Research Article | Open Access

Yu Fu, "Biharmonic and Quasi-Biharmonic Slant Surfaces in Lorentzian Complex Space Forms", *Abstract and Applied Analysis*, vol. 2013, Article ID 412709, 7 pages, 2013. https://doi.org/10.1155/2013/412709

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

**Academic Editor:**Jaeyoung Chung

#### Abstract

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 and complex index , where the complex index is defined as the complex dimension of the largest complex negative definite subspace of the tangent space. In particular, if , we say that is Lorentzian. The curvature tensor of is given by

Let denote the complex number -space with complex coordinates . The endowed with , that is, the real part of the Hermitian form defines a flat indefinite complex space form with complex index . Denote the pair by briefly, which is the flat Lorentzian complex -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 . The Hopf fibration is a submersion and there exists a unique pseudo-Riemannian matrix of complex index one on such that is a Riemannian submersion.

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

Analogously, if , consider which is an indefinite real space form of constant sectional curvature . The Hopf fibration is a submersion and there exists a unique pseudo-Riemannian matrix of complex index one on such that is a Riemannian submersion.

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

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

A real surface in a Kähler surface with almost complex structure is called slant if its Wirtinger angle is constant (see [1–3]). From -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 be a Lorentzian surface of a Lorentzian Kähler surface equipped with an almost structure and metric . Let denote the inner product associated with .

We denote the Levi-Civita connections of and by and , respectively. Gauss formula and Weingarten formula are given, respectively, by (see [1, 2]) for vector fields tangent to and normal to , where , , and are the second fundamental form, the shape operator, and the normal connection. It is well known that the second fundamental form and the shape operator are related by for tangent to and normal to .

A vector is called spacelike (timelike) if or . A vector is called lightlike if it is nonzero and it satisfies .

We define the light cone by . A curve is called null if is lightlike for any .

For each normal vector of at , the shape operator is a symmetric endomorphism of the tangent space . The mean curvature vector is defined by A Lorentzian surface in is called minimal if its mean curvature vector vanishes at each point on . And a Lorentzian surface in is called marginally trapped (or quasiminimal) if its mean curvature vector is lightlike at each point on .

For a Lorentzian surface in a Lorentzian complex space form , the Gauss and Codazzi and Ricci equations are given, respectively, by where , and are vectors tangent to and is defined by

##### 2.2. Bitension Field

For smooth maps , the tension field is a section of the vector bundle defined by where is the induced connection by on the bundle , which is the pullback of . If is an isometric immersion, then and the mean curvature vector field of are related by If at each point on , then 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 is an isometric immersion and is the complex space form , it follows from (1), (14), and (16) that where .

A smooth map is called biharmonic if at each point on . 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 is called quasi-biharmonic if is lightlike at each point on the submanifold.

#### 3. Basic Results on Lorentzian Slant Surfaces

Let be a Lorentzian surface in a Lorentzian Kähler surface . For each tangent vector of , we put where and are the tangential and the normal components of .

On the Lorentzian surface there exists a pseudoorthonormal local frame such that It follows from (19), (20), and that for some function . This function is called the Wirtinger angle of .

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

A -slant surface is called Lagrangian if and proper slant if .

If we put then we find from (19)–(22) that We call such a frame an adapted pseudoorthonormal frame for the Lorentzian surface in .

Lemma 2. *If is a slant surface in a Lorentzian Kähler surface , then with respect to an adapted pseudoorthonormal frame one has
**
for some 1-forms on . *

For a Lorentzian surface in , we put where is an adapted pseudoorthonormal frame and is the second fundamental form of .

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

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 be a -slant marginally trapped surface in a Lorentzian complex plane . There is a pseudoorthonormal local frame field such that Since the mean curvature vector is lightlike at each point, we put for some nonzero real-valued function . By putting , , we have By applying (20) and the total symmetry of , we obtain for two smooth functions , . 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 and are, respectively, given by

Lemma 5. *Let be a marginally trapped slant surface in . Then, the Gauss curvatures and are related by
*

*Proof. *It follows from (33) that the mean curvature vector is given by . By (22), (26), (27), (36), and (37), we have
Consequently, we obtain
Recall the definition of the Gauss curvature . It follows from Lemma 2 and 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 . Hence, we know from Lemma 5 that the formula also holds for slant surfaces in Lorentzian complex space forms.

Lemma 7. *Let be a marginally trapped slant surface in . Then, the normal part of 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 . Combining these with the first section of (39) completes the proof.

Lemma 8. *Let be a marginally trapped slant surface in . Then, the tangential part of 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 satisfies .*

Similarly, by the definition of quasi-biharmonic submanifolds, the bitension field 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 and satisfy and . *

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

We deduce from (26) and (38) that . There is a nonzero smooth function satisfying such that . Thus, there exist local coordinates on such that and . Then the metric tensor of is given by and the Levi-Civita connection of satisfies Moreover, it follows from (33) and (54) that Since , it follows that is a function depending only on ; that is, . Using local coordinates, (53) becomes Solving (57) gives Without loss of generality, we may assume that is only depending on variable .

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 and in .

