In 1991, Chen and Ishikawa initially studied biharmonic marginally trapped surfaces in neutral pseudo-Euclidean 4-space. Recently, biharmonic and quasi-biharmonic marginally trapped Lagrangian surfaces in Lorentzian complex space forms were studied by Sasahara in 2007 and 2011, respectively. In this paper we extend Sasahara′s results to the case of slant surfaces in Lorentzian complex space forms. By results, we completely classify biharmonic marginally trapped slant surfaces and quasi-biharmonic marginally trapped slant surfaces in Lorentzian complex space forms.

1. Introduction

Let

be a simply connected Lorentzian complex space form of complex dimension n and complex index i (i ≥ 0), where the complex index is defined as the complex dimension of the largest complex negative definite subspace of the tangent space. In particular, if i = 1, we say that

is Lorentzian. The curvature tensor

is given by

()

Let ℂⁿ denote the complex number n-space with complex coordinates z₁, …, z_n. The ℂⁿ endowed with g_s,n, that is, the real part of the Hermitian form

()

defines a flat indefinite complex space form with complex index s. Denote the pair (ℂⁿ, g_s,n) by

briefly, which is the flat Lorentzian complex n-space. In particular,

is the flat Lorentzian complex plane.

Let us consider the differentiable manifold:

()

which is an indefinite real space form of constant sectional curvature c. The Hopf fibration

()

is a submersion and there exists a unique pseudo-Riemannian matrix of complex index one on

such that π is a Riemannian submersion.

The pseudo-Riemannian manifold is a Lorentzian complex space form of positive holomorphic sectional curvature 4c.

Analogously, if c < 0, consider

()

which is an indefinite real space form of constant sectional curvature c. The Hopf fibration

()

is a submersion and there exists a unique pseudo-Riemannian matrix of complex index one on

such that π is a Riemannian submersion.

The pseudo-Riemannian manifold is a Lorentzian complex space form of negative holomorphic sectional curvature 4c.

It is well known that a complete simply connected complex space form is holomorphic isometric to , , or , according to c = 0, c > 0, or c < 0, respectively.

A real surface in a Kähler surface with almost complex structure J is called slant if its Wirtinger angle is constant (see [1–3]). From J-action point of views, slant surfaces are the simplest and the most natural surfaces of a Lorentzian Kähler surface . It should be pointed out that slant surfaces arise naturally and play important roles in the studies of surfaces of Kähler surfaces in the complex space forms; see [4].

In last years, the geometry of Lorentzian surfaces in Lorentzian complex space forms has been studied by a series of papers given by Chen and other geometers, for instance, [1, 3, 5–13]. Lorentzian geometry is a vivid field of mathematical research that represents the mathematical foundation of the general theory of relativity—which is probably one of the most successful and beautiful theories of physics. For Lorentzian surfaces immersed in Lorentzian complex space forms, Chen [7] proved that Ricci equation is a consequence of Gauss and Codazzi equations, which indicates that Lorentzian surfaces in Lorentzian complex space forms have many interesting properties.

During the last decade, the theory of biharmonic submanifolds has advanced greatly. By definition, a submanifold is called biharmonic if the bitension field of the isometric immersion defining the submanifold vanishes identically. There are a lot classification results and nonexistence results, (see, e.g., [4, 14, 15]). Recently, Sasahara introduces the notion of quasi-biharmonic submanifold in [16], which is defined with the property that the bitension field of the isometric immersion defining the submanifold is lightlike at each point. It is shown in [16] that the class of quasi-biharmonic submanifolds is quite different from the class of biharmonic submanifolds.

A surface of a pseudo-Riemannian manifold is called marginally trapped (or quasiminimal) if its mean curvature vector field is lightlike. In the theory of cosmic black holes, a marginally trapped surface in a space-time plays an extremely important role. From the viewpoint of differential geometry, some classification results on marginally trapped surfaces have been obtained by some geometers (see [1, 3, 9–12]). In particular, Chen and Dillen [9] gave a complete classification of marginally trapped Lagrangian surfaces in Lorentzian complex space forms.

In this paper, we investigate the bitension field of marginally trapped slant surfaces in Lorentzian complex space forms. In particular, we completely classify biharmonic marginally trapped slant surfaces and quasi-biharmonic marginally trapped slant surfaces in Lorentzian complex space forms, respectively (see Theorems 12 and 13). Our classification results extend Sasahara’s results from Lagrangian case to the slant case in Lorentzian complex space forms.

