[Paper Review] Hesse manifolds and Hessian symmetries of multifield cosmological models

Calin Iuliu Lazaroiu|arXiv (Cornell University)|Sep 10, 2020

Geometric Analysis and Curvature Flows12 references5 citations

TL;DR

This paper introduces Hesse manifolds—Riemannian manifolds admitting nontrivial Hesse functions, solutions to the Hesse equation—as a geometric framework for classifying hidden (Hessian) symmetries in multifield cosmological models. It establishes that complete Hesse manifolds are hyperbolic in nature, with maximal Hesse index $ n+1 $ if and only if the manifold is isometric to a Poincaré ball, and shows that Hesse functions encode geometric data via distance to characteristic subsets, generalizing hyperbolic geometry.

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 Poincaré 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 Poincaré disk, the hyperbolic punctured disk or a hyperbolic annulus. On a complete Hesse manifold $({\cal M},G)$, the value of any Hesse function $Λ$ can be expressed though the distance from a characteristic subset of ${\cal M}$ determined by $Λ$.

Motivation & Objective

• To develop a geometric framework for classifying Noether symmetries in multifield cosmological models, distinguishing between visible (isometric) and hidden (Hessian) symmetries.
• To define and characterize Hesse manifolds as Riemannian manifolds admitting nontrivial solutions to the Hesse equation.
• To establish the relationship between the Hesse index (dimension of the space of Hesse functions) and the geometry of the manifold, particularly in the complete case.
• To show that complete Hesse manifolds with maximal Hesse index are isometric to the Poincaré ball, via an isomorphism constructed from the Weierstrass map.
• To generalize hyperbolic geometry by showing that all elementary hyperbolic space forms are complete Hesse manifolds, and that maximal local Hesse index implies local hyperbolicity.

Proposed method

• Defining Hesse manifolds as Riemannian manifolds admitting nontrivial Hesse functions, which are solutions to the Hesse equation on the target space of the scalar field map.
• Introducing the Hesse index as the dimension of the space of Hesse functions, equipped with a natural symmetric bilinear pairing.
• Using the Weierstrass map to construct an isomorphism between the space of Hesse functions and $ \mathbb{R}^{1,n} $ when the Hesse index is maximal.
• Establishing that a complete Hesse manifold achieves maximal Hesse index $ n+1 $ if and only if it is isometric to the Poincaré ball.
• Applying the uniformization theorem of hyperbolic geometry to show that any complete Hesse manifold with maximal local Hesse index is locally hyperbolic.
• Deriving geometric expressions for the value and gradient flow of any Hesse function $ \Lambda $ in terms of the distance to a characteristic subset of the manifold.

Experimental results

Research questions

• RQ1 What geometric structure underlies the hidden (Hessian) symmetries in multifield cosmological models, beyond visible isometries of the target space?
• RQ2 How is the space of Hesse functions related to the geometry of the target manifold, particularly in the complete case?
• RQ3 Under what conditions does a Riemannian manifold admit a maximal Hesse index, and what does this imply about its global structure?
• RQ4 Can Hesse manifolds be used to generalize hyperbolic geometry, and if so, how do they relate to elementary hyperbolic space forms?
• RQ5 How can the value and dynamics of a Hesse function be expressed geometrically, in terms of distance functions on the manifold?

Key findings

• All Hesse $ n $-manifolds are non-compact and have a Hesse index bounded above by $ n+1 $, with the bound attained if and only if the manifold is isometric to the Poincaré ball when complete.
• For a complete Hesse manifold, the space of Hesse functions is isomorphic to $ \mathbb{R}^{1,n} $ via the Weierstrass map, establishing a deep link to Lorentzian geometry.
• Any elementary hyperbolic space form is a complete Hesse manifold, and any Hesse manifold with maximal local Hesse index is locally hyperbolic.
• The class of complete Hesse surfaces coincides exactly with the class of elementary hyperbolic surfaces, including the Poincaré disk, the hyperbolic punctured disk, and hyperbolic annuli.
• The value of any Hesse function $ \Lambda $ on a complete Hesse manifold can be expressed as a function of the distance to a specific subset of the manifold determined by $ \Lambda $.
• The gradient flow of a Hesse function $ \Lambda $ is fully characterized by the distance function to its associated characteristic subset, providing a geometric dynamical description.