Riemann curvature tensor

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In differential geometry, the Riemann curvature tensor is the most standard way to express curvature of Riemannian manifolds, or more generally, any manifold with an affine connection, torsionless or with torsion. It is one of many things named after Bernhard Riemann. The curvature tensor is given in terms of a Levi-Civita connection (more generally, an affine connection) <math>\nabla</math>(or covariant differentiation) by the following formula:

<math>R(u,v)w=\nabla_u\nabla_v w - \nabla_v \nabla_u w - \nabla_{[u,v]} w .</math>

Here <math>R(u,v)</math> is a linear transformation of the tangent space of the manifold; it is linear in each argument.

NB. Some authors define the curvature tensor with the opposite sign.

If <math>u=\partial/\partial x^i</math> and <math>v=\partial/\partial x^j</math> are coordinate vector fields then <math>[u,v]=0</math> and therefore the formula simplifies to

<math>R(u,v)w=\nabla_u\nabla_v w - \nabla_v \nabla_u w </math>

i.e. the curvature tensor measures noncommutativity of the covariant derivative.

The linear transformation <math>w\mapsto R(u,v)w</math> is also called the curvature transformation or endomorphism.

1 Coordinate expression
2 Symmetries and identities
3 For surfaces
4 See also

[edit] Coordinate expression

In local coordinates <math>x^\mu</math> the Riemann curvature tensor is given by

<math>{R^\rho}_{\sigma\mu\nu} = dx^\rho(R(\partial_{\mu},\partial_{\nu})\partial_{\sigma})</math>

where <math>\partial_{\mu} = \partial/\partial x^{\mu}</math> are the coordinate vector fields. The above expression can be written using Christoffel symbols:

<math>{R^\rho}_{\sigma\mu\nu} = \partial_\mu\Gamma^\rho_{\nu\sigma}

   - \partial_\nu\Gamma^\rho_{\mu\sigma}
   + \Gamma^\rho_{\mu\lambda}\Gamma^\lambda_{\nu\sigma}
   - \Gamma^\rho_{\nu\lambda}\Gamma^\lambda_{\mu\sigma}</math>

[edit] Symmetries and identities

The Riemann curvature tensor has the following symmetries:

<math>\langle R(u,v)w,z \rangle=-\langle R(u,v)z,w \rangle^{}_{}</math>

The last identity was discovered by Ricci, but is often called the first Bianchi identity or algebraic Bianchi identity, because it looks similar to the Bianchi identity below. These three identities form a complete list of symmetries of the curvature tensor, i.e. given any tensor which satisfies the identities above, one can find a Riemannian manifold with such a curvature tensor at some point. Simple calculations show that such a tensor has <math>n^2(n^2-1)/12</math> independent components.

Yet another useful identity follows from these three:

<math>\langle R(u,v)w,z \rangle=\langle R(w,z)u,v \rangle^{}_{}</math>

The Bianchi identity (often the second Bianchi identity or differential Bianchi identity) involves the covariant derivatives:

<math>\nabla_uR(v,w)+\nabla_vR(w,u)+\nabla_w R(u,v) = 0</math>

Given any coordinate chart about some point on the manifold, the above identities may be written in terms of the components of the Riemann tensor at this point as:

<math>R_{a[bcd]}^{}=0</math> (first Bianchi identity)

<math>R_{ab[cd;e]}^{}=0</math> (second Bianchi identity)

where the square brackets denote cyclic symmetrisation over the indices and the semi-colon is a covariant derivative.

[edit] For surfaces

For a two-dimensional surface, the Bianchi identities imply that the Riemann tensor can be expressed as

where <math>g_{ab}</math> is the metric tensor and <math>K</math> is a function called the Gauss curvature. Note that the Gauss curvature coincides with the sectional curvature of the surface. It is also exactly half the scalar curvature of the 2-manifold, while the Ricci curvature tensor of the surface is simply given by

<math>\operatorname{Ric} = Kg . \,\!</math>