PUBLICATIONS DE L’INSTITUT MATHÉMATIQUE Nouvelle série, tome 103(117) (2018), 223–236 DOI: https://doi.org/10.2298/PIM1817223T

ON BICONSERVATIVE HYPERSURFACES IN PSEUDO-RIEMANNIAN SPACE FORMS AND THEIR GAUSS MAP

Nurettin Cenk Turgay

Abstract. We first present a survey about recent results on biconservative hypersurfaces in the Minkowski space 𝔼_{1}^{4}, pseudo-Euclidean space 𝔼_{2}^{5} and Rieamnnian space-form ℍ^{4}. Then we obtain some geometrical properties of these hypersurface families concerning their mean curvature and Gauss map.

1. Introduction

Let ϕ : M → N be a smooth mapping between two Riemannian manifolds (M^{n}, g), (N^{m}, g̃) and suppose that its tension field is denoted by τ(ϕ). Then, ϕ is said to be biharmonic if it is a critical point of the bienergy functional

E_{2}(ϕ) = \frac{1}{2} ∫_{M} |τ(ϕ)|^{2}v_{g}.

In [9], Jiang proved that ϕ is biharmonic if and only if it satisfies the system of fourth order elliptic partial differential equations given by τ_{2}(ϕ) = 0, where τ_{2}(ϕ) = Δτ(ϕ) − trR̃(dϕ, τ(ϕ))dϕ is the bitension field. If ϕ : M → N is a biharmonic isometric immersion, then M is said to be a biharmonic submanifold of N.

As a generalization of biharmonic submanifolds, the following definition was given.

Definition 1.1. A submanifold M of N is said to be biconservative if the isometric immersion ϕ : M → N satisfies τ_{2}(ϕ)^{⊤} = 0, where τ_{2}(ϕ)^{⊤} denotes the tangential part of τ_{2}(ϕ).

Before we proceed, we state the following proposition (see [2, 8, 9]).

Proposition 1.1. Let ϕ : M → N be an isometric immersion between Riemannian manifolds. Then the following statements are equivalent.

(1) ϕ (or M) is biconservative;

(2) The stress-energy tensor S_{2} of ϕ satisfies divS_{2} = 0; (3) The equation

2010 Mathematics Subject Classification: Primary 53C40; Secondary 53C42, 53C50.

Key words and phrases: biharmonic submanifolds, Lorentzian hypersurfaces, biconservative hypersurfaces, finite type submanifolds.

m∇∥H∥^{2} + 4 traceA_{∇_{·}^{⊥} H} (·) + 4 trace(R̃(·, H) · )^{T} = 0

is satisfied, where H, ∇^{⊥} and A are the mean curvature vector, normal connection and the shape operator of M, respectively.

In this paper, we give a short survey of biconservative surfaces on pseudo-Riemannian space-forms 𝔼_{1}^{4}, 𝔼_{2}^{5} and ℍ^{4}. Then, we study some of geometrical properties of these hypersurface families considering their Gauss map and mean curvature. The organization of this paper is as following. In Section 2, we first describe some of the basic facts. In Section 3, we present explicit parametrization of biconservative hypersurfaces obtained in [7, 13, 14]. In Section 4, we obtain some new results of biconservative hypersurfaces in Minkowski spaces. In Section 5, we get some classification results considering Gauss map of hypersurface families presented in Section 3.

2. Preliminaries

Let 𝔼_{t}^{m} denote the semi-Euclidean m-space with the canonical semi-Euclidean metric tensor of index t given by

g̃ = ⟨, ⟩ = − ∑_{i=1}^{t} dx_{i}^{2} + ∑_{j=t+1}^{m} dx_{j}^{2}.

Pseudo-Riemannian space-forms are defined by

𝕊_{t}^{n}(r^{2}) = {x ∈ 𝔼_{t}^{n+1} : ⟨x, x⟩ = r^{−2} },
ℍ_{t−1}^{n} (−r^{2}) = {x ∈ 𝔼_{t}^{n+1} : ⟨x, x⟩ = −r^{−2} }.

For the particular case t = 1 and r = 1, we put ℍ_{0}^{n}(−1) = ℍ^{n} which is called the anti-de Sitter space-time when n = 4.

Consider an n-dimensional semi-Riemannian submanifold M of the pseudo-Euclidean space 𝔼_{t}^{m}. We denote the Levi-Civita connections of 𝔼_{t}^{m} and M by ˜∇ and ∇, respectively. Note that the Gauss and Weingarten formulas are given by

˜∇_{X} Y = ∇_{X} Y + h(X, Y),
˜∇_{X} ζ = −A_{ζ}(X) + ∇_{X}^{⊥} ζ,

respectively, for all tangent vectors fields X, Y and normal vector fields ζ, where h, ∇^{⊥} and A denote the second fundamental form, the normal connection and the shape operator of M, respectively. The Gauss and Codazzi equations are given, respectively, by

(2.1)  R(X, Y)Z = A_{h(X,Z)} Y − A_{h(Y,Z)} X,
(2.2)  ( ¯∇_{X} h)(Y, Z)= ( ¯∇_{Y} h)(X, Z),

where R is the curvature tensor associated with connection ∇ and ¯∇ h is defined by

( ¯∇_{X} h)(Y, Z) = ∇_{X}^{⊥} h(Y, Z) − h(∇_{X} Y, Z) − h(Y, ∇_{X} Z).

