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.

Last revised on July 18, 2012 at 10:40:49. See the history of this page for a list of all contributions to it.