Torsion is antisymmetric

From Diffgeom
Revision as of 17:51, 6 January 2012 by Vipul (talk | contribs) (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...")
(diff) ← Older revision | Latest revision (diff) | Newer revision → (diff)
Jump to: navigation, search


Let M be a differential manifold and \nabla be a linear connection on M (viz., \nabla is a connection on the tangent bundle TM of M).

Consider the torsion of \nabla, namely:

\tau(\nabla): \Gamma(TM) \times \Gamma(TM) \to \Gamma(TM)

given by:

\tau(\nabla)(X,Y) = \nabla_X Y - \nabla_Y X - [X,Y]

Then, \tau(\nabla) is antisymmetric, i.e.:

\tau(\nabla)(Y,X) = -\tau(\nabla)(X,Y)

Equivalently, \tau is alternating, i.e.:

\tau(\nabla)(X,X) = 0.

Related facts


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