Singularities of Slant Focal Surfaces along Lightlike Locus on Mixed Type Surfaces

: There are generally the mixed type surfaces with lightlike locus in the Lorentz-Minkowski 3-space. To investigate the geometry of lightlike locus, we deﬁne slant focal surfaces and slant evolutes associated to the oringinal mixed type surface by using a moving frame ﬁeld along the lightlike locus deﬁned by Honda etc. We obtain that singularities of slant focal surfaces and slant evolutes depend on the differential geometric properties of the lightlike locus. Furthermore, we investigate the relationship between slant focal surfaces and slant evolutes. We also consider the relationship between slant evolutes and the lightlike locus on the lightcone.


Introduction
The research aim of this paper is to investigate the differential geometric properties of the mixed type surfaces in Lorentz-Minkowski space. Let f : U → R 3 1 be a frontal, where R 3 1 is a 3-dimensional Lorentz-Minkowski space. If frontal f is an immersion, spacelike, timelike and lightlike points can be defined on it in terms of the induced metric. the lightlike points is independent from singular points of frontal f . It can be a singular point of the induced metric. When p is the lightlike point of the first kind [1,2], the lightlike locus f | L( f ) may be a spacelike regular curve in [2]. Then we have tangent vector e of f | L( f ) and two lightlike vectors L, N along L( f ). Thus they construct a moving frame along L( f ).
The focal surface and evolute of regular space curves are investigated as classical objects in differential geometry (cf. [3][4][5][6][7]). the focal surface is the envelope of family of normal planes. the evolute is not only the locus of the centre of osculating spheres but also set of singular values of the focal surfaces. Since lightlike locus is a spacelike regular curve, we can give the definitions of focal surface and evolute of lightlike locus. It is the envelope of family of normal planes which are spanned by two lightlike vectors L and N satisfying a symmetric (or, dual) condition along L( f ). Moreover, the osculating lightlike surface of lightlike locus is considered in [1]. It is the envelope of family of osculating planes which are also limiting tangent plane spanned by e and L of f along f | L( f ) .
On the other hand, the evolute of plane curve is the envelope of family of its normal lines and the envelope of family of tangent lines is the original curve. It is natural to ask what lies between normal lines and tangent lines. In [8], P. J. Giblin and J. P. Warder give the definition of straight lines L which is obtained by rotating tangent lines counterclockwise through a angle α along plane curve. Moreover, envelope of lines L is called by τ α . Then τ 0 is original curve and τπ 2 is the evolute of plane curve. Inspired by this thought, since osculating planes of f along f | L( f ) are similar to tangent lines of plane curve and normal planes of f along f | L( f ) are similar to normal lines of plane curves, we can define θ-planes which move between osculating planes and normal planes along lightlike locus and study new symmetric properties about these planes. the θ is given by angle of between θ-planes and osculating planes. It follows that the slant focal surface of lightlike locus is defined by the envelope of family of θ-planes of f along f | L( f ) . If θ = 0, the slant focal surfaces is the osculating lightlike surface. If θ = π 2 , the slant focal surface is the focal surface of lightlike locus. and the slant evolute is given by singular set of the slant focal surface.
In this paper, we give some basic notions including frame {e, L, N} along lightlike locus in Section 2. In Section 3, by using this frame, we give the definitions of the slant focal surface and the slant evolute of lightlike locus L( f ). Then geometry and singularities of them can be investigated by the moving frame. On the other hand, wave front is firstly given in [9]. Furthermore, many articles on the frontal or front have been published during the two decades [10][11][12][13][14][15][16]. Thus, by the criterions of singularities of frontal or front, we obtain that singularities of slant focal surfaces have not only cuspidal edge, swallowtail but also cuspidal beaks under this frame. But cuspidal cross cap and cuspidal lips are never appeared on the slant focal surface. Moreover, we investigate relationship between slant focal surfaces and slant evolutes from viewpoint of singularity theory. We obtain that the image of slant evolute is precisely the set of non-degenerate singular values of slant focal surface. Since the geometry of moving frame is related to the properties of f | L( f ) on the frontal f , thus the geometry and singularities of slant focal surfaces and slant evolutes are deeply depended on geometric properties of the frontal f . In Section 4, slant focal surfaces and slant evolutes are given by the discriminant set and the secondary discriminant sets of θ-function. Moreover, if the slant evolute is a constant point under a certain condition, then the lightlike locus is on a lightcone whose vertex is the slant evolute, meanwhile this lightcone is the osculating lightlike surface. Finally, if the evolute of lightlike locus is a constant point, then the lightlike locus is on a pseudo sphere. In recent years, some of the latest connected studies can be seen in [17][18][19][20][21][22][23][24][25][26]. In the future work, we are going to proceed to study some applications combine with singularity theory and submanifold theory, etc. to obtain new results and theorems about symmetry.
All manifolds and mappings are C ∞ unless otherwise stated.

