A generalized transversality in global analysis
Jipu Ma · Pacific Journal of Mathematics · 2008
Let M and N be C r Banach manifolds with r ≥ 1.Let P be a submanifold of N and f : M → N a C r map.This paper extends the well-known transversality f P mod N to the tangent map T x f with a sharper singularity by using a new characteristic of the continuity of generalized inverses of linear operators in Banach spaces under small perturbations.We introduce a concept of generalized transversality, written as f G P mod N. We show that if f P mod N, then f G P mod N, but the converse is false in general.Then Thom's famous result is expanded into a generalized transversality theorem: if f G P mod N, then the preimage S = f -1 ( P) is a submanifold of M with the tangent space T x S = (T x f ) -1 (T f (x) P) for any x ∈ S. As a consequence, when P={ y} is a single point set, f G P mod N if and only if y is a generalized regular value of f .Finally, we give an equivalent geometric description of generalized transversality without the aid of charts.