The hypersurfaces with conformal normal Gauss map in H^n+1 and S_{1}^{n+1}

In this paper we introduce the fourth fundamental form for the hypersurfaces in $H^{n+1}$ and the space-like hypersurfaces in $S_{1}^{n+1}$ and discuss the conformality of the normal Gauss maps of the hypersurfaces in $H^{n+1}$ and $S_{1}^{n+1}$. Particularly, we discuss the surfaces with conformal normal Gauss maps in $H^{3}$ and $S_{1}^{3}$ and prove a duality property. We give the Weierstrass representation formula for the space-like surfaces in $S_{1}^{3}$ with conformal normal Gauss maps. We also state the similar results for the time-like surfaces in $S_{1}^{3}.$


Introduction
It is well known that the classical Gauss map has played an important role in the study of the surface theory in R 3 and has been generalized to the submanifold of arbitrary dimension and codimension immersed into the space forms with constant sectional curvature( see [15]in detail).
Particularly, for the n-dimensional submanifold x : M → V in space V with constant sectional curvature, Obata [13] introduced the generalized Gauss map which assigns to each point p of M the totally geodesic n-subspace of V tangent to x(M) at x(p). He defined the third fundamental form of the submanifold in constant curvature space as the pullback of the metric of the set of all the totally geodesic n-subspaces in V under the generalized Gauss map. He derived a relationship among the Ricci form of the immersed submanifold and the first, the second and the third fundamental forms of the immersion. Meanwhile, Lawson [10] discussed the generalized Gauss map of the immersed surfaces in S 3 and prove a duality property between the minimal surfaces in S 3 and their generalized Gauss map image.
Epstein [4] and Bryant [3] defined the hyperbolic Gauss map for the surfaces in H 3 and Bryant [3] obtained a Weierstrass representation formula for the constant mean curvature one surfaces with conformal hyperbolic Gauss map. Using the Weierstrass representation formula, Bryant also studied the properties of constant mean curvature one surfaces. Using the hyperbolic Gauss map, Gálvez and Martĺnez and Milán [6] studied the flat surfaces in H 3 with conformal hyperbolic Gauss map with respect to the second conformal structure on surfaces (see [7] for the definition) and obtained a Weierstrass representation formula for such as surfaces.
Kokubu [8] considered the n-dimensional hyperbolic space H n as a Lie group G with a left-invariant metric and defined the normal Gauss maps of the surfaces which assigns to each point of the surface the tangent plane translated to the Lie algebra of G. He also gave a Weierstrass representation formula for minimal surfaces in H n . On the other hand, Gálvez and Martĺnez [5] studied the properties of the Gauss map of a surface Σ immersed into the Euclidean 3-space R 3 by using the second conformal structure on surface and obtained the Weierstrass representation formula for the surfaces with prescribed Gauss map. Motivated by their work, the author [16] gave a Weierstrass representation formula for the surfaces with prescribed normal Gauss map and Gauss curvature in H 3 by using the second conformal structure on surfaces. From this, the surfaces whose normal Gauss maps are conformal have been found and the translational surfaces with conformal normal Gauss maps locally are given. In [17], the author classified locally the ruled surfaces with conformal normal Gauss maps within the Euclidean ruled surfaces and studied some global properties of the ruled surfaces and translational surfaces with conformal normal Gauss maps.
Aiyama and Akutagawa [1] defined the normal Gauss map for the space-like surfaces in the de Sitter 3-space S 3 1 and gave the Weierstrass representation formula for the space-like surfaces in S 3 1 with prescribed mean curvature and normal Gauss map.
