphi).$$

At a critical point $x$, (i.e., $(d\varphi)_x = 0$) one checks that

$$X (Y \varphi) = Y (X \varphi)$$

so that for a real function germ $f$,

$$X ((f\cdot Y)\varphi) = f \cdot (X (Y \varphi))$$

and hence the symmetric bilinear form

$$Hesse_{\varphi,x} (X,Y) = X(Y \varphi)$$

descends to the tangent space at $x$, and is pronounced as the Hessian of $\varphi$ at the critical point $x$.

When $M$ carries a torsion-free connection $\nabla$, (e.g., the Levi-Civita connection of a Riemannian structure), one may define a global Hessian, defined for germs by

$$Hesse_{\varphi,\nabla} (X,Y) = X(Y \varphi) - (\nabla_X Y) \varphi$$

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