Random Lie Brackets that Induce Torsion: A Model for Noisy Vector Fields

TL;DR

This paper introduces a stochastic Lie bracket model that induces torsion in vector fields on manifolds, challenging the assumption of parallel transport in stochastic analysis and providing new tools for noisy geometric applications.

Contribution

It defines a novel stochastic Lie bracket inducing torsion, analyzes its geometric implications, and derives associated stochastic differential equations and curvature tensors.

Findings

Stochastic Lie bracket induces torsion in vector fields.

The model leads to new stochastic geodesic and parallel transport equations.

Derived curvature tensors and Laplace-Beltrami operator for the stochastic setting.

Abstract

We define and study a random Lie bracket that induces torsion in expectation. Almost all stochastic analysis on manifolds have assumed parallel transport. Mathematically this assumption is very reasonable. However, in many applied geometry and graphics problems parallel transport is not achieved, the "change in coordinates" are not exact due to noise. We formulate a stochastic model on a manifold for which parallel transport does not hold and analyze the consequences of this model with respect to classic quantities studied in Riemannian geometry. We first define a stochastic lie bracket that induces a stochastic covariant derivative. We then study the connection implied by the stochastic covariant derivative and note that the stochastic lie bracket induces torsion. We then state the induced stochastic geodesic equations and a stochastic differential equation for parallel transport. We also derive the curvature tensors for our construction and a stochastic Laplace-Beltrami operator. We close with a discussion of the motivation and relevance of our construction.