Abstract
Let M be a real hypersurface of a complex projective space. For any operator B on M and any nonnull real number k, we can define two tensor fields of type (1,2) on M, \(B_F^{(k)}\) and \(B_T^{(k)}\). We will classify real hypersurfaces in complex projective space for which \(B_F^{(k)}\) and \(B_T^{(k)}\) either take values in the maximal holomorphic distribution \(\mathbb {D}\) or are parallel to the structure vector field \(\xi \), in the particular case of \(B=A\), where A denotes the shape operator of M. We also introduce the concept of \(A_F^{(k)}\) and \(A_T^{(k)}\) being \(\mathbb {D}\)-recurrent and classify real hypersurfaces such that either \(A_F^{(k)}\) or \(A_T^{(k)}\) are \(\mathbb {D}\)-recurrent.
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1 Introduction
Consider the complex projective space \(\mathbb {C}P^{m}\), \(m \ge 2\), endowed with the Kaehlerian structure (J, g), where J denotes the complex structure and g the Fubini-Study metric of constant holomorphic sectional curvature 4 on \(\mathbb {C}P^{m}\). Let M be a connected real hypersurface in \(\mathbb {C}P^{m}\) without boundary. Denote also by g the induced metric on M and by N a local unit normal vector field on M. The Reeb (or structure) vector field on M is defined by \(\xi =-JN\). Let \(\nabla \) be the Levi-Civita connection on M and A the shape operator associated to N. For any vector field X tangent to M write \(JX=\phi X+\eta (X)N\), where \(\phi X\) is the tangential component of JX and \(\eta (X)=g(X,\xi )\). Then \((\phi , \xi , \eta , g)\) defines an almost contact metric structure on M [1].
The existence of such a structure allows us to define, for any nonnull real number k, the so-called kth Generalized Tanaka–Webster connection on M, \({\hat{\nabla }}^{(k)}\) [2, 3], given by
for any X, Y tangent to M.
This connection on M is a metric one and any element of the almost contact metric structure is parallel for such a connection. If A satisfies \(\phi A+A\phi =2k\phi \), M becomes a contact manifold and this connection coincides with the Tanaka–Webster connection on M, [14,15,16].
The tensor field of type (1, 2) obtained as the difference of both connections is called the kth Cho tensor on M (see [6, Proposition 7.10]) and it is given by \(F^{(k)}(X,Y)=g(\phi AX,Y)\xi -\eta (Y)\phi AX-k\eta (X)\phi Y\), for any X, Y tangent to M. From this, for any X tangent to M and any nonnull real number k, we define the kth Cho operator corresponding to X, as \(F_X^{(k)}Y=F^{(k)}(X,Y)\) for any Y tangent to M.
The torsion of the connection \({\hat{\nabla }}^{(k)}\) is given by \(T^{(k)}(X,Y)=F_X^{(k)}Y-F_Y^{(k)}X\) for any X, Y tangent to M, [2]. Thus we define the kth torsion operator associated to X, for any nonnull real number k and any X tangent to M, by \(T_X^{(k)}Y={\hat{T}}^{(k)}(X,Y)\), for any Y tangent to M.
If \({\mathcal {L}}\) denotes the Lie derivative on M, we now that it is given by \({{\mathcal {L}}}_XY=\nabla _XY-\nabla _YX\), for any X, Y tangent to M. If we consider the kth Generalized Tanaka–Webster connection we can also define on M a differential operator of first order, that we call the derivative of Lie type associated to such a connection, \({{\mathcal {L}}}^{(k)}\), given by
for any X, Y tangent to M.
Best known real hypersurfaces in \(\mathbb {C}P^m\) are called Hopf and satisfy that the Reeb vector field \(\xi \) is an eigenvector of the shape operator, that is, \(A\xi =\alpha \xi \), for a certain function \(\alpha \) on M, called the Reeb curvature of M. The distribution on M given by \(\mathbb {D}=Ker(\eta )\) is \(\phi \)-invariant and called the maximal holomorphic distribution on M.
Takagi classified homogeneous real hypersurfaces in complex projective space (see [11,12,13]). Kimura [4], proved that Takagi’s real hypersurfaces are the unique ones that are Hopf and have constant principal curvatures for A in \(\mathbb {C}P^m\). Takagi’s list contains the following 6 types of real hypersurfaces
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Type \((A_1)\), geodesic hyperspheres of radius r, \(0< r < \frac{\pi }{2}\), with 2 distinct constant principal curvatures, \(2\cot (2r)\) with eigenspace \(\mathbb {R}[\xi ]\) and \(\cot (r)\) with eigenspace \(\mathbb {D}\).
