arXiv subjects

Kyoya Hashibori

Publications and source records attributed to Kyoya Hashibori.

3 recordsLinked to original sources

arXiv2023

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.

math.DG↗

arXiv2023

Gauss-Bonnet type formulas for frontal bundles over surfaces with boundary and their applications

We define a frontal bundle by imposing a compatibility condition on two types of coherent tangent bundles over a surface with boundary. Since it is known that there are two Gauss-Bonnet type formulas for coherent tangent bundles, we obtain four Gauss-Bonnet type formulas for frontal bundles. Furthermore, since two coherent tangent bundles over the surface are related to each other by the compatibility condition, an appropriate combination of these Gauss-Bonnet type formulas leads to new formulas that relate a frontal to its unit normal vector field using the Gaussian curvature, the Euler characteristic, etc. If the extrinsic curvature of a frontal is non-zero bounded, these formulas lead to formulas that relate the number of singular points to the Euler characteristic.

math.DG↗

arXiv2023

On Gauss-Bonnet type formulas for mappings between surfaces with boundary and their applications

We define singular points of the first kind and singular points of the second kind as singular points of mappings between surfaces. Typical examples of these singular points are fold singular points and cusp singular points, respectively. We can construct coherent tangent bundles, which are natural intrinsic formulations of wave fronts, by using mappings between surfaces. It is known that two types of Gauss-Bonnet type formulas hold for coherent tangent bundles over surfaces (possibly with boundary). Hence, we obtain two Gauss-Bonnet type formulas for mappings between surfaces (possibly with boundary). By applying these Gauss-Bonnet type formulas, we prove the Levine formula that relates the rotation indices to the Euler characteristic, and the Quine-Fukuda-Ishikawa formula that relates the mapping degree to the Euler characteristic and the number of singular points.

math.DG↗