Commuting Structure Jacobi Operator for Real Hypersurfaces in Complex Space Forms

  , , , M be a real hypersurface of a complex space form with almost contact metric structure g Let    . In this paper, we prove that if the structure Jacobi operator   , R R      is     -parallel and R commute with the shape operator, then M is a Hopf hypersurface. Further, if R is     -parallel and R commute with the Ricci tensor, then M is also a Hopf hypersurface provided that TrR is constant.


Introduction
A complex -dimensional Kähler manifold of constant holomorphic sectional curvature is called a complex space form, which is denoted by M in M c is said to be a structure Jacobi operator on M .The structure Jacobi operator has a fundamental role in contact geometry.In [6], Cho and first author started the study on real hypersurfaces in complex space form by using the operator R  .In particular the structure Jacobi operator has been studied under the various commutative condition [7][8][9].For example, Pérez et al. [9] called that real hypersurfaces M has commuting structure Jacobi operator if X on  for any vector field M , and proved that there exist no real hypersurfaces in with commuting structure Jacobi operator.On the other hand Ortega et al. [10] have proved that there are no real hypersurfaces in n M and the tangential component of a vector.The Reeb vector   T M c R with parallel structure Jacobi operator  , that is, , where    X on M .More generally, such a result has been extended by [11].In this situation, if naturally leads us to be consider another condition weaker than parallelness.In the preceding work, we investigate the weaker condition . A real hypersurface is said to a Hopf hypersurface if the Reeb vector  of M is principal.
Hopf hypersurfaces is realized as tubes over certain submanifolds in n , by using its focal map (see Cecil and Ryan [1]).By making use of those results and the mentioned work of Takagi [2,3], Kimura [4] proved the local classification theorem for Hopf hypersurfaces of n whose all principal curvatures are constant.For the case n , Berndt [5] proved the classification theorem for Hopf hypersurfaces whose all principal curvatures are constant.Among the several types of real hypersurfaces appeared in Takagi's list or Berndt's list.under the condition that the structure Jacobi operator commute with the shape operator or the Ricci tensor.
, , This paper consists of two parts.In the first part of this paper, we prove that if the structure Jacobi operator  R -parallel and  commute with the shape operator, then M is a Hopf hypersurface (see Theorem 1 in Section 4).In the second part of this paper, we prove that if R  is     R -parallel and  commute with the Ricci tensor, then M is also a Hopf hypersurface provided that TrR  is constant (see Theo- rem 2 in Section 5).
All manifolds in this paper are assumed to be connected and of class and the real hypersurfaces are supposed to be oriented.

Fundamental Facts of Real Hypersurface
In this section the elemental factors of a real hypersurface are recalled.Let M be a real hypersurface in a complex space form for any vector fields X and Y on M , where g denotes the Riemannian metric of M induced from g  and A is the shape operator of M in   n M c .For any vector field X tangent to M , we put for the complex structure J of n   M c .We call  the Reeb vector field.Then we may see that the aggre- for any vector fields X and Y on M .From Kähler condition , and making use of Gauss and Weingarten formulas, we obtain and for any vector fields X Y tangent to M .The equations of Gauss and Codazzi are respectively given by the following: , and for a function f we denote by f  the gradient vector In this paper, we basically use the technical computations with the orthogonal triplet and their associated scalars  and  . Using (2.2) and (2.7), it is seen that Now, differentiating (2.5) covariantly along M and making use of (2.1), (2.2) and (2.4), we find which enables us to obtain By the definition of , (2.2) and (2.12), it is verified From the Gauss Equation (2.3) the structure Jacobi operator  is given by for any vector field X on M .
Let be the open subset of  M defined by At each point of , the Reeb vector field  is not principal.That is,  is not an eigenvector of the shape In what follows we assume that is not an empty set in order to prove our main theorem by reductio ad absurdum, unless otherwise stated, all discussion concerns the set .

