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. Separately, we compute the Leibniz cohomology with adjoint coefficients for a certain family of vector fields on Euclidean ${\mathbf{R}}^n$ corresponding to the affine orthogonal Lie algebra, $n\ge 3$.

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可微流形上的莱布尼茨上同调和联系

我们展示了黎曼流形上的仿射连接如何自然地作为协链在莱布尼茨上同调的向量场中以伴随表示中的系数出现。Levi-Civita 连接的莱布尼茨共边界可以表示为两项之和，一项是拉普拉斯-贝尔特拉米算子，另一项是 Ricci 曲率项。这个共同边界的消失可以用拉普拉斯算子的本征函数来解释。另外，我们使用欧几里得上特定向量场族的伴随系数计算莱布尼茨上同调${\mathbf{电阻}}^n$ 对应于仿射正交李代数， $n\ge 3$.