Curvature tensor

A tensor of type $(1,3)$ obtained by decomposing the curvature form in a local co-basis on a manifold $M^n$. In particular, in a holonomic co-basis $dx^i$, $i=1,\dots,n$, the components of the curvature tensor $R_{lij}^k$ of an affine connection are expressed in terms of the Christoffel symbols of the connection $\Gamma_{ij}^k$ and their derivatives:

$$R_{lij}^k=\partial_i\Gamma_{jl}^k-\partial_j\Gamma_{il}^k+\Gamma_{ip}^k\Gamma_{jl}^p-\Gamma_{jp}^k\Gamma_{il}^p.$$

In similar fashion one defines the curvature tensor for an arbitrary connection on a principal fibre space with structure Lie group $G$ in terms of a decomposition of the appropriate curvature form; this applies, in particular, to conformal and projective connections. It takes values in the Lie algebra of the group $G$ and is an example of a so-called tensor with non-scalar components.

For references see Curvature.