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Riemann Curvature Tensor

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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:

R_{\mu\nu\rho}^\sigma=\mbox{d}x^\sigma\left[\nabla_\mu,\nabla_\nu\right]\partial_\sigma

IdentitiesEdit this section

Correction to Proof of the Zeroth Bianchi Identity08:13

Correction to Proof of the Zeroth Bianchi Identity

More on Bianchi Identities14:57

More on Bianchi Identities

More on Bianchi Identities-014:57

More on Bianchi Identities-0

The definition of the Riemann Curvature Tensor immediately implies that:

 R_{\mu\nu\rho\sigma}=-R_{\nu\mu\rho\sigma}

And therefore;

 R_{(\mu\nu)\rho\sigma}=0

Now, below are some other identities:

R_{\mu\nu(\rho\sigma)} = 0 See [1] .

First Bianchi Identity [2] .

Second Bianchi Identity [3] .


ReferencesEdit this section

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