Curvature tensor

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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:


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.

How to Cite This Entry:
Curvature tensor. Encyclopedia of Mathematics. URL:
This article was adapted from an original article by M.I. Voitsekhovskii (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article