nLab Hessian manifold

Definition

Definition

In coordinates, a Hessian manifold is a pseudo-Riemannian manifold $(M,g)$ that admits local coordinates $\{q^a\}$, called special coordinates, that realize the metric as the Hessian of a function $H$, the Hessian potential:

$$g_{a,b} = \partial_{a} \partial_{b} H = H_{ab}$$

such a pseudo-Riemannian metric $g$ is known as a Hessian metric.

The defining relation is only invariant under affine transformations.

Equivalently, in a coordinate-free expression (see (Shima (2013))), a Hessian manifold is a pseudo-Riemannian manifold $(M,g)$ endowed with a torsion-free flat connection $\nabla$, such that the rank-3 tensor

$$S=\nabla g$$

is totally anti-symmetric.

Related concepts

• special geometry

References

General:

• Hirohiko Shima. (2013). “Geometry of Hessian Structures”. In: Nielsen, F., Barbaresco, F. (eds) Geometric Science of Information. GSI 2013. Lecture Notes in Computer Science, vol 8085. Springer, Berlin, Heidelberg. (doi)

• Gabriel Lopes Cardoso, Thomas Mohaupt. Special geometry, Hessian structures and applications. Physics Reports 855, 25 April 2020, Pages 1-141. (doi)