Flat connection

From Diffgeom
Revision as of 17:57, 7 March 2007 by Vipul (talk | contribs)
(diff) ← Older revision | Latest revision (diff) | Newer revision → (diff)
Jump to: navigation, search

Template:Connection property

Definition

Symbol-free definition

A connection on 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 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