  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.
