Morse-Darboux lemma for surfaces with boundary

TL;DR

This paper extends the classical Morse-Darboux lemma to surfaces with boundaries, providing a new theoretical framework for understanding the local behavior of functions on such surfaces.

Contribution

It introduces and proves an analog of the Morse-Darboux lemma specifically for surfaces with boundary, filling a gap in differential topology.

Findings

Established a new lemma for surfaces with boundary

Provided a theoretical foundation for boundary-sensitive Morse theory

Enhanced understanding of local surface behavior near boundaries

Abstract

We formulate and prove an analog of the classical Morse-Darboux lemma for the case of a surface with boundary.