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Gaussian curvature, mean curvature, sectional curvature, scalar curvature, curvature tensors (Riemann, Ricci, Weyl)

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Curvature tensor for a singular manifold

For obtaining the curvature tensor from holonomy it is usually considered that $C$ is an infinitesimal parallelogram. … Or can I formulate the generalized curvature tensor something mor simple? …
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