Geometric deformations of cuspidal $S_1$ singularities

TL;DR

This paper investigates the geometric properties of cuspidal S_1 singularities and their deformations, providing a form based on diffeomorphisms and isometries, and characterizing when such maps are frontals.

Contribution

It introduces a specific form for cuspidal S_1 singularities considering geometric deformations and characterizes the frontal condition for these maps.

Findings

Necessary and sufficient condition for a map to be a frontal.

Characterization of cuspidal S_1 singularities as frontals.

Analysis of geometric properties of cuspidal S_1 singularities and cross caps.

Abstract

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.