2. Preliminaries

2.1. Basic Notation, Formulas, and Definitions

Let M be a Lorentzian surface of a Lorentzian Kähler surface equipped with an almost structure J and metric . Let 〈, 〉 denote the inner product associated with .

We denote the Levi-Civita connections of M and

by ∇ and

, respectively. Gauss formula and Weingarten formula are given, respectively, by (see [1, 2])

()

for vector fields X, Y tangent to M and ξ normal to M, where h, A, and D are the second fundamental form, the shape operator, and the normal connection. It is well known that the second fundamental form h and the shape operator A are related by

()

for X, Y tangent to M and ξ normal to M.

A vector v is called spacelike (timelike) if 〈v, v〉>0 or 〈v, v〉 = 0(〈v, v〉<0). A vector v is called lightlike if it is nonzero and it satisfies 〈v, v〉 = 0.

We define the light cone by . A curve w(t) is called null if w^′ is lightlike for any t.

For each normal vector ξ of M at x ∈ M, the shape operator A_ξ is a symmetric endomorphism of the tangent space T_xM. The mean curvature vector is defined by

()

A Lorentzian surface M in

is called minimal if its mean curvature vector H vanishes at each point on M. And a Lorentzian surface M in

is called marginally trapped (or quasiminimal) if its mean curvature vector is lightlike at each point on M.

For a Lorentzian surface M in a Lorentzian complex space form

, the Gauss and Codazzi and Ricci equations are given, respectively, by

()

where X, Y, Z, and W are vectors tangent to M and

is defined by

()

2.2. Bitension Field

For smooth maps

, the tension field τ(ϕ) is a section of the vector bundle

defined by

()

where ∇^ϕ is the induced connection by ϕ on the bundle

, which is the pullback of

. If ϕ is an isometric immersion, then τ(ϕ) and the mean curvature vector field H of M are related by

()

If τ(ϕ) = 0 at each point on M, then ϕ is called a harmonic map. The harmonic maps between two Riemannian manifolds are critical points of the energy functional

()

for smooth maps

The bitension field is defined by

()

where

is the curvature tensor of

If ϕ is an isometric immersion and

is the complex space form

, it follows from (1), (14), and (16) that

()

where

A smooth map ϕ is called biharmonic if τ₂(ϕ) = 0 at each point on M. It is easy to see that harmonic maps are always biharmonic.

Biharmonic maps

between Riemannian manifolds are critical points of the bienergy functional

()

Sasahara proposed the notion of quasi-biharmonic submanifolds as follows.

Definition 1. A pseudo-Riemannian submanifold isometrically immersed in a pseudo-Riemannian manifold by ϕ is called quasi-biharmonic if τ₂ is lightlike at each point on the submanifold.

3. Basic Results on Lorentzian Slant Surfaces

Let M be a Lorentzian surface in a Lorentzian Kähler surface

. For each tangent vector X of M, we put

()

where PX and FX are the tangential and the normal components of JX.

On the Lorentzian surface M there exists a pseudoorthonormal local frame {e₁, e₂} such that

()

It follows from (19), (20), and 〈JX, JY〉 = 〈X, Y〉 that

()

for some function θ. This function θ is called the Wirtinger angle of M.

When the Wirtinger angle θ is constant on M, the Lorentzian surface M is called a slant surface (cf. [2, 3]). In this case, θ is called the slant angle; the slant surface is then called θ-slant.

A θ-slant surface is called Lagrangian if θ = 0 and proper slant if θ ≠ 0.

If we put

()

then we find from (19)–(22) that

()

We call such a frame {e₁, e₂, e₃, e₄} an adapted pseudoorthonormal frame for the Lorentzian surface M in

Lemma 2. If M is a slant surface in a Lorentzian Kähler surface , then with respect to an adapted pseudoorthonormal frame one has

()

for some 1-forms ω, Φ on M.

For a Lorentzian surface M in

, we put

()

where {e₁, e₂, e₃, e₄} is an adapted pseudoorthonormal frame and h is the second fundamental form of M.

Lemma 3 (see [3].)If M is a θ-slant surface in a Lorentzian Kähler surface , then with respect to an adapted pseudoorthonormal frame one has

()

for any X, Y ∈ TM and j = 1,2, where ω_j = ω(e_j) and Φ_j = Φ(e_j).

For Lorentzian slant surfaces in , the author with Hou has proved the following interesting result.

Theorem 4 (see [17].)Every slant surface in a nonflat Lorentzian complex space form must be Lagrangian.

