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The Riemann Curvature Tensor is a tensor in Riemannian Geometry that describes the curvature of a manifold. Specifically, it measures the failure (known as the holonomy) of the ability of parallel transport to retain the direction of a vector in a space. In other words, it can be calculated as:
Identities[]
The definition of the Riemann Curvature Tensor immediately implies that:
And therefore;
Now, below are some other identities:
See [1] .
First Bianchi Identity [2] .
Second Bianchi Identity [3] .