Real Hypersurfaces Satisfying R S SR
Combining above two equations and using (2.7), we obtain where a 1-form w is defined by

 
, X for any vector fie which shows that   (3.5) and hence Now, differentiating (3.5) covariantly along  , we find By taking the inner product with in the last equation, we obtain since is a unit vector field orthogonal to  .We also have by applying  to (3.7) and making use of (2.9) which together with the Codazzi Equation (2.4) gives Putting   in (3.8) and using (3.11), we obtain : , where we have put .Differentiating (3.4) covariantly and using (2.2) we find which together with (2.4) and (2.12) implies that X by If we replace  in (3.13) and make use of (2.4), (2.12) and the last equation, then we get Now, we define a 1-form by for any vector field X , it is, using (2.4) and (3.13), seen that to both sides of (3.15) and take account of (2.12), (3.5), (3.6), (3.9) and (3.10), then we obtain In the following we assume that M satisfies because of (2.5) and (2.7).Putting X W  in the last equation and using (2.2), we have . If we replace Y by  and make use of (2.12) and (3.5), then we obtain If not, then we have  

AW
, and then we restrict our arguments on such a place.From (3.18) we have 0   , which together with (3.5) yields 0   and hence (3.5) reformed as AW   .But, it is, using (2.8) and (3.12), that 0 where we have used (2.8) and     .
On the other hand (3.17) is reduced to by W and take account of (2.7), (2.8) and (3.10), then we obtain and consequently   . This contradicts the fact that  0 .Therefore  on  is proved.

  
If we make use of (3.18) and Remark 1, then (3.17) reformed as Using (3.5) and (3.10), we can write the last equation as ) and make use of (2.8), (3.8) and (3.12), then we obtain Taking inner product to this, and using (3.5) and (3.12), we find If we take the inner product  to (3.21) and make use of (2.7) and (3.5), then we have , which together with (3.12) and (3.22) yields

Real Hypersurfaces Satisfying and 0
In the following we assume that    , .Then we have from above equation on this subset.On the other hand, if we take the inner product to (3.20) and make use of (3.8) and  , then we obtain Comparing this with last equation, we verify that , a contradiction, Therefore (4.3) is established on whole space.
Because of (4.2) and ( 4.3), we can write (3.21) as Using (4.3), we can also write (3.20) as If we take the inner product U to (3.7) and take account of (2.4), (2.10) and (4.3), then we obtain which together with (2.8), (4.2) and (4.6) yields Now, applying by  in (2.11) and using (2.10), we find  in this and make use of (2.5), (3.5) and (4.3), then we obtain Taking the inner product to (2.11), we also obtain where we have used (2.4), (2.5) and (4.3), which together with (4.7) implies that If we apply by  to this and make use of (3.5) and (4.9), then we obtain for some function  on  .
On the other hand, differentiating (4.3) covariantly, and using itself again, we find which together with (2.4) and (2.5) gives is the exterior derivative of a 1-form is given by or putting and making use of (4.3), From this and (4.8) it follows that because is orthogonal to W . Comparing this with (4.10), we have By virtue of (2.7), (3.5) and (4.2), we can write this as If we put X U  in (4.11) and take account of the last equation, then we obtain  U If we take the inner product to this, and make use of (4.3), we deduce that Thus, (4.13) reformed as where we have put    .Now, we are going to prove that 0   on  .For this, the last equation is rewritten as Differentiating this with respect to a vector field X again, and taking the skew-symmetric parts with respect to X and Y , then we eventually have .

APM
If we take the inner product  to (4.4) and make use of (2.8), (4.2) and (4.12), then we obtain Using this, we can write (4.4) as where we have used (2.7), (2.8), (3.5), (4.2) and (4.12).Combining this with (4.14), we obtain which tells us that Using the quite same method as that used to (4.19) from (4.14), we can drive from (4.21) the following: where we have used (2.2) and (4.22).Putting   in this and using (4.14), (4.17) and (4.20), we obtain which together with (2.7) and (4.18) implies that Thus, it follows that , a contradiction.Thus,  on  is proved.Consequently we prove that  0 W  by virtue of (4.12) and Remark 1.By (4.20) we also have 0   .Therefore (4.14) and (4.21) are reduced respectively to , which together with (4.17) gives , and then we restrict our arguments on such a place.Then we have 3 0 , which together with (4.5) yields