2.1. Gauss map of hypersurfaces in semi-Euclidean spaces. Consider an oriented hypersurface M of a (semi-)Euclidean space and let N be its Gauss map. By definition, M is said to have pointwise 1-type Gauss map if the Laplacian of its Gauss map takes the form

(2.3)  ΔN = ψ(N + C)

for a smooth function ψ and constant vector C. More precisely, a pointwise 1-type Gauss map is called of the first kind if (2.3) is satisfied for C = 0, and of the second kind if C ≠ 0. Moreover, if (2.3) is satisfied for a nonconstant function ψ, then M is said to have proper pointwise 1-type Gauss map. Otherwise, G is said to be (global) 1-type, [3, 4, 6].

3. Recent classifications of biconservative hypersurfaces

In this section, we would like to present some recent results on biconservative hypersurfaces.

3.1. Biconservative hypersurfaces in the Minkowski space 𝔼_{1}^{4}. In [7], the author and Yu Fu considered biconservative hypersurfaces in the Minkowski 4-space with diagonalizable shape operator. They obtained the following results.

Proposition 3.1. [7] Let M be a hypersurface in 𝔼_{1}^{4} given by

\begin{aligned}x(s, t, u) = & ( \frac{1}{2}s(t^{2} + u^{2}) + au^{2} + s + ϕ(s), st, (s + 2a)u, \\ & \frac{1}{2}s(t^{2} + u^{2}) + au^{2} + ϕ(s)) , a ≠ 0. \end{aligned} (3.1)

Then, M is biconservative if and only if either M is Riemannian and

ϕ(s) = c_{1} ( ln(s + 2a) − lns − \frac{a}{s} − \frac{a}{s + 2a} ) − \frac{s}{2}

or it is Lorentzian and

ϕ(s) = c_{1} ∫_{s_{0}}^{s} (ξ(ξ + 2a))^{2/3} dξ − \frac{s}{2},

where c_{1} ≠ 0 and s_{0} are some constants.

Theorem 3.1. [7] Let M be a hypersurface in 𝔼_{1}^{4} with diagonalizable shape operator and three distinct principal curvatures. Then M is biconservative if and only if it is congruent to one of hypersurfaces

(1) A generalized cylinder M_{0}^{2} × 𝔼_{1}^{1} where M is a biconservative surface in 𝔼^{3};

(2) A generalized cylinder M_{0}^{2} × 𝔼^{1} where M is a biconservative Riemannian surface in 𝔼_{1}^{3};

(3) A generalized cylinder M_{1}^{2} × 𝔼^{1}, where M is a biconservative Lorentzian surface in 𝔼_{1}^{3};

(4) A Rimennian surface given by

(3.2)  x(s, t, u) = (s cosht, s sinht, f_{1}(s) cosu, f_{1}(s) sinu)

for a function f_{1} satisfying

\frac{f′′_{1}}{f_{1}^{′2} − 1} = \frac{f_{1}f′_{1} + s}{sf_{1}} ;

(5) A Lorentzian surface with the parametrization given in (3.2) for a function f_{1} satisfying

\frac{−3f′′_{1}}{f_{1}^{′2} − 1} = \frac{f_{1}f′_{1} + s}{sf_{1}} ;

(6) A Rimennian surface given by

(3.3)  x(s, t, u) = (s sinht, s cosht, f_{2}(s) cosu, f_{2}(s) sinu)

for a function f_{2} satisfying

\frac{f′′_{2}}{f_{2}^{′2} + 1} = \frac{f_{2}f′_{2} + s}{sf_{2}} ;

(7) A surface given in Proposition 3.1.

Recently, Kumari studied biconservative hypersurfaces with nondiagonalizable shape operator and obtain the following result, [10].

Theorem 3.2. [10] Let M_{1}^{n} in 𝔼_{1}^{n+1} be a biconservative Lorentz hypersurface having nondiagonalizable shape operator with complex eigenvalues and with at most five distinct principal curvatures. Then M_{1}^{n} has constant mean curvature.

3.2. Biconservative hypersurfaces in the pseudo-Euclidean space 𝔼_{2}^{5}. In [14], the author and Upadhyay studied biconservative hypersurfaces with index 2 in the pseudo-Euclidean space 𝔼_{2}^{5}. They obtained the following result.

Theorem 3.3. [14] Let M be an oriented hypersurface of index 2 in the pseudo-Euclidean space 𝔼_{2}^{5}. Assume that its shape operator has the form

S = diag(k_{1}, k_{2}, k_{2}, k_{4}), k_{4} ≠ k_{2}

for some nonvanishing smooth functions k_{1}, k_{2}, k_{4}. Then, it is congruent to one of the following eight type of hypersurfaces for some smooth functions ϕ_{1} = ϕ_{1}(s) and ϕ_{2} = ϕ_{2}(s).

