Flat connection: Difference between revisions

Latest revision as of 22:01, 24 July 2011

Definition

Symbol-free definition

A connection on a vector bundle over a differential manifold is said to be flat or integrable or curvature-free or locally flat if the curvature of the connection is zero everywhere.

Definition with symbols

A connection $\nabla$ on a vector bundle $E$ over a differential manifold $M$ is said to be flat or integrable or curvature-free or locally flat if the curvature form vanishes identically, viz for any vector fields $X$ and $Y$ :

$R (X, Y) = \nabla_{X} \nabla_{Y} - \nabla_{Y} \nabla_{X} - \nabla_{[X, Y]} = 0$

Definition in local coordinates

In local coordinates, we require that the curvature matrix should vanish identically; in other words:

$Ω : = d ω + ω \land ω = 0$

where $ω$ is the matrix of connection forms.

Alternative definitions

Further information: Flat connection equals module structure over differential operators

Recall that one alternative view of a connection is as giving the space of sections $Γ (E)$ the structure of a module over the connection algebra of $M$ . Equivalently, it is a way of giving the sheaf of sections $E$ the structure of a sheaf-theoretic module over the sheaf of connection algebras.

The connection is flat if and only if this descends to a module structure over the sheaf of differential operators. In other words, a flat connection is equivalent to a structure of $E$ as a module over the sheaf of differential operators.

@@ Line 5: / Line 5: @@
 ===Symbol-free definition===
-A [[connection]] on a [[vector bundle]] over a [[differential manifold]] is said to be '''flat''' or '''integrable''' or '''curvature-free''' or '''locally flat''' if the curvature of the connection is zero everywhere.
+A [[defining ingredient::connection]] on a [[defining ingredient::vector bundle]] over a [[defining ingredient::differential manifold]] is said to be '''flat''' or '''integrable''' or '''curvature-free''' or '''locally flat''' if the [[defining ingredient::curvature of a connection|curvature of the connection]] is zero everywhere.
 ===Definition with symbols===
-A [[connection]] <math>\nabla</math> on a [[differential manifold]] <math>M</math> is said to be '''flat''' or '''integrable''' or '''curvature-free''' or '''locally flat''' if the curvature form vanishes identically, viz for any vector fields <math>X</math> and <math>Y</math>:
+A [[connection]] <math>\nabla</math> on a [[vector bundle]] <math>E</math> over a [[differential manifold]] <math>M</math> is said to be '''flat''' or '''integrable''' or '''curvature-free''' or '''locally flat''' if the curvature form vanishes identically, viz for any vector fields <math>X</math> and <math>Y</math>:
 <math>R(X,Y) = \nabla_X\nabla_Y - \nabla_Y\nabla_X - \nabla_{[X,Y]} = 0</math>
@@ Line 15: / Line 15: @@
 ===Definition in local coordinates===
-In local coordinates, we require that the [[curvature matrix of a connection|curvature matrix]] should vanish identically; in other words:
+In local coordinates, we require that the [[defining ingredient::curvature matrix of a connection|curvature matrix]] should vanish identically; in other words:
 <math>\Omega := d\omega + \omega \wedge \omega = 0</math>
-where <math>\omega</math> is the [[matrix of connection forms]].
+where <math>\omega</math> is the [[defining ingredient::matrix of connection forms]].
+===Alternative definitions===
+{{further|[[Flat connection equals module structure over differential operators]]}}
+Recall that one alternative view of a connection is as giving the space of sections <math>\Gamma(E)</math> the structure of a module over the [[connection algebra]] of <math>M</math>. Equivalently, it is a way of giving the ''sheaf'' of sections <math>\mathcal{E}</math> the structure of a sheaf-theoretic module over the [[sheaf of connection algebras]].
+The connection is flat if and only if this descends to a module structure over the [[sheaf of differential operators]]. In other words, a flat connection is equivalent to a structure of <math>\mathcal{E}</math> as a module over the [[sheaf of differential operators]].