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

We define cuspidal curvature k(c) (resp. normalized cuspidal curvature mu(c)) 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 k(Pi) called the product curvature (resp. mu(Pi) called normalized product curvature) of k(c)(resp. mu(c)) and the limiting normal curvature k(v) 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 k(Pi) when cuspidal edges accumulate to other singularities. Moreover, several new geometric invariants of cuspidal edges and swallowtails are given.