|
|
|
|
transversality
|
(Definition)
|
|
|
Transversality is a fundamental concept in differential topology. We say that two smooth submanifolds  of a smooth manifold  intersect transversely, if at any point
 , we have
where  denotes the tangent space at  , and we naturally identify  and  with subspaces of  .
In this case,  and  intersect properly in the sense that  is a submanifold of  , and
A useful generalization is obtained if we replace the inclusion
 with a smooth map  . In this case we say that  is transverse to
 , if for each point
 , we have
In this case it turns out, that  is a submanifold of  , and
Note that if  is a single point  , then the condition of  being transverse to  is precisely that  is a regular value for  . The result is that  is a
submanifold of  . A further generalization can be obtained by replacing the inclusion of  by a smooth function as well. We leave the details to the reader.
The importance of transversality is that it's a stable and generic condition. This means, in broad terms that if  is transverse to
 , then small perturbations of  are also transverse to  . Also, given any smooth map  , it can be perturbed slightly to obtain a smooth map which is transverse to a given submanifold
 .
|
"transversality" is owned by mathcam. [ full author list (2) | owner history (1) ]
|
|
(view preamble)
| Also defines: |
transversal, transverse, transversally, transversely |
|
|
Cross-references: small perturbations, terms, generic, stable, regular value, smooth map, inclusion, submanifold, subspaces, tangent space, point, intersect, smooth manifold, smooth submanifolds, topology
There are 5 references to this entry.
This is version 3 of transversality, born on 2003-03-02, modified 2003-07-25.
Object id is 4071, canonical name is Transversality.
Accessed 7856 times total.
Classification:
| AMS MSC: | 57R99 (Manifolds and cell complexes :: Differential topology :: Miscellaneous) |
|
|
|
|
|
|
Pending Errata and Addenda
|
|
|
|
|
|
|
|
|
|
|
