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Real Hypersurfaces of Nonflat Complex Projective Planes Whose Jacobi Structure Operator Satisfies a Generalized Commutative Condition
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.
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 . The same author classified real hypersurfaces in , with constant principal curvatures in , but only when the number of distinct principal curvatures satisfies . Kimura showed in  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 , 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 .
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, satisfyingrestricted 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 .
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 :
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 yieldsThe 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) yieldMoreover, the decompositions of and combined with which holds due to (1) and (4) yieldLet 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 ruleBy 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 thatThe last equation because of the linear independency of , , and yieldsIn the same way, putting , in (8), we obtain Combining the last equation with (11) and Lemma 1, we haveSimilarly, putting , in (8), we get , which, by use of (11) and Lemma 1, implies thatWe 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 yieldThe last equations using (24), (25), and (28) yieldFollowing a similar way, we calculate and then . By equalizing the results we obtainComparing (35) with (36) and by making use of (26) we obtainDue 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 yieldLet us assume that there exists a point in , such that in a neighborhood of this point. Then (38) impliesBased on the definition of the vector field in Section 2, condition is equivalent to . The last relation, (39), and (29) yieldBy virtue of (39) and (40) we simplify (27) and obtainWe 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 acquireThe combination of (26) and (42) leads toIn addition, (27), , and (42) yieldWe 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 . We consider which satisfiesFrom (7) and (47) we obtainBy making use of (1) with , in combination with (48), we obtainIf then and is the root of the quadratic  and consequently a constant. The classification follows from . Since is of type then (due to ) satisfies . The last condition and (1) and (15) lead to , , which is combined with , resulting in , .
The author declares that there are no competing interests.
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