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 .
|Date of creation||2013-03-22 13:29:46|
|Last modified on||2013-03-22 13:29:46|
|Last modified by||mathcam (2727)|