Hesse manifolds and Hessian symmetries of multifield cosmological models

TL;DR

This paper explores the mathematical structure of hidden symmetries in multifield cosmological models, introducing Hesse manifolds characterized by solutions to the Hesse equation, with implications for the geometry of target spaces.

Contribution

It develops the theory of Hesse manifolds and their symmetries, providing a classification of these spaces and linking them to hyperbolic geometry in cosmological contexts.

Findings

Hesse manifolds are non-compact with a bounded Hesse index.

Complete Hesse manifolds include hyperbolic spaces like the Poincaré disk.

Hesse functions relate to distances from characteristic subsets in the manifold.

Abstract

I give a brief overview of the mathematical theory of Noether symmetries of multifield cosmological models, which decompose naturally into visible and Hessian (a.k.a. 'hidden') symmetries. While visible symmetries correspond to those infinitesimal isometries of the Riemannian target space of the scalar field map which preserve the scalar potential, Hessian symmetries have a much deeper theory. The latter correspond to Hesse functions, defined as solutions of the so-called Hesse equation of the target space. By definition, a Hesse manifold is a Riemannian manifold which admits nontrivial Hesse functions -- not to be confused with a Hessian manifold (the latter being a Riemannian manifold whose metric is locally the Hessian of a function). All Hesse $n$-manifolds ${\cal M}$ are non-compact and characterized by their index, defined as the dimension of the space of Hesse functions, which carries a natural symmetric bilinear pairing. The Hesse index is bounded from above by $n+1$ and, when the metric is complete, this bound is attained iff ${\cal M}$ is a Poincar\'e ball, in which case the space of Hesse functions identifies with $\mathbb{R}^{1,n}$ through an isomorphism constructed from the Weierstrass map. More generally, any elementary hyperbolic space form is a complete Hesse manifold and any Hesse manifold whose local Hesse index is maximal is hyperbolic. In particular, the class of complete Hesse surfaces coincides with that of elementary hyperbolic surfaces and hence any such surface is isometric with the Poincar\'e disk, the hyperbolic punctured disk or a hyperbolic annulus. On a complete Hesse manifold $({\cal M},G)$, the value of any Hesse function $\Lambda$ can be expressed though the distance from a characteristic subset of ${\cal M}$ determined by $\Lambda$.