(1) x(s, t, u, v) = (ϕ_{2} sinhv, ϕ_{1} cosht, ϕ_{1} sinht cosu, ϕ_{1} sinht sinu, ϕ_{2} coshv) ,

ϕ_{1}^{′2} − ϕ_{2}^{′2} = 1;
(2) x(s, t, u, v) = (ϕ_{2} cosv, ϕ_{2} sinv, ϕ_{1} cost, ϕ_{1} sint cosu, ϕ_{1} sint sinu) ,
ϕ_{1}^{′2} − ϕ_{2}^{′2} = −1;
(3) x(s, t, u, v) = (ϕ_{1} cosht sinu, ϕ_{1} cosht cosu, ϕ_{1} sinht, ϕ_{2} cosv, ϕ_{2} sinv) ,
ϕ_{1}^{′2} − ϕ_{2}^{′2} = 1;

(4) x(s, t, u, v) = (ϕ_{2} sinhv, ϕ_{1} sinht, ϕ_{1} cosht cosu, ϕ_{1} cosht sinu, ϕ_{2} coshv) ,

ϕ_{1}^{′2} + ϕ_{2}^{′2} = 1;

(5) x(s, t, u, v) = (ϕ_{2} coshv, ϕ_{1} sinht, ϕ_{1} cosht cosu, ϕ_{1} cosht sinu, ϕ_{2} sinhv) ,

ϕ_{1}^{′2} − ϕ_{2}^{′2} = −1;

(6) x(s, t, u, v) = (ϕ_{1} sinhtcosu, ϕ_{1} sinht sinu, ϕ_{1} coshu, ϕ_{2} cosv, ϕ_{2} sinv) , ϕ_{1}^{′2} + ϕ_{2}^{′2} = 1;

(7) A hypersurface given by

x(s, t, u, v) = ( \frac{s}{2}(t^{2} + u^{2} − v^{2}) − av^{2} + ψ, v(2a + s), st, su,

\frac{s}{2}(t^{2} + u^{2} − v^{2}) − av^{2} + ψ − s )

for a nonzero constants a and a smooth function ψ = ψ(s) such that 1 − 2ψ′ < 0;

(8) A hypersurface given by

x(s, t, u, v) = ( \frac{s(t^{2} − u^{2} − v^{2})}{2} + av^{2} + ψ, st, su, v(s − 2a),

\frac{s(t^{2} − u^{2} − v^{2})}{2} + av^{2} + ψ + s )

for a nonzero constants a and a smooth function ψ = ψ(s̃) such that 1 + 2ψ′ < 0.

3.3. Biconservative hypersurfaces in the Riemannian space-form ℍ^{4}. In this subsection, we want to announce the biconservative hypersurfaces in ℍ^{4} that was recently obtained in a joint work with Upadhyay in [13].

If M is a hypersurface in a 4-dimensional Riemannian space-form, then it is biconservative if and only if the equation

(3.4)  S(∇H) = −2H∇H

is satisfied, where S is the shape operator of M.

Example 3.1. Consider the hypersurface in ℍ^{4} given by

x(s, t, u) = ( \frac{aA(s)^{2} + a}{s} + asu^{2} + \frac{s}{4a}, su, A(s) cost, A(s) sint, (3.5)

\frac{aA(s)^{2} + a}{s} + asu^{2} − \frac{s}{4a} )

for a smooth nonvanishing function A. We would like to note that if A is chosen properly then the hypersurface given by (3.5) satisfies (3.4).

Example 3.2. Consider the hypersurface in ℍ^{4} given by

x(s, t, u) = ( \frac{aA(s)^{2}}{s} + as(t^{2} + u^{2}) + \frac{s}{4a} + \frac{a}{s}, st, su, A(s), (3.6)

\frac{aA(s)^{2}}{s} + as(t^{2} + u^{2}) − \frac{s}{4a} + \frac{a}{s} )

for a smooth nonvanishing function A. A direct computation yields that M has two distinct principle curvatures. Furthermore, if A is chosen properly then the hypersurface given by (3.6) satisfies (3.4).

4. Biconservative hypersurfaces in Minkowski spaces

In this section, we get new classifications of biconservative hypersurfaces in Minkowski spaces. Note that a hypersurface in the Minkowski space 𝔼^{n+1} is biconservative if and only if the differential equation

(4.1)  S(∇H) = \frac{−εnH}{2} ∇H

is satisfied, where ε = ⟨N, N⟩.

4.1. A classification of biconservative hypersurfaces with more than 3 distinct principle curvatures. In this subsection, we construct an example of biconservative hypersurfaces with arbitrary number of distinct principle curvatures.

