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In 2015, Hasegawa, Honda, Naokawa, Saji, Umehara, and Yamada definedintrinsic cross cap singularities, which are generalizations of cross capsingularities, and proved the Gauss-Bonnet type formula for surfaces withoutboundary that admit these singularities. In this paper, we prove theboundedness of geodesic curvature measures of a regular curve passing throughan intrinsic cross cap singularity and generalize the Gauss-Bonnet type formulaby Hasegawa et al. to the case of surfaces with boundary. Also, we giveconditions for the boundedness of the geodesic curvature at an intrinsic crosscap singularity.

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