2020/09/23 by Lodder, Jerry
#17A32 #53B05 #Differential Geometry (math.DG) #FOS: Mathematics #K-Theory and Homology (math.KT)
paper · doi:10.48550/arxiv.2009.11366
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 \bfRn corresponding to the affine orthogonal Lie algebra, n ≥ 3.