According to Theorem 4, we need only to consider the slant surfaces in Lorentzian complex plane because the case of Lagrangian marginally trapped surfaces has been considered in [16, 18].

4. The Bitension Field of Marginally Trapped Slant Surfaces

Let M be a θ-slant marginally trapped surface in a Lorentzian complex plane

. There is a pseudoorthonormal local frame field

such that

()

Since the mean curvature vector H is lightlike at each point, we put

for some nonzero real-valued function α. By putting

, we have

()

By applying (20) and the total symmetry of 〈h(X, Y), FZ〉, we obtain

()

for two smooth functions λ, γ. It follows from (9), (22), (25), and (33) that

()

By Lemma 2, (33), and differentiating the second fundamental form covariantly, we get

()

By the Codazzi equation, comparing coefficients gives

()

On the other hand, from Lemma 3 and (33) we have

()

Consequently, (36) becomes

()

In order to express the bitension field of marginally trapped slant surfaces in

with respect to a pseudoorthonormal frame (31), we need the following formula (cf. [1, 18]):

()

where Δ and Δ^D are, respectively, given by

()

Lemma 5. Let M be a marginally trapped slant surface in . Then, the Gauss curvatures K and Δ^DH are related by

()

Proof. It follows from (33) that the mean curvature vector H is given by H = −Fe₁. By (22), (26), (27), (36), and (37), we have

()

Consequently, we obtain

()

Recall the definition of the Gauss curvature K = −〈R(e₁, e₂)e₂, e₁〉. It follows from Lemma 2 and ω₁ = 0 that

()

which completes the proof of Lemma 5.

Remark 6. For marginally trapped Lagrangian surfaces immersed into Lorentzian complex space forms, Sasahara [15] has proved the formula Δ^DH = −KH. Hence, we know from Lemma 5 that the formula Δ^DH = −KH also holds for slant surfaces in Lorentzian complex space forms.

Lemma 7. Let M be a marginally trapped slant surface in . Then, the normal part of ΔH is expressed as

()

Proof. On one hand, it follows from (34) that

()

Combining (46) with (33) gives

()

On the other hand, it follows from (33) and the Gauss equation that the Gauss curvature is given by

()

By Lemma 5, we have Δ^DH = cosh ² θλγFe₁. Combining these with the first section of (39) completes the proof.

Lemma 8. Let M be a marginally trapped slant surface in . Then, the tangential part of ΔH is expressed as

()

Proof. By (27), (36), and (46), we obtain

()

It follows from (26), (38), and (46) that

()

Combing (50)-(51) with (37) gives the conclusion.

Hence, by (17) and Lemmas 7 and 8, we get the expression of the bitension field of marginally trapped slant surfaces in Lorentzian complex plane .

Lemma 9. Let be a marginally trapped slant immersed in . Then the bitension field is given by

()

5. Classification Results

From now on, let us consider the biharmonic and quasi-biharmonic Marginally trapped slant surfaces in Lorentzian complex plane .

By the definition of biharmonic surfaces, we can conclude the following from Lemma 9.

Lemma 10. Let be a marginally trapped slant immersed in . Then the immersion is biharmonic if and only if the function λ satisfies λ = 0.

Similarly, by the definition of quasi-biharmonic submanifolds, the bitension field τ₂(ϕ) is lightlike. Therefore, we also have the following.

Lemma 11. Let be a marginally trapped slant immersed in . Then the immersion is quasi-biharmonic if and only if the functions λ and γ satisfy λ ≠ 0 and γ = 0.

Since the Gauss curvature K is given by (48), we deduce from Lemmas 10 and 11 that K = 0 in both cases. Moreover, (44) implies that e₁(ω₂) = 0.

We deduce from (26) and (38) that [e₁, e₂] = −ω₂e₁. There is a nonzero smooth function β satisfying

()

such that [βe₁, e₂] = 0. Thus, there exist local coordinates (x, y) on M such that ∂/∂x = βe₁ and ∂/∂y = e₂. Then the metric tensor of M is given by

()

and the Levi-Civita connection of g satisfies

()

Moreover, it follows from (33) and (54) that

()

Since e₁(ω₂) = 0, it follows that ω₂ is a function depending only on y; that is, ω₂ = ω₂(y). Using local coordinates, (53) becomes

()

Solving (57) gives

()

Without loss of generality, we may assume that β is only depending on variable y.

