Difference between revisions of "Torsion form"
From Encyclopedia of Mathematics
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| − | The covariant differential of the vector-valued | + | The [[covariant differential]] of the vector-valued $1$-form of the displacement of an [[affine connection]], the $2$-form |
| − | + | $$ | |
| − | + | \Omega = D \omega = d \omega + \theta \wedge \omega | |
| − | + | $$ | |
| − | where | + | where $\theta$ is the connection form. The torsion form satisfies the first [[Bianchi identity]]: |
| − | + | $$ | |
| − | + | d \Omega = \theta \wedge \Omega + \omega \wedge \Theta | |
| − | + | $$ | |
| − | where | + | where $\Theta$ is the [[curvature form]] of the given connection. The definition of a torsion form for reductive connections is analogous. |
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====References==== | ====References==== | ||
| − | <table><TR><TD valign="top">[a1]</TD> <TD valign="top"> S. Kobayashi, K. Nomizu, "Foundations of differential geometry" , '''1–2''' , Interscience (1963)</TD></TR></table> | + | <table> |
| + | <TR><TD valign="top">[a1]</TD> <TD valign="top"> S. Kobayashi, K. Nomizu, "Foundations of differential geometry" , '''1–2''' , Interscience (1963)</TD></TR> | ||
| + | </table> | ||
| + | |||
| + | {{TEX|done}} | ||
Latest revision as of 19:51, 6 October 2016
The covariant differential of the vector-valued $1$-form of the displacement of an affine connection, the $2$-form $$ \Omega = D \omega = d \omega + \theta \wedge \omega $$ where $\theta$ is the connection form. The torsion form satisfies the first Bianchi identity: $$ d \Omega = \theta \wedge \Omega + \omega \wedge \Theta $$ where $\Theta$ is the curvature form of the given connection. The definition of a torsion form for reductive connections is analogous.
Comments
References
| [a1] | S. Kobayashi, K. Nomizu, "Foundations of differential geometry" , 1–2 , Interscience (1963) |
How to Cite This Entry:
Torsion form. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Torsion_form&oldid=18526
Torsion form. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Torsion_form&oldid=18526
This article was adapted from an original article by M.I. Voitsekhovskii (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article