Dirac structure: Difference between revisions

Latest revision as of 19:38, 18 May 2008

A Dirac structure on a Riemannian manifold $(M, g)$ is a quadruple $(E, c, \nabla^{E}, \nabla^{M})$ where:

$E$ is a Hermitian vector bundle on $M$
$c : T^{*} (M) \to E n d (E)$ is a self-adjoint Clifford structure on $E$
$\nabla^{M}$ is a metric connection on $M$
$\nabla^{E}$ is a Hermitian connection on $E$ compatible with Clifford multiplication

A geometric Dirac structure is a Dirac structure where the metric connection $\nabla^{M}$ is the Levi-Civita connection.

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+{{addlstructureonrm}}
 ==Definition==
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 * <math>E</math> is a [[Hermitian vector bundle]] on <math>M</math>
-* <math>c</math> is a [[self-adjoint Clifford structure]] on <math>E</math>
+* <math>c: T^*(M) \to End(E)</math> is a [[self-adjoint Clifford structure]] on <math>E</math>
 * <math>\nabla^M</math> is a [[metric connection]] on <math>M</math>
 * <math>\nabla^E</math> is a [[Hermitian connection]] on <math>E</math> compatible with Clifford multiplication
-A [[geometric Dirac structure]] is a Diract structuer where the metric connection <math>\nabla^M</math> is the [[Levi-Civita connection]].
+A [[geometric Dirac structure]] is a Dirac structure where the metric connection <math>\nabla^M</math> is the [[Levi-Civita connection]].