Geometric Hamiltonian matrix on the analogy between geodesic equation and Schr\"{o}dinger equation

TL;DR

This paper establishes a geometric Hamiltonian matrix on Riemannian manifolds by comparing geodesic and Schrödinger equations, linking eigenvalues to scalar curvature, and providing a novel geometric perspective.

Contribution

It introduces a new geometric Hamiltonian matrix derived from the geodesic equation and explores its eigenvalue problem on Riemannian manifolds, connecting geometry with quantum mechanics.

Findings

Derived the geometric Hamiltonian matrix from geodesic and Schrödinger equations.

Linked eigenvalues of the Hamiltonian to scalar curvature.

Proposed a geometric Hamiltonian function related solely to scalar curvature.

Abstract

By formally comparing the geodesic equation with the Schr\"{o}dinger equation on Riemannian manifold, we come up with the geometric Hamiltonian matrix on Riemannian manifold based on the geospin matrix, and we discuss its eigenvalue equation as well. Meanwhile, we get the geometric Hamiltonian function only related to the scalar curvature.