Morin singularities of coframes and frames

TL;DR

This paper extends the concept of Morin singularities to coframes and frames, analyzing their properties and singularities, and generalizes a known Euler characteristic relation for Morin maps to these new structures.

Contribution

It introduces Morin singularities for coframes and frames, and generalizes a key Euler characteristic congruence to these contexts.

Findings

Defined Morin singularities for coframes and frames

Analyzed singularities of generic 1-forms associated to Morin coframes

Generalized Euler characteristic congruence for Morin structures

Abstract

Inspired by the properties of an $n$-frame of gradients $(\nabla f_1, \ldots, \nabla f_n)$ of a Morin map $f:M\rightarrow\mathbb{R}^n$, with $\dim M\geq n$, 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.