Torsion tensor

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1 Definition

Let M/var/www/vhost/map.mpim-bonn.mpg.de/tmp/AppWikiTex/tex_JTwzuq be a smooth manifold and \nabla a covariant derivative on TM. Then the expression T : \Gamma TM \times \Gamma TM \to \Gamma TM,

(1)T(X,Y) = \nabla_XY -\nabla_YX - [X,Y]

for all vector fields X,Y\in \Gamma TM is a tensor, called torsion tensor. 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 vanishing of the torsion means

(2)\nabla_i\phi_j = \nabla_j\phi_i

for any i,j = 1,\dots,n where \nabla_i = \nabla_{\phi_i}.

2 References

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