We put by a⃗_{1} · a⃗_{2} = ∑ _{i} a_{1i}a_{2i}, where a_{j} = (a_{j1} , a_{j2} , …, a_{j(n−1)} ). Consider a hypersurface M in 𝔼_{1}^{n+1} given by

x(s,t⃗) = ( \frac{1}{2}st⃗· t⃗+ a⃗ · t⃗+ s + ϕ(s), st_{1}, (s + 2a_{2})t_{2}, …, (4.2)

\frac{1}{2}st⃗· t⃗+ a⃗ · t⃗+ ϕ(s) )

for a smooth function ϕ, where t⃗ = (t_{1}, t_{2}, …, t_{n−1} ), a⃗ = (a_{1}, a_{2}, a_{3}, …, a_{n−1} ) for some constants a_{1} = 0, a_{2}, …, a_{n}. Note that

e_{1} = \frac{1}{\sqrt{−ϵ(2ϕ′(s) + 1)}} ∂_{s}, e_{l} = \frac{1}{s + a_{l−1}} ∂_{t_{l−1}} , l = 2, 3, …, n

form an orthonormal frame field and the normal vector field of M is

N = \frac{1}{\sqrt{−ϵ(2ϕ′(s) + 1)}} (t⃗· t⃗− 2ϕ′(s), st_{1}, (s + 2a_{2})t_{2}, …,t⃗· t⃗− 2ϕ′(s) − 2) ,

where we put ε = ⟨e_{1}, e_{1}⟩.

A direct computation yields that the matrix representation of the shape operator of M is

S = ( \begin{matrix}k_{1} & 0 & … & 0 \\ 0 & k_{2} & … & 0 \\ ⋮ & ⋮ & ... & ⋮ \\ 0 & 0 & … & k_{n} \end{matrix} ) ,

where

(4.3)  \begin{aligned}k_{1} & = −\frac{ϵϕ′′(s)}{(−ϵ(2ϕ′(s) + 1))^{3/2}} , \\ k_{l} & = −\frac{1}{(2a_{l−1} + s)\sqrt{−ϵ(2ϕ′(s) + 1)}} , l = 2, 3, …, n. \end{aligned}

Note that the tangent vector fields ∂_{s} is proportional to ∇H and it is a principle direction of M. Therefore, M is biconservative if and only if −2εk_{1} = nH = k_{1} + k_{2} + ⋯+ k_{n} because of (4.1). Hence, M is biconservative if and only if either ϵ = 1 and k_{1} = k_{2} + k_{3} + ⋯+ k_{n} or ϵ = −1 and −3k_{1} = k_{2} + k_{3} + ⋯+ k_{n}. By considering (4.3), we obtain the following result.

Theorem 4.1. Let M be a hypersurface in the Minkowski space 𝔼^{n+1} given by (4.2) for a smooth function ϕ. Then, M is biconservative if and only if either M is Riemannian and ϕ is the function given by

ϕ(s) = ∫_{s_{0}}^{s} \frac{c_{1}}{ξ^{2}(2a_{2} + ξ)^{2} …(2a_{n−1} + ξ)^{2}} dξ − \frac{s}{2}

for some constants c_{1} ≠ 0, s_{0} or M is Lorentzian and ϕ is the function given by

ϕ(s) = ∫_{s_{0}}^{s} c_{2}ξ^{2/3} (2a_{2} + ξ)^{2/3} …(2a_{n−1} + ξ)^{2/3} dξ − \frac{s}{2}

for a constant c_{2} ≠ 0, s_{0}.

4.2. A characterization of biconservative hypersurfaces with nondiagonalizable shape operator. In this subsection, we give a characterization of biconservative hypersurfaces in 𝔼^{n+1} with nondiagonalizable shape operator S. We focus on the case of having minimal polynomial of P (λ) = (λ − k_{1})(λ − k_{2})^{2} of S for some smooth function k_{1}, k_{2}.

We would like to note that in this case M is Lorentzian. Thus, (4.1) becomes

(4.4)  S(∇H) = \frac{−3H}{2} ∇H.

Theorem 4.2. Let M be a hypersurface in the Minkowski space 𝔼_{1}^{4} and S its shape operator with nonconstant mean curvature. Assume that the minimal polynomial of S is P (λ) = (λ − k_{1})(λ − k_{2})^{2} for some smooth function k_{1}, k_{2}. Then, M is biconservative if and only if there exists a frame field {e_{1}, e_{2}, e_{3}} on M with ⟨e_{1}, e_{1}⟩ = 1, ⟨e_{a}, e_{b}⟩ = δ_{ab} − 1, ⟨e_{1}, e_{a}⟩ = 0, a, b = 2, 3 satisfying the following conditions.

