HESSE MANIFOLDS AND HESSIAN SYMMETRIES OF MULTIFIELD COSMOLOGICAL MODELS

Abstract

© 2021, Publishing House of the Romanian Academy. All rights reserved.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 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 M is a Poincaré ball, in which case the space of Hesse functions identifies with ℝ1,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 Poincaré disk, the hyperbolic punctured disk or a hyperbolic annulus. Thus Hesse manifolds generalize hyperbolic manifolds. On a complete Hesse manifold (M, G), the value of any Hesse function Λ can be expressed though the distance from a characteristic subset of M determined by Λ. Moreover, the gradient flow of Λ can be described using the distance function.