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 GG on germs X,YX,Y of TMT M at a point xx of a twice-differentiable manifold MM is C 2C^2-linear in one argument and symmetric, then it descends to a bilinear form on the tangent space T xMT_x M.


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

X(Yφ). X (Y \varphi).

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

X(Yφ)=Y(Xφ) X (Y \varphi) = Y (X \varphi)

so that for a real function germ ff,

X((fY)φ)=f(X(Yφ)) X ((f\cdot Y)\varphi) = f \cdot (X (Y \varphi))

and hence the symmetric bilinear form

Hesse φ,x(X,Y)=X(Yφ) Hesse_{\varphi,x} (X,Y) = X(Y \varphi)

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

When MM 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 φ,(X,Y)=X(Yφ)( XY)φ 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.

Revised on July 18, 2012 10:40:49 by Toby Bartels (