Torsion tensor
From Manifold Atlas
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{{Authors|Jost Eschenburg}}{{Stub}} | {{Authors|Jost Eschenburg}}{{Stub}} | ||
− | == Definition == | + | ==Definition == |
<wikitex>; | <wikitex>; | ||
Let $M$ be a smooth manifold and $\nabla$ a [[Covariant derivative|covariant derivative]] on $TM$. | Let $M$ be a smooth manifold and $\nabla$ a [[Covariant derivative|covariant derivative]] on $TM$. | ||
Then the expression $T : \Gamma TM \times \Gamma TM \to \Gamma TM$, | Then the expression $T : \Gamma TM \times \Gamma TM \to \Gamma TM$, | ||
− | + | \begin{equation} T(X,Y) = \nabla_XY -\nabla_YX - [X,Y] \end{equation} | |
for all vector fields $X,Y\in \Gamma TM$ is a [[Tensor|tensor]], called ''torsion tensor''. | for all vector fields $X,Y\in \Gamma TM$ is a [[Tensor|tensor]], called ''torsion tensor''. | ||
A covariant derivative $\nabla$ is called ''torsion free'' if $T = 0$. | A covariant derivative $\nabla$ is called ''torsion free'' if $T = 0$. | ||
In terms of coordinates, using a local parametrization $\phi : \R^n_o \to M$, the | In terms of coordinates, using a local parametrization $\phi : \R^n_o \to M$, the | ||
vanishing of the torsion means | vanishing of the torsion means | ||
− | + | \begin{equation} \nabla_i\phi_j = \nabla_j\phi_i \end{equation} | |
for any $i,j = 1,\dots,n$ where $\nabla_i = \nabla_{\phi_i}$. | for any $i,j = 1,\dots,n$ where $\nabla_i = \nabla_{\phi_i}$. | ||
</wikitex> | </wikitex> | ||
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[[Category:Definitions]] | [[Category:Definitions]] | ||
+ | [[Category:Connections and curvature]] |
Latest revision as of 10:52, 15 May 2013
The user responsible for this page is Jost Eschenburg. No other user may edit this page at present. |
This page has not been refereed. The information given here might be incomplete or provisional. |
1 Definition
Let be a smooth manifold and a covariant derivative on . Then the expression ,
(1)
for all vector fields is a tensor, called torsion tensor. A covariant derivative is called torsion free if . In terms of coordinates, using a local parametrization , the vanishing of the torsion means
(2)
for any where .