Abstract. We define cuspidal curvature κc (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 κ Π called the product curvature (resp. µ Π called normalized product curvature) of κc (resp. µc) and the limiting normal curvature κν 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.

We study the geometry of surfaces in R 3 with corank 1 singularities. At a singular point we define the curvature parabola using the first and second fundamental forms of the surface, which contains all the local second order geometrical information about the surface. The curvature parabola is used to introduce the concepts of asymptotic directions and umbilic curvature, which are related to contact properties of the surface with planes and spheres.

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We study differential geometric properties of cuspidal edges with boundary. There are several differential geometric invariants which are related with the behavior of the boundary in addition to usual differential geometric invariants of cuspidal edges. We study the relation of these invariants with several other invariants. Maps from manifolds with boundaryThere are several studies for C ∞ map-germs f :There is also several studies for the case that the source space has a boundary. In [2], mapgerms from 2-dimensional manifolds with boundaries into R 2 are classified, and in [8], map-germs from 3-dimensional manifolds with boundaries into R 2 are considered. Let W ⊂ (R m , 0) be a closed submanifold-germ such that 0 ∈ ∂W and dim W = m. We call f | W a map-germ with boundary, and we call interior points of W interior domain of f | W . Since ∂W is an (m − 1)-dimensional submanifold, regarding ∂W = B, map-germs from manifolds with boundaries can be treated as a map-germ f : (R m , 0) → (R n , 0) with a codimension one oriented submanifold B ⊂ (R m , 0). We consider (R m , 0) has an orientation and the submanifold B is considered as the boundary. We define the interior domain of such map-germ f is the component of (R m , 0)\B such that positively oriented normal vectors of B points. With this terminology, an equivalent relation for mapgerms with boundary is the following. Let f, g : (R m , 0) → (R n , 0) be map-germs with codimension one submanifolds B, B ′ ⊂ (R n , 0) which contain 0. Then f and g are Bequivalent if there exist an orientation preserving diffeomorphism ϕ : (R m , 0) → (R m , 0) such that ϕ(B) = B ′ , and a diffeomorphism Φ : (R n , 0) → (R n , 0) satisfiesWe say that f is a cuspidal edge with boundary B ⊂ (R 2 , 0) if B is a codimension one oriented submanilfold, that is, there exists a parametrization b : (R, 0) → (R 2 , 0) to B satisfying b ′ (0) = (0, 0). The domain 2010 Mathematics Subject classification. Primary 53A05; Secondary 58K05, 58K50.

We study germs of pairs of codimension one regular foliations in R 3 . We show that the discriminant of the pair determines the topological type of the pair. We also consider various classifications of the singularities of the discriminant.