Weyl tensor
The curvature tensor is the most standard way to express curvature of Riemannian manifolds. The curvature tensor is given in terms of a Levi-Civita 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.
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 anticommutativity of the covariant derivative.
The linear transormation <math>w\mapsto R(u,v)w<math> is also called the curvature transformation.
Symmetries and identities
The curvature tensor has the following symmetries:
- <math>R(u,v)=-R(v,u)^{}_{}<math>
- <math>\langle R(u,v)w,z \rangle=-\langle R(u,v)z,w \rangle^{}_{}<math>
- <math>R(u,v)w+R(v,w)u+R(w,u)v=0 ^{}_{}<math>
The last identity was discovered by Ricci, but is often called the first 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)
involves the covariant derivatives:
- <math>\nabla_uR(v,w)+\nabla_vR(w,u)+\nabla_w R(u,v) = 0<math>
See also:
Curvature, Curvature of Riemannian manifolds