Theorem 12. *Up to rigid motions of , every biharmonic marginally trapped -slant surface in is given by a flat slant surface defined by
**
where is lightlike vector and is a null curve in satisfying
**
for some nonzero real-valued function . *

*Proof. *Let be a biharmonic marginally trapped -slant surface in . According to Lemma 10, we have . Substituting (62) into (59), we have
which yields for two constant vectors and in . Hence, the immersion becomes
Note that 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 -slant surface in is given by a flat slant surface defined by
**
where is a null curve lying in the light cone satisfying
*

*Proof. *Let be a quasi-biharmonic marginally trapped -slant surface in . According to Lemma 11, we have and . In this case, the second and third equations of (38) become
Solving , we get
By applying (71), (58) yields
for some nonzero constant number . Consequently, with the previous information, the PDE system (59)–(61) becomes
Substituting (62) into (76), we have
which yields
for two constant vectors and in . We denote by . 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 . 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

- B.-Y. Chen,
*Pseudo-Riemannian Geometry,*, World Scientific Publishing, Hackensack, NJ, USA, 2011.*δ*-Invariants and Applications - B.-Y. Chen,
*Geometry of Slant Submanifolds*, Katholieke Universiteit Leuven, Leuven, Belgium, 1990. View at: MathSciNet - B. Y. Chen and I. Mihai, “Classification of quasi-minimal slant surfaces in Lorentzian complex space forms,”
*Acta Mathematica Hungarica*, vol. 122, no. 4, pp. 307–328, 2009. View at: Publisher Site | Google Scholar | Zentralblatt MATH | MathSciNet - K. Kenmotsu and D. Zhou, “The classification of the surfaces with parallel mean curvature vector in two-dimensional complex space forms,”
*American Journal of Mathematics*, vol. 122, no. 2, pp. 295–317, 2000. View at: Publisher Site | Google Scholar | Zentralblatt MATH | MathSciNet - K. Arslan, A. Carriazo, B.-Y. Chen, and C. Murathan, “On slant submanifolds of neutral Kaehler manifolds,”
*Taiwanese Journal of Mathematics*, vol. 14, no. 2, pp. 561–584, 2010. View at: Google Scholar | Zentralblatt MATH | MathSciNet - B.-Y. Chen, “Minimal flat Lorentzian surfaces in Lorentzian complex space forms,”
*Publicationes Mathematicae Debrecen*, vol. 73, no. 1-2, pp. 233–248, 2008. View at: Google Scholar | Zentralblatt MATH | MathSciNet - B.-Y. Chen, “Dependence of the Gauss-Codazzi equations and the Ricci equation of Lorentz surfaces,”
*Publicationes Mathematicae Debrecen*, vol. 74, no. 3-4, pp. 341–349, 2009. View at: Google Scholar | Zentralblatt MATH | MathSciNet - B.-Y. Chen, “Nonlinear Klein-Gordon equations and Lorentzian minimal surfaces in Lorentzian complex space forms,”
*Taiwanese Journal of Mathematics*, vol. 13, no. 1, pp. 1–24, 2009. View at: Google Scholar | Zentralblatt MATH | MathSciNet - B.-Y. Chen and F. Dillen, “Classification of marginally trapped Lagrangian surfaces in Lorentzian complex space forms,”
*Journal of Mathematical Physics*, vol. 48, no. 1, Article ID 013509, 23 pages, 2007. View at: Publisher Site | Google Scholar | Zentralblatt MATH | MathSciNet - B.-Y. Chen and S. Ishikawa, “Biharmonic surfaces in pseudo-Euclidean spaces,”
*Memoirs of the Faculty of Science. Kyushu University A*, vol. 45, no. 2, pp. 323–347, 1991. View at: Publisher Site | Google Scholar | Zentralblatt MATH | MathSciNet - B.-Y. Chen and S. Ishikawa, “Biharmonic pseudo-Riemannian submanifolds in pseudo-Euclidean spaces,”
*Kyushu Journal of Mathematics*, vol. 52, no. 1, pp. 167–185, 1998. View at: Publisher Site | Google Scholar | Zentralblatt MATH | MathSciNet - B.-Y. Chen, “Black holes, marginally trapped surfaces and quasi-minimal surfaces,”
*Tamkang Journal of Mathematics*, vol. 40, no. 4, pp. 313–341, 2009. View at: Google Scholar | Zentralblatt MATH | MathSciNet - L. Vrancken, “Minimal Lagrangian submanifolds with constant sectional curvature in indefinite complex space forms,”
*Proceedings of the American Mathematical Society*, vol. 130, no. 5, pp. 1459–1466, 2002. View at: Publisher Site | Google Scholar | Zentralblatt MATH | MathSciNet - S. Montaldo and C. Oniciuc, “A short survey on biharmonic maps between Riemannian manifolds,”
*Revista de la Unión Matemática Argentina*, vol. 47, no. 2, pp. 1–22, 2006. View at: Google Scholar | Zentralblatt MATH | MathSciNet - T. Sasahara, “Quasi-minimal Lagrangian surfaces whose mean curvature vectors are eigenvectors,”
*Demonstratio Mathematica*, vol. 38, no. 1, pp. 185–196, 2005. View at: Google Scholar | Zentralblatt MATH | MathSciNet - T. Sasahara, “Quasi-biharmonic Lagrangian surfaces in Lorentzian complex space forms,”
*Annali di Matematica Pura ed Applicata*, vol. 192, no. 2, pp. 191–201, 2013. View at: Publisher Site | Google Scholar - Y. Fu and Z. H. Hou, “Classification of pseudo-umbilical slant surfaces in Lorentzian complex space forms,”
*Taiwanese Journal of Mathematics*, vol. 15, no. 5, pp. 1919–1938, 2011. View at: Google Scholar | Zentralblatt MATH | MathSciNet - T. Sasahara, “Biharmonic Lagrangian surfaces of constant mean curvature in complex space forms,”
*Glasgow Mathematical Journal*, vol. 49, no. 3, pp. 497–507, 2007. View at: Publisher Site | Google Scholar | Zentralblatt MATH | MathSciNet

#### Copyright

Copyright © 2013 Yu Fu. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.