Morin singularities of coframes and frames

Abstract

Inspired by the properties of an n-frame of gradients (∇f 1 , . . . , ∇fn) of a Morin map f : M → R n , with dim M ≥ n, we introduce the notion of Morin singularities in the context of singular n-coframes and singular n-frames. We also study the singularities of generic 1-forms associated to a Morin n-coframe, in order to generalize a result of T. Fukuda [4, Theorem 1], which establishes a modulo 2 congruence between the Euler characteristic of a compact manifold M and the Euler characteristics of the singular sets of a Morin map defined on M , to the case of Morin n-coframes and Morin n-frames.