The purpose of this paper is to study the conformality of the normal Gauss maps for the hypersurfaces in H n+1 and the space-like hypersurfaces in S n+1 1 and to prove a duality property between the surfaces in H 3 and the space-like surfaces in S 3 1 with conformal normal Gauss maps. The rest of this paper is organized as follows. In the second section, we describe the generalized definition of the normal Gauss map for the hypersurfaces in H n+1 and the space-like hypersurfaces in S n+1 1 (cf. [1] [8]). The third section introduces the fourth fundamental form for the hypersurfaces in H n+1 and S n+1 1 and obtains a relation among the first, the second, the third and the fourth fundamental forms of the hypersurfaces. As a application, we discuss the conformality of the normal Gauss map for the hypersurfaces in H n+1 and the space-like hypersurfaces in S n+1 1 . By means of the generalized Gauss map of the surfaces in H 3 and S 3 1 , the fourth one proves a duality property between the surfaces in H 3 and the space-like surfaces in S 3 1 with conformal normal Gauss maps. The fifth one gives the Weierstrass representation formula for the space-like surfaces in S 3 1 with conformal normal Gauss map and the sixth one derives the PDE for the space-like graphs in S 3 1 with conformal normal Gauss map and classifies locally the translational surfaces and the Euclidean ruled surfaces in S 3 1 with conformal normal Gauss map. In the last section, we state the similar results for time-like surfaces in S 3 1 with conformal normal Gauss map. Acknowledgement The author would like to express his sincere gratitude to Prof. Detang Zhou for his enthusiastic encouragement, support and valuable help as well as for his significant suggestions and heuristic discussions with the author and for his providing the author with Omori and Yau's paper [14][18].
Proof. At first we prove the Theorem for H n+1 . Choose the normal coordinates u 1 , u 2 , · · · , u n near p ∈ M. By the Weingarten formula, we get [13] and by the Gauss equa- Next, similar to the above proof, for S n+1 Similar to the proof of (3.1), we can prove (3.2).
Next, we consider the applications of these formulas (3.1)−(3.4). In the following of this paper, that the normal Gauss map is conformal means that the fourth fundamental form is proportional to the second fundamental form, i.e. IV = ρII for some smooth function ρ on M.
, where the vectors X and Y belong to different principal direction spaces.
Proof. The case of H n+1 . For any point p ∈ M, let {e 1 , e 2 , · · · , e n } be a local frame field so that (h ij ) is diagonalized at this point, i.e. h ij (p) = λ i δ ij . By IV = ρII and (3.3), we get, for i = 1, 2, · · · , n, that Because x(M) has no umbilics, the equation (3.6) with respect to λ i has exactly two distinct solutions λ and µ and λµ = η 2 n+1 . By the Gauss equation, one Conversely, choose the local tangent frame {e 1 , e 2 , · · · , e n } and the dual frame {ω 1 , ω 2 , · · · , ω n } near p, such that h ij = 0, i = j and h 11 The sufficiency has been proved for H n+1 . Similarly, we can prove Theorem 3.2 for S n+1 Remark. By (3.5), we know that the normal Gauss maps of all totally umbilics hypersurfaces except the totally geodesic hyperspheres in H n+1 are conformal. Similarly, for the space-like hypersurfaces in S n+1 1 , since η n+1 = 0, the normal Gauss maps of all totally umbilic space-like hypersurfaces except totally geodesic space-like hypersurfaces are conformal.
For H 3 and S 3 1 , by Theorem 3.2, we immediately get THEOREM 3.3. Let M be a 2-dimensional Riemannian manifold and x : M → H 3 (resp. x : M → S 3 1 ) be an immersed surface (resp. space-like surface) without umbilics. Then the normal Gauss map of x(M) is conformal if and only if the Gauss curvature K = −1 + η 2 3 (resp.K = 1 − η 2 3 ). Remark. In [16][17], we assume that the second fundamental form is positive definite and induces the conformal structure on the surfaces in H 3 . Here, the assumption with respect to the positive definite second fundamental form is dropped. THEOREM 3.4. Let M be a n-dimensional Einstein manifold and x : M → H n+1 (resp. x : M → S n+1 1 ) be an immersed hypersurface (resp. space-like hypersurface) with the non-degenerate second fundamental form and without umbilics. If the normal Gauss map of x(M) is conformal map,i.e. IV = ρII, then n = 2 and ρ = 2(H − η 3 ) (resp.ρ = 2(H + η 3 )).

