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

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.