Difference between revisions of "Curvature tensor"
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| − | A tensor of type | + | {{TEX|done}} |
| + | A tensor of type $(1,3)$ obtained by decomposing the [[Curvature form|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 | + | 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|Curvature]]. | For references see [[Curvature|Curvature]]. | ||
Latest revision as of 10:01, 20 September 2014
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.
Curvature tensor. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Curvature_tensor&oldid=16964