A duality for the surfaces in H 3 and S 3 1 with conformal normal Gauss maps
Let L 4 be the Minkowski 4-space with the canonical coordinates X 0 , X 1 , X 2 , X 3 and the Lorentz-Minkowski scalar product −X 2 0 +X 2 1 +X 2 2 +X 2 3 . The Minkowski model of H 3 is given by Accordingly, the space-like normal vector of the surface in the Minkowski model of We get The Minkowski model of the de Sitter 3-space is defined as and can be divided into three components as follows(cf. [1]), Identify S − and S + with the upper half-space model R 3 + of the de Sitter 3-space by (cf. [1]) For the space-like surface X : We get Remark. In [1], the normal Gauss map of the space-like surface X : M → S 3 1 is defined globally on M. Because of the density of U − and U + in M, in this paper, we may consider that the normal Gauss map of the space-like surface X : M → S 3 1 is defined on U − and U + . Let X : M → H 3 (resp. X : M → S 3 1 ) be an immersed surface(resp. spacelike surface). Parallel translating the space-like (resp. time-like) unit normal vector N to the origin of L 4 , one gets the map N : M → S 3 1 (resp.N : M → H 3 ) which is usually called generalized Gauss map of X : M → H 3 (resp. X : M → S 3 1 ). The generalized Gauss map image can be considered as the surface in S 3 1 (resp.H 3 ). Proof. In the context of this paper, we prove (2). For any p ∈ M, let{e 0 , e 1 , e 2 , e 3 } be the orthonormal frame near p, such that e 0 = X, e 3 = N. Let {ω 0 , ω 1 , ω 2 , ω 3 } be the dual frame. The connection 1-forms is ω β α , α, β = 0, 1, 2, 3. The coefficients of the second fundamental form of X : M → S 3 1 is given by Choose the local tangent frame {e 1 , e 2 } near p , such that h ij = λ i δ ij . Then ds 2 * = λ 2 1 ω 2 1 +λ 2 2 ω 2 2 . So, when λ 1 λ 2 = 0, i.e. K = 1, N(M) is an immersed surface into H 3 . Its space-like unit normal vector is X and the second fundamental form is II = − dX, dN = −λ 1 ω 2 1 − λ 2 ω 2 2 . By the Gauss equation, K * = −1 + 1 λ 1 λ 2 = K 1−K . By Theorem 3.3, (4.1),(4.2) and Theorem 4.1, we get the following duality.

Weierstrass representation formula
In this section, we give the Weierstrass representation formula for the spacelike surfaces in S 3 1 with conformal normal Gauss maps. At first, we describe the normal Gauss map and the de Sitter Gauss map of the space-like surfaces in S 3 1 . Take the upper half-space model R 3 + of S 3 1 . The normal Gauss map of the space-like surface x : M → S 3 1 is given by By means of the stereographic projection from the north pole (0, 0, 1) of H 2 (−1) to the (x 1 , x 2 )−plane identified with C, we get which is also called the normal Gauss map of the space-like surface x : M → S 3 1 .
N can be written as Next, we describe the definition of the de Sitter Gauss map for the spacelike surfaces in S 3 1 (in [11], it is still called hyperbolic Gauss map), which is the analogue of Epstein and Bryant's hyperbolic Gauss map for the surfaces in H 3 (cf [3][4] [16]). The time-like geodesic is either the Euclidean equilateral half-hyperbola consisting of two branches which is orthonormal to the coor- Using the Euclidean geometry, as similar as done in the Theorem 5.1 of [16], we get 3 . By the duality given in section 4, the generalized Gauss map of x : M → H 3 is given, when η 3 > 0, by and when η 3 < 0, by and in the Minkowski model of the de Sitter 3-space, their time-like unit normal vector is X : M → H 3 . Again by the duality given in section 4, a straightforward computation shows us that the normal Gauss map of N : M → S 3 1 is given by So,  [17]). From this, we also prove the Theorem 4.2. By (5.1)-(5.4) and the Theorem 5.1 of [16], we get that when η 3 > 0, i.e. |g S | > 1, and when η 3 < 0, i.e. |g S | < 1, where G H is exactly the hyperbolic Gauss map of x : M → H 3 (cf [3][4] [16]). In the following, we write respectively g S and G S as g and G.
