integral manifold

In the following we will C when we say smooth.


Let M be a smooth manifoldMathworldPlanetmath of dimension m and let Δ be a distribution of dimension n on M. Suppose that N is a connected submanifoldMathworldPlanetmath of M such that for every xN we have that Tx(N) (the tangent space of N at x) is contained in Δx (the distribution at x). We can abbreviate this by saying that T(N)Δ. We then say that N is an integral manifold of Δ.

Do note that N could be of lower dimension then Δ and is not required to be a regular submanifold of M.


We say that a distribution Δ of dimension n on M is completely integrable if for each point xM there exists an integral manifold N of Δ passing through x such that the dimension of N is equal to the dimension of Δ.

An example of an integral manifold is the integral curve of a non-vanishing vector field and then of course the span of the vector field is a completely integrable distribution.


  • 1 William M. Boothby. , Academic Press, San Diego, California, 2003.
Title integral manifold
Canonical name IntegralManifold
Date of creation 2013-03-22 14:52:00
Last modified on 2013-03-22 14:52:00
Owner jirka (4157)
Last modified by jirka (4157)
Numerical id 6
Author jirka (4157)
Entry type Definition
Classification msc 53B25
Classification msc 52-00
Classification msc 37C10
Related topic FrobeniussTheorem
Defines completely integrable
Defines completely integrable distribution