Purity and hybridness of two tensors on a real hypersurface in complex projective space

1 Introduction

Let M be a connected real hypersurface without boundary in the complex projective space C P m , m ≥ 2 , equipped with the Kaehlerian structure ( J , g ) , g being the Fubini-Study metric of constant holomorphic sectional curvature 4. We will continue denoting by g the restriction of the metric on C P m to M and by ∇ the Levi-Civita connection on M . The structure vector field (or Reeb vector field) on M is defined as ξ = − J N , for a unit local normal vector field N on M . Then M inherits an almost contact metric structure ( ϕ , ξ , η , g ) from the Kaehlerian structure of C P m , where ϕ is a tensor field of type (1, 1), and η an 1-form on M . Let A be the shape operator of M associated to N . M is said to be Hopf if ξ is an eigenvector of A , A ξ = α ξ , and α is called the Reeb curvature of M . We also call D to the maximal holomorphic distribution on M , D p = { X ∈ T p M ∣ g ( X , ξ ) = 0 } , for any p ∈ M .

Takagi, see [1–3] classified homogeneous real hypersurfaces in C P m , obtaining six types. Among them, type ( A 1 ) real hypersurfaces are geodesic hyperspheres of radius r , 0 < r < π 2 , and type ( A 2 ) are tubes of radius r , 0 < r < π 2 , over totally geodesic complex projective spaces C P n , 0 < n < m − 1 . We will call both types of real hypersurfaces type ( A ) real hypersurfaces. They are the unique real hypersurfaces in C P m such that A ϕ = ϕ A [4]. Kimura [5] proved that the real hypersurfaces appearing in Takagi’s list are the unique isoparametric Hopf real hypersurfaces, that is, real hypersurfaces all whose principal curvatures are constant.

Ruled real hypersurfaces in C P m satisfy that D is integrable and its integral manifolds are totally geodesic C P m − 1 . Equivalently, g ( A D , D ) = 0 . Examples of ruled real hypersurfaces can be found in [6] or [7].

The Tanaka-Webster connection [8,9] is the canonical affine connection defined on a non-degenerate, pseudo-Hermitian CR-manifold. As a generalization of this connection, Tanno [10] defined the generalized Tanaka-Webster connection for contact metric manifolds. Using the almost contact metric structure on M and the naturally extended affine connection of Tanno’s generalized Tanaka-Webster connection, Cho defined, for any nonnull real number k , the k th generalized Tanaka-Webster connection ∇ ˆ ( k ) for a real hypersurface M in C P m , see [11,12], by

for any X , Y tangent to M . Then ∇ ˆ ( k ) η = 0 , ∇ ˆ ( k ) ξ = 0 , ∇ ˆ ( k ) g = 0 , ∇ ˆ ( k ) ϕ = 0 . In particular, if the shape operator of a real hypersurface satisfies ϕ A + A ϕ = 2 k ϕ , M is a contact manifold and the kth generalized Tanaka-Webster connection coincides with the Tanaka-Webster connection.

The tensor field of type (1, 2) given by the difference of both connections F ( k ) ( X , Y ) = g ( ϕ A X , Y ) ξ − η ( Y ) ϕ A X − k η ( X ) ϕ Y , for any X , Y tangent to M , see [13] Proposition 7.10, pages 234–235, is called the k th Cho tensor on M . Then, for any X tangent to M and any nonnull real number k , we can consider the tensor field of type (1, 1) F X ( k ) , given by F X ( k ) Y = F ( k ) ( X , Y ) for any Y ∈ T M . This operator will be called the k th Cho operator corresponding to X . Notice that if X ∈ D , the corresponding Cho operator does not depend on k , and we simply write it F X . The torsion of the connection ∇ ˆ ( k ) is given by T ( k ) ( X , Y ) = F X ( k ) Y − F Y ( k ) X for any X , Y tangent to M . We define the k th torsion operator associated to X as the operator given by T X ( k ) Y = T ( k ) ( X , Y ) , for any X , Y tangent to M .

If ℒ denotes the Lie derivative on M , we know that ℒ X Y = ∇ X Y − ∇ Y X for any X , Y tangent to M . Analogously, we can define on M a differential operator of first order associated with the k th generalized Tanaka-Webster connection, given by

for any X , Y tangent to M . We will call it the derivative of Lie type associated to the k th generalized Tanaka-Webster connection.

Let now B be a symmetric tensor of type (1, 1) defined on M . Then we can consider the type (1, 2) tensor B F ( k ) associated to B as follows:

for any X , Y tangent to M . We also can consider another tensor of type (1, 2), B T ( k ) , associated with B , by

for any X , Y tangent to M . Notice that if X ∈ D , B F ( k ) does not depend on k . We will write it simply B F .

