International Journal of Mathematics and Mathematical Sciences
Volume 2016 (2016), Article ID 3089298, 6 pages
http://dx.doi.org/10.1155/2016/3089298

Research Article

Real Hypersurfaces of Nonflat Complex Projective Planes Whose Jacobi Structure Operator Satisfies a Generalized Commutative Condition

Theocharis Theofanidis

School of Second Chance Delta, 57008 Thessaloniki, Greece

Received 20 May 2016; Accepted 29 June 2016

Academic Editor: Luc Vrancken

Copyright © 2016 Theocharis Theofanidis. 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.

Abstract.

Real hypersurfaces satisfying the condition have been studied by many authors under at least one more condition, since the class of these hypersurfaces is quite tough to be classified. The aim of the present paper is the classification of real hypersurfaces in complex projective plane satisfying a generalization of under an additional restriction on a specific function.

1. Introduction

An -dimensional Kaehlerian manifold of constant holomorphic sectional curvature is called complex space form, which is denoted by . A complete and simply connected complex space form is a projective space if , a hyperbolic space if , or a Euclidean space if . The induced almost contact metric structure of a real hypersurface of will be denoted by ().

Real hypersurfaces in which are homogeneous were classified by Takagi [1]. The same author classified real hypersurfaces in , with constant principal curvatures in [2], but only when the number of distinct principal curvatures satisfies . Kimura showed in [3] that if a Hopf real hypersurface in has constant principal curvatures, then the number of distinct principal curvatures of is 2, 3, or 5. Berndt gave the equivalent result for Hopf hypersurfaces in [4], where he divided real hypersurfaces into four model spaces, named , , , and . Real hypersurfaces of types and in and of types , , and in are said to be hypersurfaces of type A for simplicity. Another class of real hypersurfaces that appears quite often is the Hopf hypersurfaces where the structure vector field is a principal vector field. For more details and examples on real hypersurfaces of type and Hopf, we refer to [5].

A Jacobi field along geodesics of a given Riemannian manifold () plays an important role in the study of differential geometry. It satisfies a well-known differential equation which inspires Jacobi operators. For any vector field , the Jacobi operator is defined by , where denotes the curvature tensor and is a vector field on . is a self-adjoint endomorphism in the tangent space of and is related to the Jacobi differential equation, which is given by along a geodesic on , where denotes the velocity vector along on .

In a real hypersurface of a complex space form , , the Jacobi operator on with respect to the structure vector field is called the structure Jacobi operator and is denoted by .

Real hypersurfaces have been studied from many points of view. Certain authors have studied real hypersurfaces under conditions which include the operator [6–9]. Other authors have studied real hypersurfaces under the condition , equipped with one or two additional conditions [10–15], proving that these hypersurfaces are Hopf and classifying them as type .

In the present paper we classify real hypersurfaces of complex projective planes, satisfying

restricted in the subspace of for every point , where consists of all vector fields orthogonal to the Reeb flow vector field and the form is assumed to be nonlinear with respect to scalar product. If is linear, then by replacing with , we obtain which implies . So (1) takes the simpler form .

Since this class is rather difficult to classify, a second condition is imposed. However it is not a condition acting in vector fields, but only in the function , where is the shape operator. Geometrically speaking, we demand the function to be constant in the direction of the integral curves of . Namely, we prove the following.

Main Theorem. A real hypersurface of a complex projective plane , satisfying , ( is nonlinear), and , is Hopf. Furthermore, if then is locally congruent to a model space of type and .

For the case of in order to determine real hypersurface of type , the technical assumption is needed. Actually, there is a nonhomogeneous tube with (of radius ) over a certain Kaehler submanifold in , when its focal map has constant rank on [16].

2. Preliminaries

Let be a Kaehlerian manifold of real dimension , equipped with an almost complex structure and a Hermitian metric tensor . Then for any vector fields and on , the following relations hold:

where denotes the Riemannian connection of of .

Now, let be a real -dimensional hypersurface of , and denote by a unit normal vector field on a neighborhood of a point in (from now on we will write M instead of ). For any vector field tangent to we have , where is the tangent component of , is the normal component, and

By properties of the almost complex structure and the definitions of and , the following relations hold [17]:

The above relations define an almost contact metric structure on which is denoted by . By virtue of this structure, we can define a local orthonormal basis , called -basis. Furthermore, let be the shape operator in the direction of , and denote by the Riemannian connection of on . Then is symmetric and the following equations are satisfied:

As the ambient space is of constant holomorphic sectional curvature , the equations of Gauss and Codazzi are, respectively, given by

The tangent space , for every point , is decomposed as follows:

where and is defined as follows:

Based on the above decomposition, by virtue of (6), we decompose the vector field in the following way:

where and , provided that .

