Leibniz Cohomology and Connections on Differentiable Manifolds

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, $n \geq 3$.