In [14,15], we proved non-existence of real hypersurfaces in C P m , m ≥ 3 , such that the tensors of type (1, 2) associated to the shape operator, A F ( k ) = 0 and A T ( k ) = 0 , respectively, for any nonnull real number k . Further results on such tensors were presented in [16,17].

Let S be an operator on M and C a tensor of type (1, 2) on M . C is said to be pure with respect to S [18], if

for any X , Y tangent to M . If C and S satisfy the same condition for any X , Y ∈ D , C is said to be η -pure with respect to S .

If with the same conditions

for any X , Y tangent to M , C is said to be hybrid with respect to S . If we have the same equality for any X , Y ∈ D , C is said to be η -hybrid with respect to S .

Then we can consider purity and hybridness of any tensor field of type (1, 2) with respect to different operators as the shape operator, the Ricci operator, the structure Jacobi operator, or the structure Lie operator, among others. The purpose of the present article is to begin the study of purity and hybridness of the tensor fields A F ( k ) and A T ( k ) with respect to one of such operators, as the shape one, A . We will prove the following:

Also we will prove the following:

Two real hypersurfaces in the complex projective space are congruent if there exists an isometry of the ambient space applying one real hypersurface into the other. As a consequence of this theorem, we can prove the following:

In the case of the tensor field A T k , we will obtain

2 Preliminaries

Throughout this article, all manifolds, vector fields, etc., will be considered of class C ∞ unless otherwise stated. Let M be a connected real hypersurface in C P m , m ≥ 2 , without boundary. Let N be a locally defined unit normal vector field on M . Let ∇ be the Levi-Civita connection on M and ( J , g ) be the Kaehlerian structure of C P m .

For any vector field X tangent to M , we write J X = ϕ X + η ( X ) N , and − J N = ξ , where ϕ X denotes the tangential component of J X . Then ( ϕ , ξ , η , g ) is an almost contact metric structure on M [19]. That is, we have

for any vector fields X , Y tangent to M . From (2.1), we obtain

From the parallelism of J , we obtain

and

for any X , Y tangent to M , where A denotes the shape operator of the immersion. As the ambient space has holomorphic sectional curvature 4, the equation of Codazzi is given by

for any vector fields X , Y tangent to M .

In the sequel, we need the following result:

3 Proof of Theorem 1.1

Let us suppose that A F ( k ) is pure with respect to A . Then F A X ( k ) A Y − A F A X ( k ) Y = F X ( k ) A 2 Y − A F X ( k ) A Y , for any X , Y tangent to M . Then

for any X , Y tangent to M .

Let us suppose first that M is Hopf with Reeb curvature α . If X , Y ∈ D from (3.1), we obtain g ( ϕ A 2 X , A Y ) ξ − α g ( ϕ A 2 X , Y ) ξ = g ( ϕ A X , A 2 Y ) ξ − α g ( ϕ A X , A Y ) ξ . Its scalar product with ξ gives A ϕ A 2 X − α ϕ A 2 X = A 2 ϕ A X − α A ϕ A X , for any X ∈ D . If we suppose that A X = λ X , we will obtain λ 2 μ − α λ 2 = λ μ 2 − α λ μ , where μ = α λ + 2 2 λ − α . Therefore,

If now we take X ∈ D , Y = ξ in (3.1) it follows − α ϕ A 2 X + A ϕ A 2 X = − α 2 ϕ A X + α A ϕ A X . If we suppose again that A X = λ X , this yields

On the other hand, if in (3.1)  X = ξ , Y ∈ D , we have, supposing that A Y = λ Y , that − α λ + α μ = − λ 2 + λ μ . Thus,

From (3.4) either λ = α or λ = μ . If we suppose that λ = α , (3.3) is satisfied and from (3.2), we should have λ ( λ − μ ) 2 = 0 . Thus, either λ = 0 or λ = μ . As λ = α , if λ = μ M should be totally umbilical, which is impossible. Therefore, λ = α = 0 . This is also impossible if you look at the value of μ . Thus, we obtain that λ = μ . In this case, (3.2) is satisfied, but then (3.3) yields λ ( λ − α ) 2 = 0 . As λ = α is impossible, we should have λ = μ = 0 . This implies 0 = − 2 α , which also is impossible.