As stated before, if the vector field is a principal vector field, the real hypersurface is called hypersurface. In this case the vector field is expressed as , .

Finally, differentiation of a function along a vector field will be denoted by . All manifolds of this paper are assumed to be connected and of class .

3. Auxiliary Relations

In the study of real hypersurfaces of a complex space form , , it is a crucial condition that the structure vector field is principal. The purpose of this paragraph is to establish relations that will help us prove this condition.

Let in a neighborhood of . If we had at least one point of , where , then from (7) we would obtain and . Combining the last two equations with (1) we would take which is a contradiction. Therefore in .

Lemma 1. Let M be a real hypersurface of a complex projective plane , satisfying (1). Then the following relations hold in :

where , , and are smooth functions in .

Proof. From (7) we get

which for and yields

The scalar products of (17)(i) with and yield, respectively,

where and . From (18), (19), and we obtain . From (19), , and we obtain . In order to prove (12) we need to show that and . Combining the analysis of and with (11) and (17) we obtain and . The last two equations and which holds due to (1) yield

Moreover, the decompositions of and combined with which holds due to (1) and (4) yield

Let us assume that in a neighborhood of a point in . Then (20)(i) and (21)(i) give . Apparently ; otherwise (20)(ii) would yield . As a result, (20) and (21) lead, respectively, to and . The last two relations are added and result in which is a contradiction. This means that holds, and (20) and (21) imply .
Equation (13) is obtained from (12) and relation (6) for , , and . Next we recall the rule

By virtue of (22) for , and for , , it is shown, respectively, that and , which means . In a similar way, (22) for and , yields, respectively, and . So holds. Finally, (22) for , and , , (with the aid of (13)) yields, respectively, and . Therefore we have and (14) has been proved. In order to prove (15) we use the second of (6) with the combinations, (i) , , (ii) , and (iii) , , and make use of (11), (12), and (14).

Lemma 2. Let M be a real hypersurface of a complex projective plane , satisfying (1). Then in one has .

Proof. Putting , in (8), we obtain Combining the last equation with (11) and Lemma 1, it follows that

The last equation because of the linear independency of , , and yields

In the same way, putting , in (8), we obtain Combining the last equation with (11) and Lemma 1, we have

Similarly, putting , in (8), we get , which, by use of (11) and Lemma 1, implies that

We expand (30) and then replace the terms , , and from (26), (27), and (29), respectively. The final equation is

Lemma 3. Let be a real hypersurface of a complex projective plane , satisfying (1). Then, holds in .

Proof. Because of (14), (15), (28), (31), and Lemma 2, the well-known relation takes the form

On the other hand (27), (29), (31), and Lemma 2 yield

The last equations using (24), (25), and (28) yield

Following a similar way, we calculate and then . By equalizing the results we obtain

Comparing (35) with (36) and by making use of (26) we obtain

Due to (32), we have . So if we had in neighborhood of point in then (37) would imply . By differentiating the last relation with respect to , in combination with and (28), we would obtain which is a contradiction.
Therefore holds in .

4. Proof of Main Theorem

We first prove the following proposition.

Proposition 4. Let be a real hypersurface of a complex projective plane , satisfying (1) and . Then is Hopf.

Proof. We keep working in . By virtue of Lemma 3 and (11), (28), and (31), we obtain However, from Lemmas 1, 2, and 3 and (11) we calculate The two expressions of yield

Let us assume that there exists a point in , such that in a neighborhood of this point. Then (38) implies

Based on the definition of the vector field in Section 2, condition is equivalent to . The last relation, (39), and (29) yield

By virtue of (39) and (40) we simplify (27) and obtain

We modify (32) by replacing the term from (40). Then we make calculations with the aid of (41), , to obtain . The last equation is differentiated with respect to , giving . As a result, (41) yields which is a contradiction and holds in .
Combining and (29) we acquire

The combination of (26) and (42) leads to

In addition, (27), , and (42) yield

We estimate the vector field from (7) and Lemma 1 as . The same vector field is calculated from , (42), , and . The two expressions of are equalized and the outcome is modified as follows: first we multiply with , then we replace the term from (43), and finally we divide with . After these steps, we are led to

We differentiate the above relation and utilize (otherwise (44) yields which is a contradiction) in order to get . The last two equations give which is a contradiction. Therefore we have a contradiction in ; hence and is Hopf.

From Proposition 4 we have on

and is a constant [5]. We consider which satisfies

From (7) and (47) we obtain

By making use of (1) with , in combination with (48), we obtain

If then and is the root of the quadratic [5] and consequently a constant. The classification follows from [3]. Since is of type then (due to [5]) satisfies . The last condition and (1) and (15) lead to , , which is combined with , resulting in , .

Competing Interests

The author declares that there are no competing interests.

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