(1) The Levi-Civita connection of M is

∇_{e_{1}} e_{1} = 0,  ∇_{e_{1}} e_{2} = \frac{3(25AH^{2} + 4e_{1}(H))}{40H} e_{2},
∇_{e_{2}} e_{1} = −\frac{3e_{1}(H)}{5H} e_{2},  ∇_{e_{2}} e_{2} = 0,
(4.5) ∇_{e_{3}} e_{1} = −Ae_{2} − \frac{3e_{1}(H)}{5H} e_{3},  ∇_{e_{3}} e_{2} = −Be_{2} − \frac{3e_{1}(H)}{5H} e_{1},
∇_{e_{1}} e_{3} = −\frac{3(25AH^{2} + 4e_{1}(H))}{40H} e_{3}, ∇_{e_{2}} e_{3} = −\frac{3e_{1}(H)}{5H} e_{1},

∇_{e_{3}} e_{3} = −Ae_{1} + Be_{3},

for some functions A, B and H satisfying

e_{2}(H) = e_{3}(H) = e_{2}(A) = 0,
40He_{1}e_{1}(H) − 64e_{1}(H)^{2} + 225H^{4} = 0,
(4.6)  \frac{9e_{1}(H)^{2}}{25H^{2}} + \frac{81H^{2}}{16} = e_{2}(B),
\frac{3e_{1}(H)}{5H} A − \frac{3}{4}H(5A^{2} + 2) = e_{1}(A),

−75H^{2}AB + 12e_{1}(H)B = 40He_{1}(B) + 75H^{2}e_{3}(A).

(2) The shape operator of M has the matrix representation

(4.7)  S = ( \begin{matrix}−\frac{3H}{2} & 0 & 0 \\ 0 & \frac{9H}{4} & 1 \\ 0 & 0 & \frac{9H}{4} \end{matrix} )

Proof. In order to prove the neessary condition, we assume that M is a biconservative hypersurface in 𝔼_{1}^{n+1} with the minimal polynomial P (λ) = (λ − k_{1})(λ − k_{2})^{2} for some smooth function k_{1}, k_{2}. Furthermore, the results obtained in [12] yields that ∇H can not be light-like. Therefore, (4.1) implies that e_{1} = \frac{∇H}{⟨∇H,∇H⟩^{1/2}} is an eigenvector of S with corresponding eigenvalue k_{1} = −3H/2, where H is the mean curvature of M and we have X(H) = 0 whenever ⟨X, e_{1}⟩ = 0. Moreover, because of (4.4), if a frame field {e_{1}, e_{2}, e_{3}} with ⟨e_{1}, e_{1}⟩ = 1, ⟨e_{a}, e_{b}⟩ = δ_{ab} − 1, ⟨e_{1}, e_{a}⟩ = 0, a, b = 2, 3 is chosen properly, then the matrix representation of S becomes as given in (4.7) (See [11]), where H is the mean curvature of M. Note that we have e_{2}(H) = e_{3}(H) = 0.

The Levi-Civita connection ∇ of M satisfies

∇_{e_{i}} e_{1} = −ω_{13}(e_{i})e_{2} − ω_{12}(e_{i})e_{3},
(4.8)  ∇_{e_{i}} e_{2} = −ω_{12}(e_{i})e_{1} − ω_{23}(e_{i})e_{2},
∇_{e_{i}} e_{3} = −ω_{13}(e_{i})e_{1} + ω_{23}(e_{i})e_{3},

where we put ω_{ij} (e_{k}) = ⟨∇_{e_{k}} e_{i}, e_{j} ⟩.

We apply the Codazzi equation (2.2) for X = e_{i}, Y = e_{j}, Z = e_{k} for X = e_{i}, Y = e_{j}, Z = e_{k} for different choices of (i, j, k) and combine equations obtained with (4.8) to get

(4.9)  \begin{aligned}ω_{12}(e_{1}) & = ω_{13}(e_{1}) = 0, & ω_{23}(e_{2}) & = ω_{12}(e_{2}) = 0, \\ ω_{12}(e_{3}) & = \frac{3e_{1}(H)}{5H} , & ω_{23}(e_{1}) & = −\frac{3 (25H^{2}ω_{13}(e_{3}) + 4e_{1}(H))}{40H} \end{aligned}

By combining (4.9) with (4.8), we obtain (4.5) for A = ω_{13}(e_{3}) and B = ω_{23}(e_{3}).

Next, by taking into account (4.5), we use Gauss equation (2.1) for X = e_{i}, Y = e_{j}, Z = e_{k} for different triplets of (i, j, k) to get (4.6). □

Remark 4.1. We would like to note that obtaining hypersurfaces given in Theorem 4.2 is still an open problem. However, it was proved in [1] that there is no biharmonic hypersurface in the Minkowski space-time 𝔼_{1}^{4} with nondiagonalizable shape operator.

5. Gauss map of hypersurfaces in 𝔼_{1}^{4} and ℍ^{4}

In this section, we consider some of hypersurface families mentioned in Section 3. We get some classification results considering their Gauss map and obtain hypersurfaces whose Gauss map N satisfies (2.3).

5.1. Gauss map of hypersurfaces in the Minkowski space 𝔼_{1}^{4}. In [5], Dursun studied rotational hypersurfaces with pointwise 1-type Gauss map in a Minkowski space with arbitrary dimension. In this subsection, we firstly obtain the following theorem by considering Gauss map of hypersurfaces in 𝔼_{1}^{4} given by (3.2).

