  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, $n \geq 3$.
