commuting vector fields

Vector fieldsMathworldPlanetmath X, Y on a manifold are commuting at pM if


where [,] is the Lie bracket.

If S is a subset of M, then we say that vector fields X and Y commute on S if they commute at every pont of S. In the case where S=M, i.e. when the vector fields commute at every point of the manifold, then we simply say that X and Y are commute.

A set V of vector fields on a manifold is said to be commuting on a set S if, for every pair of vector fields AV and BV, it is the case that A and B commute.

If S is an open set and V is a set of commuting vector fields on S, then the cardinality of V is not greater than the dimensionPlanetmathPlanetmathPlanetmath of the manifold and one can find a local coordinate system about any point of S for which these vector fields are coordinate vector fields.

Title commuting vector fields
Canonical name CommutingVectorFields
Date of creation 2013-03-22 15:22:37
Last modified on 2013-03-22 15:22:37
Owner matte (1858)
Last modified by matte (1858)
Numerical id 6
Author matte (1858)
Entry type Definition
Classification msc 53-00