Behavior of Gaussian Curvature and Mean Curvature Near Non-degenerate Singular Points on Wave Fronts

Abstract

We define cuspidal curvature $\kappa_c$ (resp. normalized cuspidal curvature $\mu_c$) along cuspidal edges (resp. at swallowtail singularity) in Riemannian $3$-manifolds, and show that it gives a coefficient of the divergent term of the mean curvature function. Moreover, we show that the product $\kappa_\Pi$ called the product curvature (resp. $\mu_\Pi$ called normalized product curvature) of $\kappa_c$ (resp. $\mu_c$) and the limiting normal curvature $\kappa_\nu$ is an intrinsic invariant of the surface, and is closely related to the boundedness of the Gaussian curvature. We also consider the limiting behavior of $\kappa_\Pi$ when cuspidal edges accumulate to other singularities. Moreover, several new geometric invariants of cuspidal edges and swallowtails are given.