Preliminaries
In this section, we prepare some basic notions about frontal and lightlike locus in R 3 1 . Firstly, we give the definition of the frontal in R 3 1 . About lightlike locus on mixed surfaces, please see [2] for details.

Frontals in
For any x = (x 0 , x 1 , x 2 ), y = (y 0 , y 1 , y 2 ) ∈ R 3 , the pseudo scalar product of x and y is defined by we call (R 3 , , ) the Minkowski space and write R 3 1 instead of (R 3 , , ). A non-zero vector x ∈ R 3 1 is spacelike (respectively, timelike, lightlike) if x, x is positive (respectively, negative, zreo). The norm of a non-zero vector x is defined by 1 | x, v = c} is called spacelike, timelike or lightlike when v is timelike, spacelike or lightlike respectively. We can define the vector product where x = (x 0 , x 1 , x 2 ) and y = (y 0 , y 1 , y 2 ) ∈ R 3 1 . We can consider a subbundle R 3 1 × Gr(2, 3) ⊆ R 3 1 × R 3 = TR 3 1 , where Gr(2,3) is the Grassmannian of 2-planes in R 3 1 . We can identify it with the projective tangent bundle PTR 3 1 via PT q R 3 1 V q → (V q ) ⊥ by the scalar product , , where x ⊥ = {y ∈ R 3 1 | x, y = 0}. As we know that PTR 3 , where ν ⊥ : U → Gr(2, 3) be a 2-dimensional subspace-valued map. We can call ν ⊥ (p) the limiting tangent plane at p ∈ U. A frontal f is a front if the isotropic lift F is an immersion. If f is an immersion and [ν] = [ f u × f v ], then f is a front. We can call [ν] a lightcone Gauss map of f [27,28].
It is known that a frontal (, front) may have singularities. the generic singularites of frontals are cuspidal edges, swallowtails and cuspidal cross cap. Cuspidal edges and swallowtails can appear on the fronts, but cuspidal cross cap is a frontal which is not a front. A singular point of a frontal germ f at p is a cuspidal edge (briefly, at the origin. A singular point of a frontal germ f at p is a cuspidal cross cap (briefly, CBK) if f is A-equivalent to (u, v) → (u, v 2 , uv 3 ) at the origin. In this decade, differential geometric properties of frontals or fronts in R 3 are investigated (see [12,14,15,29] for example). On the other hand, a point p of frontal f is said to be spacelike , timelike or lightlike point if ν ⊥ (p) is sapcelike, timelike or lightlike. the sets of spacelike, timelike or lightlike points can be denoted by U + , U − or L( f ), respectively . Then f is said to be mixed type if U + , U − or L( f ) are non-empty. We can find many studies on differential geometric property of spacelike surfaces or surfaces whose lightlike point set L( f ) is non-empty in R 3 1 (cf. [27,28,[30][31][32][33][34][35][36][37][38][39][40]).

Frame Field along Lightlike Locus
Let f : U → R 3 1 be a frontal with non-empty lightlike point set L( f ). Then F = ( f , [ν]) is an isotropic lift of f . We can assume that L( f ) can be parameterized by a regular curve γ : (−ε, ε) → U near p = γ(0) ∈ L( f ). Under the assumption we call f (L( f )) the lightlike locus and setγ = f • γ. Furthermore, we also assume thatγ (u) = dγ du is non-zero and spacelike. This indicates that the lightlike locusγ is a spacelike regular curve in R 3 1 . Then we call frontal f which satisfies the above assumptions an admissible frontal.
We consider a frame along L( f ) as a regular curve on the frontal f (cf. [2]). We take a parameter u which satisfies |e(u)| = 1, where e is a unit tangent vector field ofγ. Then we have a frame {e, L, N} alongγ(u) satisfying e, e = 1, e, L = 0, e, N = 0, L, L = 0, L, N = 1, N, and the following Frenet-Serret type formula where Moreover, we can set L(u) = ψ(u)L(u), where ψ : I → R is a non-vanishing function. Thus, we assume N(u) = N(u)/ψ(u) in order to make the frame {e, L, N} satisfy (1) along L( f ). By (3), we can also define three functions α L , α N , α G relating to the frame {e, L, N} and have the following relationship: Then α L , α N , α G can be regarded as invariants of f as follows: We consider a vector field η on U satisfying that d f (η(q)) = L(q). We set function β = η d f (η), d f (η) | L( f ) which does not vanish along L( f ). Let κ L , κ N and κ G the lightlike singular curvature, the lightlike normal curvature and the lightlike geodesic torsion of f along L( f ), respectively ([2], [Definition 3.2]). Then by [2], [Proposition 3.5], we obtain that If we take vector field η satisfying that β = 1, then To simplify the expressions in Section 3, we define four smooth functions as follows:

