Extending a Morse function to a non-orientable $3-$manifold

TL;DR

This paper investigates conditions under which a smooth, non-singular function defined on the boundary collaring of a solid Klein bottle can be extended to the entire non-orientable 3-manifold without introducing critical points, using Reeb graph analysis.

Contribution

It provides a necessary and sufficient condition based on the Reeb graph for extending a non-singular function to a non-orientable 3-manifold.

Findings

Reeb graph characterizes extendability of functions

Necessary and sufficient condition established

Extension without critical points is characterized

Abstract

Considering a solid 3-dimensional Klein bottle and a collaring of its boundary, can we extend a generic $C^\infty$ non-singular function defined on the collaring to the full solid Klein bottle without critical points? We give a condition on the Reeb graph of the given function that is necessary and sufficient for the existence of such a non-singular extension.