Therefore, M must be non Hopf. We can write A ξ = α ξ + β U , where U is a unit vector field in D and α and β are functions on M such that β ≠ 0 at least on a neighborhood of a point p ∈ M . We will make the following calculations on such a neighborhood. Then (3.1) becomes

for any X , Y tangent to M . Take X = Y = ξ in (3.5) to obtain β g ( ϕ A 2 ξ , U ) ξ − α ϕ A 2 ξ − k α β ϕ U + A ϕ A 2 ξ = β g ( ϕ U , A 2 ξ ) ξ − β g ( A ξ , A ξ ) ϕ U − k ϕ A 2 ξ + α β A ϕ U + k β A ϕ U . From the expression of A ξ and bearing in mind that β ≠ 0 , we obtain

The scalar product of (3.6) and ξ implies 3 β g ( A ϕ U , U ) = 0 . As β ≠ 0 , we have

Taking in (3.5)  X = U , Y = ϕ U and its scalar product with ϕ Z , Z ∈ D U = { W ∈ D ∣ g ( W , U ) = g ( W , ϕ U ) = 0 } , we obtain k β g ( A ϕ U , Z ) − k β g ( A U , ϕ Z ) = 0 . As k β ≠ 0 , it follows

for any Z ∈ D U .

Take now X , Y ∈ D U in (3.5) and its scalar product with ϕ U . We obtain − g ( A Y , A ξ ) g ( A X , U ) = 0 . Therefore, − g ( A Y , α ξ + β U ) = 0 = − β g ( A U , Y ) g ( A U , X ) for any X , Y ∈ D U . If X = Y , we obtain

for any X ∈ D U . Then (3.8) and (3.9) also yield

for any X ∈ D U . From (3.7), (3.9), and (3.10), we can write A U = β ξ + γ U , A ϕ U = δ ϕ U , for some functions γ and δ .

If we take Y = U , X = ϕ U in (3.5), we have g ( ϕ A 2 ϕ U , A U ) ξ − β ϕ A 2 ϕ U − g ( ϕ A 2 ϕ U , U ) A ξ = g ( ϕ A ϕ U , A 2 U ) ξ − g ( A U , A ξ ) ϕ A ϕ U − g ( ϕ A ϕ U , A U ) A ξ + β A ϕ A ϕ U . Thus, − δ 2 γ ξ + β δ 2 U + δ 2 A ξ = − δ g ( A U , A U ) ξ + δ g ( A U , A ξ ) U + γ δ A ξ − δ β A U . Its scalar product with U yields 2 β δ 2 = δ β ( α + γ ) . As β ≠ 0 , 2 δ 2 = δ ( α + γ ) . Therefore, either δ = 0 or δ = α + γ 2 . The scalar product of the same expression and ξ gives − δ 2 γ + δ 2 α = − δ ( β 2 + γ 2 ) + α γ δ − δ β 2 . If we suppose δ ≠ 0 , δ ( α − γ ) = − ( β 2 + γ 2 ) + α γ − β 2 = − 2 β 2 + γ ( α − γ ) . That is, ( δ − γ ) ( α − γ ) = − 2 β 2 . As we suppose δ ≠ 0 , δ = α + γ 2 , and then ( α − γ ) 2 2 = − 2 β 2 . This yields β = 0 , which is impossible. We have proved that

If now we take Y = U in (3.5), we obtain − β ϕ A 2 X − k α η ( X ) ϕ A U − k β g ( X , U ) ϕ A U = g ( ϕ A X , A 2 U ) ξ − g ( A U , A ξ ) ϕ A X − k η ( X ) ϕ A 2 U + β A ϕ A X for any X tangent to M . Its scalar product with ϕ U implies

for any X tangent to M . Take X = U in (3.12) to have, bearing in mind that β ≠ 0 , g ( A U , A U ) + k g ( A U , U ) = ( α + γ ) g ( A U , U ) . Therefore, β 2 + γ 2 + k γ = α γ + γ 2 . This means that β 2 = ( α − k ) γ . If α = k , β = 0 , which is impossible. Thus,

If in (3.12)  X = ξ , we obtain − β g ( A ξ , A U ) − k α g ( A U , U ) = − β g ( A ξ , A U ) − k g ( A U , A U ) . As k ≠ 0 , α g ( A U , U ) = g ( A U , A U ) and α γ = β 2 + γ 2 . As β 2 = ( α − k ) γ , we obtain γ ( γ − k ) = 0 . If γ = 0 , β 2 = 0 , which is impossible. Then γ = k , and we can write A ξ = α ξ + β U , A U = β ξ + k U , A ϕ U = 0 , and β 2 + k 2 = α k . This also yields that D U is A -invariant.