Singularities of Slant Focal Surfaces of the Lightlike Locus
In this section, we investigate singularities of slant focal surfaces of the lightlike locus and give the relationship between the slant focal surface and the slant evolute from the viewpoint of singulartity theory.
Let f : U → R 3 1 be an admissible frontal. Under the notations in Section 2.2, we can define slant focal surfaces of the lightlike locusγ as follows: At least locally, we can easily see that (σ θ , sin θα L )(u) = (0, 0). If σ θ (u) = 0, the slant focal surface F S θγ : I × R → R 3 1 is given by If sin θα L (u) = 0, the slant focal surface F S θγ : I × R → R 3 1 is given by In the case when θ = 0, we consider the slant focal surface under the assumption σ 0 = 0. It is given by then we have the followings.
Assume that σ θ (u) = 0 and U = I × R, by Equations (2) and (10), we have )e(u) To simplify (13), we define three functions a θ 1 , a θ 2 , a θ 3 : Furthermore, we define a mapping ν θ : I → R 3 1 for a fixed θ ∈ [0, π/2] by, It follows that To investigate singularities of F S θγ , we define a mapping T θ : We can define a smooth function λ θ : U → R (cf. [1]) for a fixed θ ∈ [0, π/2] by then p is a singular point if and only if λ θ (p) = 0. Moreover, p is non-degenerate if and only if dλ θ (p) = 0. By implicit function theorem, the singular set S( is parameterized by a regular curve ξ θ : I → U in a neighborhood of p. For the singular set S(F S θγ ), there exists a non-zero vector field η θ near p satisfying η θ R = kerdF S θγ at p ∈ S(F S θγ ). We call η θ null vector field. Furthermore, we call (ξ θ ) u (u) singular direction and η θ (u) null direction if ξ θ is parametered by u. When ξ θ is parametered by m, we also call (ξ θ ) m (m) singular direction and η θ (m) null direction.
(3) F S θγ at p 0 is never A-equivalent to the cuspidal cross cap.
(3) F S θγ at p 0 is never A-equivalent to the swallowtail, cuspidal cross cap and cuspidal lips.
(3) F S 0γ at p 0 is never cuspidal cross cap, cuspidal beaks and cuspidal lips.

Theorem 2.
Under the assumption α L = 0 and θ = 0, we assume that p 0 = (u 0 , n 0 ) is a singular point of F S θγ and have the followings.
(3) F S θγ at p 0 is never A-equivalent to the cuspidal cross cap (2) F S θγ at p 0 is A-equivalent to the cuspidal beaks if and only if at p 0 is never A-equivalent to swallowtail, cuspidal cross cap and cuspidal lips.

Remark 2.
Since the proof of Theorem 2 is similar to the one of Theorem 1 under the assumption θ = 0. Then we omit it here.
On the other hand, we give the definition of slant evolutes of lightlike locus. Then we give that the image of the set of non-degenerate singular points of the slant focal surfaces coincide with the image of the slant evolute. Moreover, we give relationships between singularities of the slant evolutes and singularities of slant focal surfaces.
If sin θα L = 0, we have the similar conclusions to those under the assumption σ θ = 0. Since this proof is similar to the above proof, so we omit it.

Properties of Non-Degenerate Singular Set of Slant Focal Surfaces F S θ γ
In this section, we consider the properties of non-degenerate singular set of F S 0γ under h θ (u) = 0.
by the formula (a), formula (b) and Equation (2), a long but straightforward calculation gives that then the assertion (3) holds.
For θ-functions F θ under a fixed θ ∈ [0, π/2], its discriminant set is defined as follows and its second discriminant set is We can easily see that the slant focal surface F S θγ coincides with the discriminant set D F θ , and the slant evolute E θ γ coincides with the second discriminant set D 2 F θ . Furthermore, the second discriminant set is also the set of non-degenerate singular values of the discriminant set by Theorem 3.
Conversely, we can easily see that v =γ(u)

Examples
We give an example in order to understand the slant focal surfaces and slant evolutes of lightlike locus from intuitional viewpoint. We can see the mixed surfaces (yellow surfaces) and lightlike locus (red curves) in Figure 3.

Conclusions
To investigate the geometry of lightlike locus on the mixed type surfaces in the Lorentz-Minkowski 3-space, we define slant focal surfaces and slant evolutes by using a moving frame field along the lightlike locus. By the criterions of singularites of front, the classification theorem of singularities of the slant focal surface is given. This theorem deeply depends on geometric properties of lightlike locus. Furthermore, we obtain the relationship between slant focal surfaces and slant evolutes from the viewpoint of singularity theory. When the slant evolute is a constant point under a certain condition, the lightlike locus is on the lightcone whose vertex is the slant evolute (cf. Example 1). Meanwhile, this lightcone is the osculating lightlike surfaces. Finally, if the evolute of lightlike locus is a constant point, then the lightlike lcous is on a pseudo sphere.