Relationship between stochastic flows and connection forms

TL;DR

This paper introduces a novel representation of the Levi-Civita connection using stochastic flows associated with Brownian motion on manifolds, bridging stochastic analysis and differential geometry.

Contribution

It provides a new way to express the Levi-Civita connection through stochastic flow representations, enhancing understanding of geometric structures via stochastic processes.

Findings

New representation of Levi-Civita connection in terms of stochastic flows

Bridging stochastic analysis with differential geometry

Potential applications in geometric analysis and stochastic differential equations

Abstract

In this article I will prove new representation for the Levi-Civita connection in terms of the stochastic flow corresponding to Brownian motion on manifold.