On the boundedness of the geodesic curvature measure of a regular curve passing through a cross cap singularity

TL;DR

This paper investigates the behavior of geodesic curvature measures of regular curves passing through intrinsic cross cap singularities, extending Gauss-Bonnet formulas to surfaces with boundaries and establishing conditions for boundedness at singularities.

Contribution

It proves the boundedness of geodesic curvature measures at intrinsic cross cap singularities and generalizes the Gauss-Bonnet formula to surfaces with boundary.

Findings

Geodesic curvature measures are bounded at intrinsic cross cap singularities.

Extended Gauss-Bonnet formula applies to surfaces with boundary.

Provided conditions for boundedness of geodesic curvature at singularities.

Abstract

In 2015, Hasegawa, Honda, Naokawa, Saji, Umehara, and Yamada defined intrinsic cross cap singularities, which are generalizations of cross cap singularities, and proved the Gauss-Bonnet type formula for surfaces without boundary that admit these singularities. In this paper, we prove the boundedness of geodesic curvature measures of a regular curve passing through an intrinsic cross cap singularity and generalize the Gauss-Bonnet type formula by Hasegawa et al. to the case of surfaces with boundary. Also, we give conditions for the boundedness of the geodesic curvature at an intrinsic cross cap singularity.