Theorem 5.1. Let M be a hypersurface in 𝔼_{1}^{4} given by (3.2) for a smooth function f_{1}. Then, M has pointwise 1-type Gauss map if and only if it has constant mean curvature.

Proof. Let M be a hypersurface in 𝔼_{1}^{4} given by (3.2). We put f = f_{1}. Note that tangent vector fields given by

e_{1} = \frac{1}{\sqrt{ε (f′(s)^{2} − 1)}} ∂_{s}, e_{2} = \frac{1}{s}∂_{t}, e_{3} = \frac{1}{f(s)} ∂_{u}

form an orthonormal frame field for the tangent space of M and the unit normal vector field of M is

(5.1)  N = \frac{1}{\sqrt{ε(f′(s)^{2} − 1)}} (coshtf′(s), sinhtf′(s), cosu, sinu) ,

where we put ε = ⟨e_{1}, e_{1}⟩. By a direct computation, we obtain

∇_{e_{1}} e_{i} = 0, i = 1, 2, 3,
(5.2)  ∇_{e_{2}} e_{1} = Ae_{2},  ∇_{e_{2}} e_{2} = −εAe_{1}, ∇_{e_{2}} e_{3} = 0,
∇_{e_{3}} e_{1} = Be_{3},  ∇_{e_{3}} e_{2} = 0,  ∇_{e_{3}} e_{3} = −Be_{1}

for smooth functions A = \frac{1}{s\sqrt{ε (f′(s)^{2} − 1)}} and B = \frac{f′(s)}{f(s)\sqrt{ε (f′(s)^{2} − 1)}} . Fur-

thermore, the shape operator of M becomes

(5.3)  S = ( \begin{matrix}\frac{εf′′(s)}{(ε(f′(s)^{2} −1))^{3/2}} & 0 & 0 \\ 0 & −\frac{f′(s)}{s\sqrt{ε(f′(s)^{2} −1)}} & 0 \\ 0 & 0 & −\frac{1}{f(s)\sqrt{ε(f′(s)^{2} −1)}} \end{matrix}  )

By considering (5.1), (5.2) and (5.3), we obtain

(5.4)  ΔN = e_{1}(k_{1} + k_{2} + k_{3})e_{1} + (k_{1}^{2} + k_{2}^{2} + k_{3}^{2})N,

where k_{1}, k_{2}, k_{3} are principle curvatures of M given in (5.3). Note that the mean curvature of M is H = \frac{1}{3}(k_{1} + k_{2} + k_{3}).

Now, in order to prove the necessary condition, we assume that (2.3) is satisfied for a constant vector C and a smooth function ψ. Then, by considering (5.4), we obtain

(5.5)  3e_{1}(H)e_{1} + (k_{1}^{2} + k_{2}^{2} + k_{3}^{2})N = ψ(N + C)

which yields ⟨e_{a}, C⟩ = 0, a = 2, 3. By applying e_{2} and e_{3} to this equation, we get ⟨ ˜∇_{e_{2}} e_{2}, C⟩ = ⟨ ˜∇_{e_{3}} e_{3}, C⟩ = 0. By considering (5.2), we obtain C = 0. Therefore, (5.5) becomes 3e_{1}(H)e_{1} + (k_{1}^{2} + k_{2}^{2} + k_{3}^{2})N = ψN which implies e_{1}(H) = 0. Hence, we obtain that H is constant.

Proof of the sufficient condition follows from (5.4). □

By a similar way, we also obtain

Theorem 5.2. Let M be a hypersurface in 𝔼_{1}^{4} given by (3.3) for a smooth function f_{1}. Then, M has pointwise 1-type Gauss map if and only if it has constant mean curvature.

Proof. Let M be a hypersurface given by (3.3). Then, similar to hypersurfaces given by (3.2), we have

∇_{e_{1}} e_{i} = 0,  i = 1, 2, 3,
∇_{e_{2}} e_{1} = Ae_{2}, ∇_{e_{2}} e_{2} = εAe_{1}, ∇_{e_{2}} e_{3} = 0,
∇_{e_{3}} e_{1} = Be_{3}, ∇_{e_{3}} e_{2} = 0,  ∇_{e_{3}} e_{3} = −Be_{1}

for smooth functions

A = \frac{1}{s\sqrt{f′(s)^{2} + 1}} , B = \frac{f′(s)}{f(s)\sqrt{f′(s)^{2} + 1}} ,

where

e_{1} = \frac{1}{\sqrt{f′(s)^{2} + 1}} ∂_{s}, e_{2} = \frac{1}{s}∂_{t}, e_{3} = \frac{1}{f(s)} ∂_{u}.

