Vipul: Created page with "==Statement== Let $M$ be a differential manifold and $\nabla$ be a fact about::linear connection on $M$ (viz., $\nabla$ is..."

2012-01-06T17:51:04Z

Created page with "==Statement== Let <math>M</math> be a differential manifold and <math>\nabla</math> be a fact about::linear connection on <math>M</math> (viz., <math>\nabla</math> is..."

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==Statement==

Let <math>M</math> be a [[differential manifold]] and <math>\nabla</math> be a [[fact about::linear connection]] on <math>M</math> (viz., <math>\nabla</math> is a [[connection]] on the [[tangent bundle]] <math>TM</math> of <math>M</math>).

Consider the [[fact about::torsion of a linear connection|torsion]] of <math>\nabla</math>, namely:

<math>\tau(\nabla): \Gamma(TM) \times \Gamma(TM) \to \Gamma(TM)</math>

given by:

<math>\tau(\nabla)(X,Y) = \nabla_X Y - \nabla_Y X - [X,Y]</math>

Then, <math>\tau(\nabla)</math> is antisymmetric, i.e.:

<math>\tau(\nabla)(Y,X) = -\tau(\nabla)(X,Y)</math>

Equivalently, <math>\tau</math> is alternating, i.e.:

<math>\tau(\nabla)(X,X) = 0</math>.

==Related facts==

* [[Torsion is tensorial]]
* [[Curvature is tensorial]]
* [[Curvature is antisymmetric in first two variables]]

==Proof==

The proof follows directly from the definition, and the fact that the [[Lie bracket of derivations]] is antisymmetric.

Torsion is antisymmetric - Revision history

Vipul: Created page with "==Statement== Let M be a differential manifold and \nabla be a fact about::linear connection on M (viz., \nabla is..."

Vipul: Created page with "==Statement== Let $M$ be a differential manifold and $\nabla$ be a fact about::linear connection on $M$ (viz., $\nabla$ is..."