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

Abstract

We define cuspidal curvature Kc (resp. normalized cuspidal curvature μc) along cuspidal edges (resp. at a 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 KΠ called the product curvature (resp. μΠ called normalized product curvature) of Kc (resp. μc) and the limiting normal curvature Kv 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 KΠ when cuspidal edges accumulate to other singularities. Moreover, several new geometric invariants of cuspidal edges and swallowtails are given.