Take X , Y ∈ D U in (3.5) to obtain g ( ϕ A 2 X , A Y ) ξ − g ( ϕ A 2 X , Y ) A ξ = g ( ϕ A X , A 2 Y ) ξ − g ( ϕ A X , A Y ) A ξ . As β ≠ 0 , its scalar product with U implies − g ( ϕ A 2 X , Y ) = − g ( ϕ A X , A Y ) for any X , Y ∈ D U , Therefore, ϕ A 2 X = A ϕ A X for any X ∈ D U . If such an X satisfies A X = λ X , it follows λ 2 ϕ X = λ A ϕ X . Then, either λ = 0 or for any λ ≠ 0 , A ϕ X = λ ϕ X . This means that the eigenspace associated to any non-zero eigenvalue is ϕ -invariant. But then, also the eigenspace corresponding to 0 must be ϕ -invariant. Let us suppose that there exists a unit Z ∈ D U such that A Z = A ϕ Z = 0 . Then the Codazzi equation implies that − A ∇ Z ϕ Z + A ∇ ϕ Z Z = − 2 ξ . Its scalar product with ξ gives β g ( [ ϕ Z , Z ] , U ) = − 2 , and we can see that g ( [ ϕ Z , Z ] , U ) ≠ 0 . But its scalar product with U yields γ g ( [ ϕ Z , Z ] , U ) = 0 . Therefore, k = γ = 0 , which is impossible.

Take X ∈ D U , Y = U in (3.5) and its scalar product with ϕ Z , Z ∈ D U . Then we obtain − β g ( A 2 X , Z ) = − g ( A U , A ξ ) g ( A X , Z ) + β g ( ϕ A X , A ϕ Z ) . As β ≠ 0 , this implies A 2 X = ( α + k ) A X + ϕ A ϕ A X for any X ∈ D U . If we suppose A X = λ X , we obtain λ 2 = ( α + k ) λ − λ 2 . Now, λ ( 2 λ − ( α + k ) ) = 0 . As we know that λ ≠ 0 , the unique principal curvature on D U is α + k 2 . Take a unit Z ∈ D U . From ( ∇ Z A ) ϕ Z − ( ∇ ϕ Z A ) Z = − 2 ξ and its scalar product with ϕ Z , we obtain

for any Z ∈ D U . Bearing in mind that β 2 + k 2 = k α , we also have from (3.14)

Now ( ∇ ϕ U A ) ξ − ( ∇ ξ A ) ϕ U = U implies ( ϕ U ) ( α ) ξ + ( ϕ U ) ( β ) U + β ∇ ϕ U U + A ∇ ξ ϕ U = U . Its scalar product with ξ gives

and its scalar product with U implies

Substracting to the product of (3.17) and β the product of (3.16) and k , we obtain β ( ϕ U ) ( β ) − k ( ϕ U ) ( α ) = β + β 3 − k α β . On the other hand, as β 2 + k 2 = k α , 2 β ( ϕ U ) ( β ) − k ( ϕ U ) ( α ) = 0 . Therefore, ( ϕ U ) ( β ) = − 1 − β 2 + k α . By using again that β 2 = k α − k 2 , we obtain

Moreover, ( ∇ ϕ U A ) U − ( ∇ U A ) ϕ U = 2 ξ gives ( ϕ U ) ( β ) ξ + k ∇ ϕ U U − A ∇ ϕ U U + A ∇ U ϕ U = 2 ξ . Its scalar product with U yields − β g ( ϕ U , ϕ A U ) + k g ( ∇ U ϕ U , U ) = 0 and, as k ≠ 0 ,

and its scalar product with ξ implies ( ϕ U ) ( β ) − α g ( ϕ U , ϕ A U ) + β g ( ∇ U ϕ U , U ) = 2 . From (3.19), we obtain ( ϕ U ) ( β ) = 2 + α k − β 2 = 2 + k 2 . This and (3.18) give 3 = 0 , which is impossible and finishes the proof.

4 Proof of Theorem 1.2

If we suppose that M satisfies that A F ( k ) is η -hybrid with respect to A , we should have F A X ( k ) A Y − A F A X ( k ) Y + F X A 2 Y − A F X A Y = 0 for any X , Y ∈ D . Therefore,

