What is the definition of $R_{ijkl}$ in terms of metrics on a manifold?

Question

What is the definition of $R_{ijkl}$ in terms of metrics on a manifold?

I know what the definition of the riemann tensor, $R^l_{ink}$, is. But what exactly is meant by $R_{ijlk}$?

Answer 1

$R_{ijkl}$ is the covariant version of the curvature tensor $$ R_{ijkl} = g_{im} R^m{}_{jkl} $$

If one defines the curvature tensor by $$ R^\ell{}_{ijk}= \frac{\partial}{\partial x^j} \Gamma^\ell{}_{ik}-\frac{\partial}{\partial x^k}\Gamma^\ell{}_{ij} +\Gamma^\ell{}_{js}\Gamma_{ik}^s-\Gamma^\ell{}_{ks}\Gamma^s{}_{ij} $$ lowering indices with $R_{\ell ijk}=g_{\ell s}R^s{}_{ijk}$ one gets $$ R_{ik\ell m}=\frac{1}{2}\left( \frac{\partial^2g_{im}}{\partial x^k \partial x^\ell} + \frac{\partial^2g_{k\ell}}{\partial x^i \partial x^m} - \frac{\partial^2g_{i\ell}}{\partial x^k \partial x^m} - \frac{\partial^2g_{km}}{\partial x^i \partial x^\ell} \right) +g_{np} \left( \Gamma^n{}_{k\ell} \Gamma^p{}_{im} - \Gamma^n{}_{km} \Gamma^p{}_{i\ell} \right). $$ The symmetries of the tensor are

• $R_{ik\ell m}=R_{\ell mik}$
• $R_{ik\ell m}=-R_{ki\ell m}=-R_{ikm\ell}$.

that is, it is symmetric in the exchange of the first and last pair of indices, and antisymmetric in the flipping of a pair.

The cyclic permutation sum (First Bianchi identity) is $$R_{ik\ell m}+R_{imk\ell}+R_{i\ell mk}=0.$$ This is often written $R_{i[jk\ell]}^{}=0$, where the brackets denote the antisymmetric part on the indicated indices.

The Second Bianchi identity is $$R_{ijk\ell;m}^{}+R_{ij\ell m;k}^{}+R_{ijmk;\ell}^{}=0$$ or equivalently, $R_{ij[k\ell;m]}^{}=0$ where the semi-colon denotes a covariant derivative.

Answer 2

It's just the Riemann tensor with an index lowered, i.e. contracted with the metric:

$$ R^{i}{}_{jkl} = g^{im} R_{mjkl}.$$

If the indices are not spaced out like this, it is ambiguous which index is being raised; so it's possible that it could also be the last index (or very rarely one of the inner ones).

Answer 3

The operacion raising and lowering indices is a contracción of the tensorial product of metric tensor $g_ {ij}$ or co-metrico tensor $g^{ij}$, with other arbitrary tensor.

This tensors allow define a isomorphism, call musical isomorphism, between the tangent space $T^1(M)$ and the cotangent space $T_1(M)$.