A generalized transversality in global analysis

Jipu Ma

Vol. 236 (2008), No. 2, 357–371

DOI: 10.2140/pjm.2008.236.357

Abstract

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_xf 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 ⋔_GP mod N. We show that if f ⋔ P mod N, then f ⋔_GP mod N, but the converse is false in general. Then Thom’s famous result is expanded into a generalized transversality theorem: if f ⋔_GP mod N, then the preimage S = f⁻¹(P) is a submanifold of M with the tangent space T_xS = (T_xf)⁻¹(T_f(x)P) for any x ∈ S. As a consequence, when P={y} is a single point set, f ⋔_GP 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.

Keywords

transversality, perturbation analysis of generalized inverse, Banach manifold, global analysis

Mathematical Subject Classification 2000

Primary: 46T05, 47A55, 58C15, 58K99

Milestones

Received: 30 August 2007

Accepted: 7 April 2008

Published: 1 June 2008