Transversality is a fundamental concept in differential topology. We say that two smooth submanifolds A,B of a smooth manifoldMathworldPlanetmath M intersect transversely, if at any point xAB, we have


where Tx denotes the tangent spaceMathworldPlanetmath at x, and we naturally identify TxA and TxB with subspacesMathworldPlanetmathPlanetmath of TxX.

In this case, A and B intersect properly in the sense that AB is a submanifold of M, and


A useful generalization is obtained if we replace the inclusion AM with a smooth map f:AM. In this case we say that f is transverse to BM, if for each point af-1(B), we have


In this case it turns out, that f-1(B) is a submanifold of A, and


Note that if B is a single point b, then the condition of f being transverse to B is precisely that b is a regular value for f. The result is that f-1(b) is a submanifold of A. A further generalization can be obtained by replacing the inclusion of B by a smooth function as well. We leave the details to the reader.

The importance of transversality is that it’s a stable and genericPlanetmathPlanetmath condition. This means, in broad terms that if f:AM is transverse to BM, then small perturbations of f are also transverse to B. Also, given any smooth map AM, it can be perturbed slightly to obtain a smooth map which is transverse to a given submanifold BM.

Title transversality
Canonical name Transversality
Date of creation 2013-03-22 13:29:46
Last modified on 2013-03-22 13:29:46
Owner mathcam (2727)
Last modified by mathcam (2727)
Numerical id 6
Author mathcam (2727)
Entry type Definition
Classification msc 57R99
Defines transversal
Defines transverse
Defines transversally
Defines transversely