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Type \((A_2)\), tubes of radius r, \(0< r < \frac{\pi }{2}\), over totally geodesic complex projective spaces \(\mathbb {C}P^n\), \(0< n < m-1\), with 3 distinct constant principal curvatures, \(2\cot (2r)\) with eigenspace \(\mathbb {R}[\xi ]\), \(\cot (r)\) and \(-\tan (r)\). The corresponding eigenspaces of \(\cot (r)\) and \(-\tan (r)\) are complementary and \(\phi \)-invariant distributions in \(\mathbb {D}\).
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Type (B), tubes of radius r, \(0< r < \frac{\pi }{4}\), over the complex quadric \(Q^{m-1}\), with 3 distinct constant principal curvatures, \(2\cot (2r)\) with eigenspace \(\mathbb {R}[\xi ]\), \(\cot (r-\frac{\pi }{4})\) and \(-\tan (r-\frac{\pi }{4})\) whose corresponding eigenspaces are complementary and equal dimensional distributions in \(\mathbb {D}\) such that \(\phi V_{\cot (r-\frac{\pi }{4})}=V_{-\tan (r-\frac{\pi }{4})}\).
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Type (C), tubes of radius r, \(0< r < \frac{\pi }{4}\), over the Segre embedding of \(\mathbb {C}P^1 \times \mathbb {C}P^n\), where \(2n+1=m\) and \(m \ge 5\), with 5 distinct constant principal curvatures, \(2\cot (2r)\) with eigenspace \(\mathbb {R}[\xi ]\), \(\cot (r-\frac{\pi }{4})\) with multiplicity 2, \(\cot (r-\frac{\pi }{2})=-\tan (r)\) with multiplicity \(m-3\), \(\cot (r-\frac{3\pi }{4})\), with multiplicity 2 and \(\cot (r-\pi )=\cot (r)\) with multiplicity \(m-3\). Moreover \(\phi V_{\cot (r-\frac{\pi }{4})}=V_{\cot (r-\frac{3\pi }{4})}\) and \(V_{-\tan (r)}\) and \(V_{\cot (r)}\) are \(\phi \)-invariant.
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Type (D), tubes of radius r, \(0< r < \frac{\pi }{4}\), over the Plucker embedding of the complex Grassmannian manifold G(2, 5) in \(\mathbb {C}P^9\), with the same principal curvatures as type (C), \(2\cot (2r)\) with eigenspace \(\mathbb {R}[\xi ]\), and the other 4 principal curvatures have the same multiplicity 4 and their eigenspaces have the same behaviour with respect to \(\phi \) as in type (C).
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Type (E), tubes of radius r, \(0< r < \frac{\pi }{4}\), over the canonical embedding of the Hermitian symmetric space SO(10)/U(5) in \(\mathbb {C}P^{15}\). They also have the same principal curvatures as type (C), \(2 \cot (2r)\) with eigenspace \(\mathbb {R}[\xi ]\), \(\cot (r-\frac{\pi }{4})\) and \(\cot (r-\frac{3\pi }{4})\) have multiplicities equal to 6 and \(-\tan (r)\) and \(\cot (r)\) have multiplicities equal to 8. Their corresponding eigenspaces have the same behaviour with respect to \(\phi \) as in type (C).
We will call type (A) real hypersurfaces to both types \((A_1)\) or \((A_2)\).
Ruled real hypersurfaces in \(\mathbb {C}P^m\) were introduced by Kimura [5]. The maximal holomorphic distribution \(\mathbb {D}\) of such real hypersurfaces is integrable with integral manifolds \(\mathbb {C}P^{m-1}\). Equivalently, \(g(A\mathbb {D},\mathbb {D})=0\). Kimura gave some minimal examples of this kind of real hypersurfaces.
Let B be a symmetric operator on M. Then we can define on M a couple of tensor fields of type (1,2), for any nonnull real number k, \(B_F^{(k)}\) and \(B_T^{(k)}\), given, respectively, by
and
for any X, Y tangent to M.
In [10] we considered the case \(A=B\) in (1.1), and proved non-existence of real hypersurfaces in \(\mathbb {C}P^m\), \(m \ge 3\), such that \(A_F^{(k)}=0\), for any nonnull real number k. A similar result for \(A=B\) in (1.2) was obtained in [9]. Such conditions imply commutativity of A and either \(F_X^{(k)}\) or \(T_X^{(k)}\), for any X tangent to M, respectively.
In this paper we want to generalize such results. Then, we will consider the conditions \(g(A_F^{(k)}(X,Y),\xi )=0\), (respectively, \(g(A_F^{(k)}(X,Y),Z)=0\)) for any X, Y tangent to M (respectively, for any X, Y tangent to M, \(Z \in \mathbb {D}\)), obtaining the following
Theorem 1.1
Let M be a real hypersurface in \(\mathbb {C}P^{m}\), \(m \ge 3\), and k a nonnull real number. Then \(g(A_F^{(k)}(X,Y),\xi )=0\) for any X, Y tangent to M if and only if M is locally congruent to a ruled real hypersurface such that \(g(A\xi ,\xi )=-k\).
