Mixed type surfaces with bounded Gaussian curvature in three-dimensional Lorentzian manifolds

Isometric deformations of mixed type surfaces in Lorentz-Minkowski space

“…See Example 6.4 for the case that γ is a non-Frenet curve. As pointed out in [14], lightlike points of the first kind of mixed type surfaces are similar to the cuspidal edge singularity of wave fronts [18]. The invariants κ L and κ N of lightlike points of the first kind have several properties similar to the invariants of cuspidal edges, called the singular curvature κ s and the limiting normal curvature κ ν introduced in [29] (cf.…”

“…Invariants of lightlike points. We review here the fundamental properties of the invariants of the lightlike points of the first kind introduced in [14].…”

“…The first fundamental form of a mixed type surface is a type-changing metric. Type-changing metrics are of interest both in mathematics and theoretical physics, see references in [25,14]. Recently, mixed type surfaces of zero mean curvature, especially the so-called Bernstein-type property, have been investigated intensively (see [6,1,2,3] and the references therein).…”

“…[27,15,16,26]. In [14], the author with K. Saji and K. Teramoto investigated lightlike points of mixed type surfaces in the way similar to the case of wave fronts [29]. To study the behavior of the Gaussian curvature at lightlike points, in [14], the invariants named the lightlike singular curvature κ L and the lightlike normal curvature κ N were introduced (cf.…”

“…To determine the intrinsicity and extrinsicity of invariants is one fundamental problem. In [14], it is proved that the lightlike singular curvature κ L is an intrinsic invariant. Moreover, it is also proved that the lightlike normal curvature κ N is also intrinsic if κ L is identically zero along LD ([14, Corollary C], Fact 6.5).…”