Furthermore, the shape operator of M becomes

(5.6)  S = ( \begin{matrix}−\frac{f′′(s)}{(f′(s)^{2} +1)^{3/2}} & 0 & 0 \\ 0 & −\frac{f′(s)}{s\sqrt{f′(s)^{2} +1}} & 0 \\ 0 & 0 & \frac{1}{f(s)\sqrt{f′(s)^{2} +1}} . \end{matrix})

By a further computation, we see that (5.4) is satisfied for k_{1}, k_{2}, k_{3} given in (5.6).

In order to prove the necessary condition, we use exactly the same way that we did in the proof of Theorem 5.1 and we obtain that if (5.4) is satisfied, then C must be zero. Furthermore, similar to Theorem 5.1, the proof of the sufficient condition follows from (5.4). □

Remark 5.1. The author would like to announce that he has recently obtained analogous results for the hypersurface families given in cases (1)–(6) of Theorem 3.3.

Now, we want to consider Gauss map of hypersurface family given by (3.1). Let M be a hypersurface given by (3.1) for a constant a and a smooth function

ϕ. We consider the local orthonormal frame field {e_{1}, e_{2}, e_{3}} of the tangent bundle of M such that

e_{1} = \frac{1}{\sqrt{−ε(2ϕ′(s) + 1)}} ∂_{s}, e_{2} = \frac{1}{s}∂_{t}, e_{3} = \frac{1}{2a + s} ∂_{u}

and the unit normal vector field of M is

N = \frac{1}{\sqrt{−ε(2ϕ′(s) + 1)}} ( \frac{t^{2} + u^{2}}{2} − ϕ′(s), t, u, \frac{t^{2} + u^{2}}{2} − ϕ′(s) − 1) ,

where we put ε = ⟨e_{1}, e_{1}⟩ = ±1. We want to note the equation

(5.7)  (1, 0, 0, 1) = −\frac{ε}{\sqrt{−ε (2ϕ′(s) + 1)}} (e_{1} − N).

By a direct computation, we obtain that the Levi-Civita connection satisfies (5.2) for some smooth functions

A = \frac{1}{s\sqrt{−ε (2ϕ′(s) + 1)}} , B = \frac{1}{(2a + s)\sqrt{−ε (2ϕ′(s) + 1)}} .

Furthermore, the shape operator of M becomes

S = ( \begin{matrix}−\frac{εϕ′′(s)}{(−ε(2ϕ′(s)+1))^{3/2}} & 0 & 0 \\ 0 & −\frac{1}{s\sqrt{−ε(2ϕ′(s)+1)}} & 0 \\ 0 & 0 & −\frac{1}{(2a+s)\sqrt{−ε(2ϕ′(s)+1)}} \end{matrix}  )

We also put

(5.8)  s_{1} = traceS and s_{2} = traceS^{2}.

By a further computation, we have

(5.9)  ΔN = e_{1}(s_{1})e_{1} + s_{2}N

(see [5, Lemma 3.2]). We first obtain the following lemma.

Lemma 5.1. Let M be a hypersurface given by (3.1) with pointwise 1-type Gauss map. Then, (2.3) is satisfied for the function

(5.10)  ψ = ψ(s) = e_{1}(s_{1}) + s_{2}

and a constant vector with the form

(5.11)  C = C_{1}(s)(e_{1} − N),

where C_{1} is an appropriately chosen function and s_{1}, s_{2} are functions given by (5.8).

Proof. If M has pointwise 1-type Gauss map, then (2.3) is satisfied for some C, ψ. From (2.3) and (5.9), we have

(5.12)  e_{1}(s_{1})e_{1} + s_{2}N = ψ(N + C)

from which we obtain

(5.13)  C = C_{1}e_{1} + C_{4}N.

Since C is constant, we have e_{a}(C) = 0, a = 2, 3. By combining this equation with (5.2) and (5.13), we obtain

e_{2}(C_{1})e_{1} + e_{2}(C_{4})N + \frac{C_{1} + C_{4}}{s\sqrt{−ε(2ϕ′(s) + 1)}} e_{2} = 0,
e_{3}(C_{1})e_{1} + e_{3}(C_{4})N + \frac{C_{1} + C_{4}}{(2a + s)\sqrt{−ε(2ϕ′(s) + 1)}} e_{3} = 0

which yields C_{1} + C_{4}=0 and e_{2}(C_{1}) = e_{3}(C_{1}) = 0. Therefore, we obtained (5.11). Now, (5.12) becomes e_{1}(s_{1})e_{1} + s_{2}N = ψ(N + C_{1}(s)(e_{1} − N)) from which we have e_{1}(s_{1}) = ψC_{1}(s) and s_{2} = ψ(1 − C_{1}(s)). By combining these equations, we get (5.10). □

Theorem 5.3. Let M be a hypersurface given by (3.1). Then M has pointwise 1-type Gauss map if and only if the differential equation

(5.14)  s′_{1} = c(e_{1}(s_{1}) + s_{2})

is satisfied for a constant c.

Proof. Let M has pointwise 1-type Gauss map. Then, (2.3) is satisfied for the constant vector C and smooth function ψ given in Lemma 5.1. Since C is proportional to e_{1} − N, (5.7) implies C = (−εc, 0, 0, −εc) for a constant c. Thus, (5.7) implies

C = \frac{c}{\sqrt{−ε (2ϕ′(s) + 1)}} (e_{1} − N).