By applying (23), (55), (56), (58), and Gauss formula (7), we have the following PDE system:

()

By solving (60), we obtain that

()

for some vector-valued functions f(x) and z(y) in

Theorem 12. Up to rigid motions of , every biharmonic marginally trapped θ-slant surface in is given by a flat slant surface defined by

()

where c₁ is lightlike vector and w(y) is a null curve in

satisfying

()

for some nonzero real-valued function β(y).

Proof. Let M be a biharmonic marginally trapped θ-slant surface in . According to Lemma 10, we have λ = 0. Substituting (62) into (59), we have

()

which yields f(x) = c₁x + c₂ for two constant vectors c₁ and c₂ in

. Hence, the immersion becomes

()

Note that w(y) = c₂e^{(i−sinh θ)y} + z(y) here. Moreover, (66) yields

()

It follows from (54) and (67) that

()

This completes the proof of Theorem 12.

Theorem 13. Up to rigid motions of , every quasi-biharmonic marginally trapped θ-slant surface in is given by a flat slant surface defined by

()

where p(x) is a null curve lying in the light cone ℒ𝒞 satisfying

()

Proof. Let M be a quasi-biharmonic marginally trapped θ-slant surface in . According to Lemma 11, we have γ = 0 and λ ≠ 0. In this case, the second and third equations of (38) become

()

Solving λ, we get

()

By applying (71), (58) yields

()

for some nonzero constant number a. Consequently, with the previous information, the PDE system (59)–(61) becomes

()

Substituting (62) into (76), we have

()

which yields

()

for two constant vectors c₂ and c₃ in

. We denote p(x) by p(x) = f(x) + c₂. Hence, up to rigid motions of

the immersion becomes

()

Substituting (79) into (74), we obtain

()

Moreover, (79) gives

()

It follows from (54), (73), and (81) that

()

In view of (73), we can assume a = 1. Hence, we get the conclusion.

Remark 14. According to Theorem 4, combining Theorems 12 and 13 with Sasahara’s results in [16, 18], we finish the complete classifications of biharmonic marginally trapped slant surfaces and quasi-biharmonic marginally trapped slant surfaces in Lorentzian complex forms, respectively.

Acknowledgments

This work is supported by the Natural Science Foundation of China (no. 71271045), Program for Liaoning Excellent Talents in University (no. LJQ2012099), and General Project for Scientific Research of Liaoning Educational Committee (no. W2012186).

