Torsion tensor

Revision as of 13:17, 15 March 2013

1 Definition

Let ${{Authors|Jost Eschenburg}}{{Stub}} == Definition == <wikitex>; Let $M$ be a smooth manifold and $\nabla$ a [[Covariant derivative|covariant derivative]] on $TM$. Then the expression $T : \Gamma TM \times \Gamma TM \to \Gamma TM$, {{equation|$T(X,Y) = \nabla_XY -\nabla_YX - [X,Y]$|1}} 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 $T = 0$. In terms of coordinates, using a local parametrization $\phi : \R^n_o \to M$, the vanishing of the torsion means {{equation|$\nabla_i\phi_j = \nabla_j\phi_i$|2}} for any $i,j = 1,\dots,n$ where $\nabla_i = \nabla_{\phi_i}$. </wikitex> == References == {{#RefList:}} [[Category:Definitions]]M$ be a smooth manifold and $\nabla$ a covariant derivative on $TM$. Then the expression $T : \Gamma TM \times \Gamma TM \to \Gamma TM$,

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

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

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

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

2 References