Curvature matrix of a connection: Difference between revisions

Revision as of 23:59, 11 April 2008

Definition

Suppose $M$ is a differential manifold, $E$ a vector bundle over $M$ , and $\nabla$ a connection over $M$ . Let $p \in M$ and $U ∋ p$ be an open set such that the vector bundle $E$ , restricted to $U$ , is trivial. The curvature matrix of $p$ , denoted $Ω (p)$ is defined by:

$Ω = d ω + ω \land ω$

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 ==Definition==
-Suppose <math>M</math> is a [[differential manifold]], <math>E</math> a [[vector bundle]] over <math>M</math>, and <math>\nabla</math> a connection over <math>M</math>. Let <math>p \in M</math> and <math>U \ni p</math> be an open set such that the vector bundle <math>E</math>, restricted to <math>U</math>, is trivial. The curvature matrix of <math>p</math> is a matrix of
+Suppose <math>M</math> is a [[differential manifold]], <math>E</math> a [[vector bundle]] over <math>M</math>, and <math>\nabla</math> a connection over <math>M</math>. Let <math>p \in M</math> and <math>U \ni p</math> be an open set such that the vector bundle <math>E</math>, restricted to <math>U</math>, is trivial. The curvature matrix of <math>p</math>, denoted <math>\Omega(p)</math> is defined by:
+<math>\Omega = d\omega + \omega \wedge \omega</math>