Lightlike developables in Minkowski 3-space

Abstract

Abstract. We say that a surface in Minkowski 3-space is a lightlike developable if all pseudo-normal vectors of the regular part of the surface are lightlike. The tangent surface of a lightlike curve is one of the lightlike developables. We give a generic classification of such surfaces. The all arguments in this paper are elementary. However, we discovered the H 3 type singularity appears in generic for such a class of surfaces. Since the H 3 type singularity usually appears in non-generic situation, this is a quite interesting phenomenon. IntroductionA surface in Euclidean space whose Gauss curvature vanishes on the regular part is called a developable surface. It has been known that a developable surface is a part of a conical surface, a cylindrical surface, the tangent surface of a space curve or the glue of such surfaces. Developable surfaces have singularities in general. The tangent surface of a space curve has the most interesting singularities in the above three kinds of surfaces. In fact Cleave [2] showed that the germ of the tangent surface of a generic cuspdialedge cuspidal cross cap 2000 Mathematics Subject Classification: 53A35, 58C27, 58C28.

Recognition Problem of Frontal Singularities

“…Therefore (3) (1) and (1') is proved in [6]. The condition (2) implies that f is an opening of becà bec. Using the same notations in the proof of Theorem 3.17, we write f 3 as f 3…”

“…Hence f = F 1 is A -equivalent to F 0 , that is the normal form of (2). Therefore (3) implies (2). The converse is clear.…”

“…Then by the condition (2), we see that f is a versal opening (Definition 4.4) of g. On the other hand the map-germ of (1') is a versal opening of g ( [8]). By Theorem 4.5, we see that (2) implies (1'). The converse implication (1') to (2) is clear.…”

“…By the condition ord η a ( f 3 ) = 4, we have B(0, 0) = 0. Then, by a change of adapted system of coordinates, We may suppose f = (g, f 3 ) with 2 with B(0, 0) = 0. Then we set the family F s = (g,U 1 + sΦ.…”

“…In [24], the map-germ which is A -equivalent to the germ g : (t 1 ,t 2 ) → (t 1 , t 4 2 + t 1 t 2 ) at the origin is called a swallowtail and it is shown that a map-germ g : (R 2 , a) → (R 2 , g(a)) is a swallowtail if and only if λ is K -equivalent to the germ (t 1 ,t 2 ) → t 1 at the origin and ord η a (λ ) = 3. Suppose f satisfies (2). Then f is an opening of swallowtail.…”