Behavior of Gaussian curvature and mean curvature near non-degenerate singular points on wave fronts

Abstract

We define cuspidal curvature $κ_c$ (resp. normalized cuspidal curvature $μ_c$) 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 $κ_Π$ called the product curvature (resp. $μ_Π$ called normalized product curvature) of $κ_c$ (resp. $μ_c$) and the limiting normal curvature $κ_ν$ 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 $κ_Π$ when cuspidal edges accumulate to other singularities. Moreover, several new geometric invariants of cuspidal edges and swallowtails are given.