Real Hypersurfaces with
In this section, we will continue our arguments under the same hypotheses as those stated in section 3, namely wi nd w rguments th the aid of (5.1) and (5.3).A e restrict a on such a place.Then (3.5) becomes .
Because of (2.8), (3.12) and ( as 5.1), we can write (3.16) Using (2.8) and (5.4), the Equation (3.21) reformed as  .10),e also deduce that If we p X U  in this and use (5.4) and (5.6), then w Now, differentiating (5.4) covariantly an .e get   and using (2.2) d (5.7), we find If we take the inner product U to this d make use of en w  an (2.4), (5.4), (5.6) and (5.7), th e obtain By the way, we have from (3.20) where we have used 0, 0 W    , (5.6) and .7), ) gives (5 which together with (2.4 Substituting this into (5.10) and using (5.7), we find From this an 0. d (5.9) we verify that U U   Differentiating (5.6) covaria we get (5.11)ntly, and using itself again, 2c or, using (2.4) and (2.5) Taking skew-symmetric part with respect to X and Y , we find If we put Y U  in this and make use of (5.6) and (5.11 tain , U U (5.12) which enables us to obtain 0 ), then we ob ake the inner product (3.7)(2.4), (2.10) and Lemma 2,

 
, , , On the other hand, i with U , and make en we get  f we t use of th where the function  is given by We notice here that the following: because of (5.17).Thus, (5.18) implies that Because of (5.17) an ) and (2.5) implies th which together with (2.4 at (5.20) If we put X   in this, and make use of (2.7) (2.10), then we get and By putting Y U  in this and using Lemma have because U and W are mutually orthogonal.From this and (5.16) we verify that


We are now going to prove the followi is constant, then we have , where the function is given by f  From this and Lemma 2, we obtain Using (5.12), Lemma 2 and the last equat , .
By virtue of this and (5.15), it is verified that Since  and W are mutually orthogonal, we see fr e last equation the following: om th

Elim
 from above two equations, we evenave Putting X   in this and making use of (2.12) and (3.23), we find which together with (5.22), (5.24), (5.25) and Lemma 2 gi By the way, if we replace X by  in (3.7) and take account of (3.11), (3.23) and Lemma 1, then we obtain which connected to the last equation and Lemma 1 Using this and (3.24), we can write (5.14) as From (5.2) we have because of (2.8) and (3.23), we verif equation, that y, using the last 2), it is seen that   .By vir- tue of this and (5.2 If we take the inner product U to (5.32), then we


In the same way we see from (5.36 Therefore, we obtain   which together with the Codazzi Equation (2.4) implies that Differentiating this covariantly and using (2.2), we find If we take the skew-symmetric part with respect to X and Y , and using the Ricci identity, we obtain By putting Y W  and using (2.9), (3.5), (5.42), (5.43) and (5.45), we find (5.46) On the other hand, using (2.7), (3.5) and Lemma 4, we can write (5.15) as From (2.3), we have which together with (2.8), (3.6) and (5.44) implies that (5.47)


Substituting above four equations into (5.46),we find So naturally there exists a Kähler structure J and Kähler metric g on n   M c .Now let us consider a real hypersurface M in   n M c .Then we also denote by g the induced Riemannian metric of M and by a local unit normal vector field of N M in n   M c .Further, A denotes by the shape operator of M in of M is naturally induced from the Kähler structure of M as follows: bundle of The Reeb vector field  plays an important role in the theory of real hypersurfaces in a complex space form   n M c .Related to the Reeb vector field  the Jacobi operator R  defined by 


orthogonal to  .Further we invetigate the structure Jacobi operator is    -parallel we denote the Levi-Civita connection with respect to the Fubini-Study metric   g  of   n M c .Then the Gauss and Weingarten formulas are respectively given by subset, which together with (4.20) yields c product U to .11) and make use of (2.4), (5.4), (5.6) and (5.7) hen we obtain where we have used (2.8) and    of Lemma 4, we also ve , Because rify that   and  are constant on  by virtue o (5.36)-(5.38).Using these and 44) f(5.we can write (5.13) as 4) 6) contradiction because of Remark 1.
  .Fromthis we conclude that Theorem 1.Let M be a real hypersurface in 22)