Therefore, we have C_{1} = \frac{c}{\sqrt{−ε(2ϕ′(s)+1)}} . Hence, e_{1}(s_{1}) = ψC_{1}(s) becomes

e_{1}(s_{1}) = \frac{c(e_{1}(s_{1}) + s_{2})}{\sqrt{−ε (2ϕ′(s) + 1)}}

which gives (5.14).

The proof of the sufficient condition follows from a direct computation. □

Remark 5.2. We announce that he have recently generalized the result obtained in the previous theorem by considering hypersurfaces given by (4.2) with pointwise 1-type Gauss map.

5.2. Hyperbolic Gauss map of hypersurfaces in ℍ^{4}. In this subsection, we consider a family of hypersurfaces in ℍ^{4} considering their mean curvature and hyperbolic Gauss map N. Let M be a hypersurface in ℍ^{4} given by (3.5) for a smooth function A. We consider the local orthonormal frame field {e_{1}, e_{2}, e_{3}} of the tangent bundle of M such that

e_{1} = \frac{s}{\sqrt{(A(s) − sA′(s))^{2} + 1}} ∂_{s}, e_{2} = \frac{1}{A(s)}∂_{t}, e_{3} = \frac{1}{s}∂_{u}.

Note that we have

N = \frac{1}{\sqrt{(A − sA′)^{2} + 1}} ( 2aAA′− \frac{a}{s}(A^{2} + 1) + asu^{2} + \frac{s}{4a}, su, sA′cost,

sA′sint, 2aAA′− \frac{a}{s}(A^{2} + 1) + asu^{2} − \frac{s}{4a} ) .

By a direct computation, we obtain that the principle directions of M are

(5.15)  \begin{matrix}k_{1} = \frac{A(3s^{2}A^{′2} + 1) − 3sA^{2}A′− s(sA′′+ s^{2}A^{′3} + A′) + A^{3}}{((A − sA′)^{2} + 1)^{3/2}}  , \\ k_{2} = \frac{−sAA′+ A^{2} + 1}{A\sqrt{(A − sA′)^{2} + 1}} , k_{3} = \frac{A − sA′}{\sqrt{(A − sA′)^{2} + 1}} . \end{matrix}

Furthermore the Laplacian of the Gauss map N is

(5.16)  ΔN = e_{1}(k_{1} + k_{2} + k_{3})e_{1} + (k_{1}^{2} + k_{2}^{2} + k_{3}^{2})N + (k_{1} + k_{2} + k_{3})x.

By combining (5.15) and (5.16), we obtain the following theorem.

Theorem 5.4. Let M be a hypersurface in ℍ^{4} given by (3.5) for a smooth function A. Then, the following statements are equivalent to each other

(1) M has pointwise 1-type hyperbolic Gauss map of the first kind, i.e., N satisfies (2.3) for C = 0;

(2) M is minimal;

(3) A = A(s) satisfies

−s^{2}AA′′− 3s^{3}AA^{′3} + (9s^{2}A^{2} + s^{2}) A^{′2} + (−9sA^{3} − 5sA) A′+ 3A^{4} + 4A^{2} + 1 = 0.

Now, let M be a hypersurface given in (3.6) for a smooth function A. Note that the principle curvatures of M are

(5.17)  \begin{matrix}k_{1} = \frac{A(3s^{2}A′^{2} + 1) − 3sA^{2}A′− s(sA′′+ s^{2}A′^{3} + A′) + A^{3}}{((A − sA′)^{2} + 1)^{3/2}}  , \\ k_{2} = k_{3} = \frac{A − sA′}{\sqrt{(A − sA′)^{2} + 1}} . \end{matrix}

Also the Laplacian of the Gauss map N takes the form given in (5.16). By combining (5.16) and (5.17), we obtain the following theorem.

Theorem 5.5. Let M be a hypersurface in ℍ^{4} given by (3.6) for a smooth function A. Then the following statements are equivalent to each other

(1) M has pointwise 1-type hyperbolic Gauss map of the first kind, i.e., N satisfies (2.3) for C = 0;

(2) M is minimal;

(3) A = A(s) satisfies

−s^{2}A′′− 3s^{3}A′^{3} + 9s^{2}AA′^{2} + (−9sA^{2} − 3s) A′+ 3A^{3} + 3A = 0.

Acknowledgements. This work was presented at the XIX Geometrical seminar held in Zlatibor, Serbia in 2016. The author would like to express his deep gratitude for the warm hospitality of the local committee of the conference. The author also wants to note that he is supported by The Scientific and Technological Research Council of Turkey (TÜBİTAK) (Project Name: ‘Y_EUCL2TIP’ Project Number: 114F199).

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Istanbul Technical University Faculty of Science and Letters Department of Mathematics Istanbul Turkey

 turgayn@itu.edu.tr