References

1 Chen B.-Y., Pseudo-Riemannian Geometry, δ-Invariants and Applications, 2011, World Scientific Publishing, Hackensack, NJ, USA.
10.1142/8003
Google Scholar
2 Chen B.-Y., Geometry of Slant Submanifolds, 1990, Katholieke Universiteit Leuven, Leuven, Belgium, MR1099374.
Google Scholar
3 Chen B. Y. and Mihai I., Classification of quasi-minimal slant surfaces in Lorentzian complex space forms, Acta Mathematica Hungarica. (2009) 122, no. 4, 307–328, https://doi.org/10.1007/s10474-008-8033-6, MR2481783, ZBL1199.53167.
10.1007/s10474-008-8033-6
Web of Science® Google Scholar
4 Kenmotsu K. and Zhou D., The classification of the surfaces with parallel mean curvature vector in two-dimensional complex space forms, American Journal of Mathematics. (2000) 122, no. 2, 295–317, MR1749050, https://doi.org/10.1353/ajm.2000.0012, ZBL0997.53006.
10.1353/ajm.2000.0012
Web of Science® Google Scholar
5 Arslan K., Carriazo A., Chen B.-Y., and Murathan C., On slant submanifolds of neutral Kaehler manifolds, Taiwanese Journal of Mathematics. (2010) 14, no. 2, 561–584, MR2655787, ZBL1202.53022.
10.11650/twjm/1500405807
Web of Science® Google Scholar
6 Chen B.-Y., Minimal flat Lorentzian surfaces in Lorentzian complex space forms, Publicationes Mathematicae Debrecen. (2008) 73, no. 1-2, 233–248, MR2429037, ZBL1199.53115.
10.5486/PMD.2008.4247
Web of Science® Google Scholar
7 Chen B.-Y., Dependence of the Gauss-Codazzi equations and the Ricci equation of Lorentz surfaces, Publicationes Mathematicae Debrecen. (2009) 74, no. 3-4, 341–349, MR2521379, ZBL1199.53117.
10.5486/PMD.2009.4374
Web of Science® Google Scholar
8 Chen B.-Y., Nonlinear Klein-Gordon equations and Lorentzian minimal surfaces in Lorentzian complex space forms, Taiwanese Journal of Mathematics. (2009) 13, no. 1, 1–24, MR2489305, ZBL1172.53033.
Web of Science® Google Scholar
9 Chen B.-Y. and Dillen F., Classification of marginally trapped Lagrangian surfaces in Lorentzian complex space forms, Journal of Mathematical Physics. (2007) 48, no. 1, 23, 013509, https://doi.org/10.1063/1.2424553, MR2292624, ZBL1121.53047.
10.1063/1.2424553
Web of Science® Google Scholar
10 Chen B.-Y. and Ishikawa S., Biharmonic surfaces in pseudo-Euclidean spaces, Memoirs of the Faculty of Science. Kyushu University A. (1991) 45, no. 2, 323–347, https://doi.org/10.2206/kyushumfs.45.323, MR1133117, ZBL0757.53009.
10.2206/kyushumfs.45.323
Google Scholar
11 Chen B.-Y. and Ishikawa S., Biharmonic pseudo-Riemannian submanifolds in pseudo-Euclidean spaces, Kyushu Journal of Mathematics. (1998) 52, no. 1, 167–185, https://doi.org/10.2206/kyushujm.52.167, MR1609044, ZBL0892.53012.
10.2206/kyushujm.52.167
Google Scholar
12 Chen B.-Y., Black holes, marginally trapped surfaces and quasi-minimal surfaces, Tamkang Journal of Mathematics. (2009) 40, no. 4, 313–341, MR2604833, ZBL1194.53060.
10.5556/j.tkjm.40.2009.600
Google Scholar
13 Vrancken L., Minimal Lagrangian submanifolds with constant sectional curvature in indefinite complex space forms, Proceedings of the American Mathematical Society. (2002) 130, no. 5, 1459–1466, https://doi.org/10.1090/S0002-9939-01-06213-X, MR1879970, ZBL0999.53052.
10.1090/S0002-9939-01-06213-X
Web of Science® Google Scholar
14 Montaldo S. and Oniciuc C., A short survey on biharmonic maps between Riemannian manifolds, Revista de la Unión Matemática Argentina. (2006) 47, no. 2, 1–22, MR2301373, ZBL1140.58004.
Google Scholar
15 Sasahara T., Quasi-minimal Lagrangian surfaces whose mean curvature vectors are eigenvectors, Demonstratio Mathematica. (2005) 38, no. 1, 185–196, MR2123733, ZBL1076.53075.
10.1515/dema-2005-0121
Google Scholar
16 Sasahara T., Quasi-biharmonic Lagrangian surfaces in Lorentzian complex space forms, Annali di Matematica Pura ed Applicata. (2013) 192, no. 2, 191–201, https://doi.org/10.1007/s10231-011-0218-x.
10.1007/s10231-011-0218-x
Web of Science® Google Scholar
17 Fu Y. and Hou Z. H., Classification of pseudo-umbilical slant surfaces in Lorentzian complex space forms, Taiwanese Journal of Mathematics. (2011) 15, no. 5, 1919–1938, MR2880384, ZBL1230.53052.
10.11650/twjm/1500406414
Web of Science® Google Scholar
18 Sasahara T., Biharmonic Lagrangian surfaces of constant mean curvature in complex space forms, Glasgow Mathematical Journal. (2007) 49, no. 3, 497–507, https://doi.org/10.1017/S0017089507003886, MR2371514, ZBL1132.53310.
10.1017/S0017089507003886
Web of Science® Google Scholar

All articles

Biharmonic and Quasi-Biharmonic Slant Surfaces in Lorentzian Complex Space Forms

Abstract

1. Introduction

2. Preliminaries

2.1. Basic Notation, Formulas, and Definitions

2.2. Bitension Field

3. Basic Results on Lorentzian Slant Surfaces

4. The Bitension Field of Marginally Trapped Slant Surfaces

5. Classification Results

Acknowledgments

References

References

Information

About Wiley Online Library

Help & Support

Opportunities

Connect with Wiley

Biharmonic and Quasi-Biharmonic Slant Surfaces in Lorentzian Complex Space Forms

Abstract

1. Introduction

2. Preliminaries

2.1. Basic Notation, Formulas, and Definitions

2.2. Bitension Field

3. Basic Results on Lorentzian Slant Surfaces

4. The Bitension Field of Marginally Trapped Slant Surfaces

5. Classification Results

Acknowledgments

References

References

Related

Information