Torsion tensor

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1 Definition

Let $M$${{Authors|Jost Eschenburg}}{{Stub}} ==Definition == <wikitex>; Let $M$ be a smooth manifold and $\nabla$ a [[Covariant derivative|covariant derivative]] on $TM$. Then the expression $T : \Gamma TM \times \Gamma TM \to \Gamma TM$, \begin{equation} T(X,Y) = \nabla_XY -\nabla_YX - [X,Y] \end{equation} for all vector fields $X,Y\in \Gamma TM$ is a [[Tensor|tensor]], called ''torsion tensor''. A covariant derivative $\nabla$ is called ''torsion free'' if $T = 0$. In terms of coordinates, using a local parametrization $\phi : \R^n_o \to M$, the vanishing of the torsion means \begin{equation} \nabla_i\phi_j = \nabla_j\phi_i \end{equation} for any $i,j = 1,\dots,n$ where $\nabla_i = \nabla_{\phi_i}$. </wikitex> == References == {{#RefList:}} [[Category:Definitions]] [[Category:Connections and curvature]]M$ be a smooth manifold and $\nabla$$\nabla$ a covariant derivative on $TM$$TM$. Then the expression $T : \Gamma TM \times \Gamma TM \to \Gamma TM$$T : \Gamma TM \times \Gamma TM \to \Gamma TM$,

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

for all vector fields $X,Y\in \Gamma TM$$X,Y\in \Gamma TM$ is a tensor, called torsion tensor. A covariant derivative $\nabla$$\nabla$ is called torsion free if $T = 0$$T = 0$. In terms of coordinates, using a local parametrization $\phi : \R^n_o \to M$$\phi : \R^n_o \to M$, the vanishing of the torsion means

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

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