for any X , Y ∈ D . Let us suppose first that M is Hopf with A ξ = α ξ . Then from (4.1), we have g ( A ϕ A 2 X , Y ) ξ − α g ( ϕ A 2 X , Y ) ξ + g ( A 2 ϕ A X , Y ) ξ − α g ( A ϕ A X , Y ) ξ = 0 for any X , Y ∈ D . Thus for any X ∈ D , A ϕ A 2 X − α ϕ A 2 X + A 2 ϕ A X − α A ϕ A X = 0 . If we suppose that X ∈ D satisfies A X = λ X , then A ϕ X = μ ϕ X with μ = α λ + 2 2 λ − α . Now λ 2 μ − α λ 2 + λ μ 2 − λ μ α = 0 . This implies ( μ − α ) λ ( λ + μ ) = 0 . If we suppose λ + μ = 0 , we should have λ + α λ + 2 2 λ − α = 2 λ 2 + 2 2 λ − α = 0 , which is impossible. Therefore, λ ( μ − α ) = 0 . This yields either λ = 0 , μ = − 2 α or μ = α , λ = α 2 + 2 2 . In any case, M must have three distinct constant principal curvatures. In both cases, λ ≠ μ . Therefore, looking at the principal curvatures of the real hypersurfaces in Takagi’s list, we see that M must be locally congruent to a real hypersurface of type (B). The second case is impossible. In the first case, α = 2 cot ( 2 r ) and either cot r − π 4 = 0 or tan r − π 4 = 0 [21]. Both possibilities give contradictions, and we have proved that M must be non-Hopf.

As in the previous section, we write locally A ξ = α ξ + β U , with the same conditions, and also make the following computations locally. The scalar product of (4.1) and ϕ U gives

for any X , Y ∈ D . Take X = Y = ϕ U in (4.2) to obtain − β g ( A U , ϕ U ) 2 = 0 . As β ≠ 0 , we deduce

If in (4.2)  X = Y ∈ D U we have − β g ( A U , X ) 2 = 0 for any X ∈ D U , Then, (4.2) and (4.3) imply

for a certain function γ . If we take X = U , Y ∈ D U in (4.2), we obtain k β g ( A ϕ U , ϕ Y ) = 0 for any Y ∈ D U . Taking ϕ Y instead of Y , as k β ≠ 0 , it follows g ( A ϕ U , Y ) = 0 for any Y ∈ D U . Together with (4.3), this yields

for some function δ . The scalar product of (4.1) and U gives β g ( Y , U ) g ( A X , A ϕ U ) + k β g ( X , U ) g ( Y , A ϕ U ) − β g ( ϕ A 2 X , Y ) + k β g ( X , U ) g ( ϕ Y , A U ) + g ( A Y , A ξ ) g ( A ϕ U , X ) − β g ( ϕ A X , A Y ) + β g ( Y , U ) g ( ϕ A X , A U ) = 0 for any X , Y ∈ D . If in this equation Y = U , X = ϕ U , bearing in mind (4.4) and (4.5), it follows δ ( 2 δ + α + γ ) = 0 . Therefore, either δ = 0 or δ = − ( α + γ 2 ) . If in the expression above (4.5)  X , Y ∈ D U , we obtain, β being non-zero, g ( ϕ A 2 X , Y ) + g ( ϕ A X , A Y ) = 0 for any X , Y ∈ D U . As from (4.4) and (4.5), D U is A -invariant, ϕ A 2 X + A ϕ A X = 0 for any X ∈ D U . Taking a unit X ∈ D U such that A X = λ X , we have λ ( A ϕ X + λ ϕ X ) = 0 . Therefore, either λ = 0 or if λ ≠ 0 , A ϕ X = − λ ϕ X . This tells us that if 0 is a principal curvature on D U its corresponding eigenspace is ϕ -invariant.

The scalar product of (4.1) and Z ∈ D U implies − η ( A Y ) g ( ϕ A 2 X , Z ) − k η ( A X ) g ( ϕ A Y , Z ) + k η ( A X ) g ( A ϕ Y , Z ) − η ( A 2 Y ) g ( ϕ A X , Z ) + η ( A Y ) g ( A ϕ A X , Z ) = 0 . If X = U , Y ∈ D U we obtain − k β g ( ϕ A Y , Z ) + k β g ( A ϕ Y , Z ) = 0 , for any Y , Z ∈ D U . As k β ≠ 0 , this yields A ϕ Y = ϕ A Y for any Y ∈ D U . This and the aforementioned discussion prove that the unique principal curvature on D U is 0. Therefore, A W = 0 for any W ∈ D U . For such a W , the equation of Codazzi gives ( ∇ W A ) ϕ W − ( ∇ ϕ W A ) W = − 2 ξ , and from this, − A ∇ W ϕ W + A ∇ ϕ W W = − 2 ξ . Its scalar product with ξ implies − g ( ∇ W ϕ W , α ξ + β U ) + g ( ∇ ϕ W W , α ξ + β U ) = − 2 . That is,

and therefore, g ( [ ϕ W , W ] , U ) ≠ 0 . But the scalar product of the aforementioned expression and U implies − g ( ∇ W ϕ W , β ξ + γ U ) + g ( ∇ ϕ W W , β ξ + γ U ) = 0 and