By (5.2)-(5.6) and the Weierstrass representation for the surfaces in H 3 with conformal normal Gauss map [16], we get the Weierstrass representation formula for the space-like surfaces in S 3 1 with conformal normal Gauss map. (1) When the holomorphic map g : M → C ∪ {∞}\{|z| = 1} satisfies |g| > 1 and 10) 11) where the conformal structure on M is induced by the negative definite second fundamental form.

Graphs and examples
In this section,we give the examples of surfaces in S 3 1 with conformal normal Gauss maps within the translational surfaces and the Euclidean ruled surfaces.
In H 3 , the graph (u, v, f (u, v)) with conformal normal Gauss map satisfies the following fully nonlinear PDE (cf. [16] [17]) Take the upper half-space model of S 3 1 . Consider the space-like graph (u, v, f (u, v)) in S 3 1 with f 2 u + f 2 v < 1. Its Gauss curvature is given by K = where f 2 u + f 2 v < 1. This is the fully nonlinear PDE which the space-like graph in S 3 1 with K = 1 − η 2 3 must satisfy. Remark. There exists a nice duality between the solutions of minimal surface equation in R 3 and the ones of maximal surface equation in Lorentz-Minkowski 3-space L 3 (cf. [2]). Here, by the duality given by (5.2)(or (5.3)), we know that if f (u, v) is the solution of (6.1), then the local graph of in H 3 satisfies (6.1). Next,as similar as done in section 6 of [16], we get the following Theorem.
where a and b are nonzero constants. The parameter form of these translational surfaces are locally given by Considered as surfaces in 3-dimensional Minkowski space L 3 , the space-like ruled surfaces in S 3 1 can be represented as is a parameter domain and α(v) and β(v) are two vector value functions into L 3 corresponding to two curves in L 3 . When β is locally nonconstant, without loss of generality we can assume that either β, β = 1, β ′ , β ′ = ±1, and α ′ , β ′ = 0 or β, β = 1, β ′ , β ′ = 0, and α ′ , β = 0, where ·, · is the scalar product in L 3 . As similar as done in Theorem 2 of [16], we have THEOREM 6.2. Up to an isometric transformation in S 3 1 , every space-like ruled surface in S 3 1 with conformal normal Gauss map is locally a part of one of the following, (1) ordinary Euclidean space-like planes in S 3 1 , We should note that in the proof of Theorem 6.2, only when β ′ , β ′ = −1, we may get the nontrivial cases (2) and (3).
Remark. Every geodesic of H 3 , corresponding respectively to u = 0,u = π, v = 0 and v = π on surfaces (6.6) and to v = π 2 on surfaces (6.7) and to v = ± π 2 on surfaces (6.8) follow which K = −1 is mapped to a simple point in S 0 by the generalized Gauss map.
7 Time-like surfaces in S 3 1 with conformal normal Gauss map In this section, we state the similar results as the aboved for the time-like surfaces in S 3 1 without proofs. Take the upper-half space model of S 3 1 . Let M be a 2-dimensional Lorentz surface and x : M → S 3 1 be the time-like immersiom with the local coordinates u 1 , u 2 . The first and the second fundamental forms are given, respectively, by I = g ij du i du j and II = h ij du i du j . The space-like unit normal vector is  is the solution of (6.2) with f 2 u + f 2 v > 1, then the local graph of the surface (f f u − u, f f v − v, f f 2 u + f 2 v − 1) in S 3 1 also satisfies (6.2). Locally, the ruled surfaces (4) and (5) in Theorem 7.4 can be represented as the graph (u, v, f (u, v)) as follows,