Mapping Among Line Elements

TL;DR

This paper develops mappings between generalized quadratic Hamiltonians and constructs Calabi's Riemannian line elements for non-quadratic Hamiltonians, providing exact solutions for manifold mappings and Riccati equations.

Contribution

It introduces a novel approach to mapping among manifolds using generalized Hamiltonians and constructs new Riemannian line elements for complex Hamiltonian systems.

Findings

Derived analytical solutions for manifold mappings.

Obtained infinite solutions for matrix Riccati equations.

Extended Calabi's Riemannian line elements to non-quadratic Hamiltonians.

Abstract

In this paper, we revisit our paper "Matrix Riccati Equations, Kaluza-Klein, Finsler Spaces, and Mapping Among Manifolds"[1]. We will build mapping among generalized quadratic Hamiltonians and we construct Calabi's Riemmannian Line Elements for non-quadratic and generalized Hamiltonians. As an application, we use conformally flat forms of two general pseudo-Riemannian line elements embedded in two flat manifolds and obtain an analytical and exact solution of the mapping between these two manifolds as well as an infinite set of exact solutions of the associated matrix Riccati equation.