arXiv subjects

Runa Shimada

Publications and source records attributed to Runa Shimada.

4 recordsLinked to original sources

arXiv2026

Geometry on principal curvature surfaces

We study the geometry of the principal curvature surface associated with a Whitney umbrella. This surface is obtained by extending the Whitney umbrella in the normal direction by its bounded principal curvature and admits a natural geometric interpretation. We prove that its intersection with the normal plane coincides with the pedal curve of the curvature parabola and deduce that the focal conic is the inversion of this pedal curve. We then investigate the geometry of the principal curvature surface along the exceptional set by studying its geodesic and normal curvatures, as well as its Gaussian and mean curvatures, and determining all possible generic numbers of zeros of these curvature functions.

math.DG↗

arXiv2025

Criteria for codimension two singularities of surfaces and their applications

We give simple criteria for the singularities appearing on surfaces codimension less than or equal to two. As applications, we give conditions for codimension two singularities that appear in ruled surfaces and center maps of surfaces in the Euclidean space.

math.DG↗

arXiv2025

Geometric deformations of cuspidal $S_1$ singularities

To study a deformation of a singularity taking into consideration their differential geometric properties, a form representing the deformation using only diffeomorphisms on the source space and isometries of the target space plays a crucial role. Such a form for an $S_1$ singularity is obtained by the author's previous work. On this form, we give a necessary and sufficient condition for such a map is being a frontal. The form for an $S_1$ singularity with the frontal condition can be considered such a form for a cuspidal $S_1$ singularity. Using this form, we investigate geometric properties of cuspidal $S_1$ singularities and the cuspidal cross caps appearing in the deformation.

math.DG↗

arXiv2024

Geometry on deformations of S1 singularities

To study a one parameter deformation of an $S_1$ singularity taking into consideration its differential geometric properties, we give a form representing the deformation using only diffeomorphisms on the source and isometries of the target. Using this form, we study differential geometric properties of $S_1$ singularities and the Whitney umbrellas appearing in the deformation.

math.DG↗