Leibniz Cohomology and Connections on Differentiable Manifolds

TL;DR

This paper explores the relationship between affine connections on Riemannian manifolds and Leibniz cohomology, revealing how geometric structures relate to cohomological properties and eigenfunctions of the Laplacian.

Contribution

It introduces a novel interpretation of affine connections as Leibniz cochains and computes Leibniz cohomology for specific vector fields on Euclidean space.

Findings

Leibniz coboundary of Levi-Civita connection expressed via Laplace-Beltrami and Ricci curvature.

Vanishing coboundary linked to eigenfunctions of the Laplacian.

Computed Leibniz cohomology for vector fields related to affine orthogonal Lie algebra.

Abstract

We show how an affine connection on a Riemannian manifold occurs naturally as a cochain in the complex for Leibniz cohomology of vector fields with coefficients in the adjoint representation. The Leibniz coboundary of the Levi-Civita connection can be expressed as a sum of two terms, one the Laplace-Beltrami operator and the other a Ricci curvature term. The vanishing of this coboundary has an interpretation in terms of eigenfunctions of the Laplacian. Additionally, we compute the Leibniz cohomology with adjoint coefficients for a certain family of vector fields on Euclidean ${\bf{R}}^n$ corresponding to the affine orthogonal Lie algebra, $n \geq 3$.