And
Theorem 1.2
Let M be a real hypersurface in \(\mathbb {C}P^m\), \(m \ge 3\), and k a nonnull constant. Then \(g(A_F^{(k)}(X,Y),Z)=0\) for any X, Y tangent to M, \(Z \in \mathbb {D}\), if and only if M is locally congruent to a real hypersurface of type (A)
Similar conditions for \(A_T^{(k)}\) give us the following results
Theorem 1.3
There does not exist any real hypersurface M in \(\mathbb {C}P^m\), \(m \ge 3\), such that \(g(A_T^{(k)}(X,Y),\xi )=0\), for any X, Y tangent to M and any nonnull real number k.
And
Theorem 1.4
Let M be a real hypersurface in \(\mathbb {C}P^m,m \ge 3\) and k a nonnull real number. Then \(g(A_T^{(k)}(X,Y),Z)=0\), for any X, Y tangent to M, \(Z \in \mathbb {D}\), if and only if M is locally congruent to a real hypersurface of type (A).
On the other hand, we will say that A is \(({\hat{\nabla }}^{(k)},\nabla )\)-recurrent if \((({\hat{\nabla }}_X^{(k)}-\nabla _X)A)Y=\omega (X)AY\), for any X, Y tangent to M, where \(\omega \) is a nonnull 1-form on M. This is equivalent to have \(A_F^{(k)}(X,Y)=\omega (X)AY\).
Similarly, we will say that A is \(({{\mathcal {L}}}^{(k)},\mathcal L)\)-recurrent if \((({{\mathcal {L}}}_X^{(k)}-{{\mathcal {L}}}_X)A)Y=\delta (X)AY\), for any X, Y tangent to M and a nonnull 1-form \(\delta \) on M. This is equivalent to have \(A_T^{(k)}(X,Y)=\delta (X)AY\).
If we consider \({\mathbb {D}}-({\hat{\nabla }}^{(k)},\nabla )\)-recurrency or \({\mathbb {D}}-({{\mathcal {L}}}^{(k)},{\mathcal {L}})\)-recurrency (the same conditions as above for \(X,Y \in \mathbb {D}\)) we obtain
Theorem 1.5
Let M be a real hypersurface in \(\mathbb {C}P^m\), \(m \ge 3\), and k a nonnull real number. Then \(A_F^{(k)}(X,Y)=\omega (X)AY\), for any \(X,Y \in \mathbb {D}\) and a nonnull 1-form \(\omega \) on M if and only if M is locally congruent either to a real hypersurface of type (A) or to a ruled real hypersurface.
and
Theorem 1.6
There does not exist any real hypersurface M in \(\mathbb {C}P^m\), \(m \ge 3\), such that \(A_T^{(k)}(X,Y)=\delta (X)AY\), for any \(X,Y \in \mathbb {D}\), and a nonnull 1-form \(\delta \) on M, k being a nonnull real number.
2 Preliminaries
Any mathematical object in the sequel will be considered of class \(C^{\infty }\) unless otherwise stated. Let M be a connected real hypersurface without boundary in \(\mathbb {C}P^m, m \ge 2\), and N a locally defined normal unit vector field on M. Let\(\nabla \) be the Levi-Civita connection on M and (J, g) the Kaehlerian structure of \(\mathbb {C}P^m\).
For any vector field X tangent to M, we write \(JX=\phi X+\eta (X)N\), where \(\phi X\) denotes the tangential component of JX, and \(-JN=\xi \). Then \((\phi ,\xi ,\eta ,g)\) is an almost contact metric structure on M (see [1]). Therefore,
for any tangent vectors X, Y to M. From (2.1) we get
From the parallelism of J we obtain
for any X, Y tangent to M, where A denotes the shape operator of the immersion. As \(\mathbb {C}P^m\) has holomorphic sectional curvature 4, the equations of Gauss and Codazzi are given, respectively, by
and
for any tangent vectors X, Y, Z to M, where R is the curvature tensor of M.
In the sequel the following result owed to Maeda [7], is needed.
Theorem 2.1
Let M be a Hopf real hypersurface in \(\mathbb {C}P^m\), \(m\ge 2\). Then \(\alpha =g(A\xi ,\xi )\) is constant and if W is a vector field which belongs to \(\mathbb {D}\) such that \(AW=\lambda W\), then \(2\lambda -\alpha \ne 0\) and \(A\phi W=\mu \phi W\), where \(\mu =\frac{\alpha \lambda +2}{2\lambda -\alpha }\).
We will also need the following theorem proved by Okumura [8]
Theorem 2.2
Let M be a real hypersurface in \(\mathbb {C}P^m\), \(m \ge 2\). Then \(\phi A=A\phi \) if and only if M is locally congruent to a real hypersurface of type (A).
3 Proofs of Theorems 1.1 and 1.2
Let us suppose that \(g(A_F^{(k)}(X,Y),\xi )=0\) for any X, Y tangent to M. This yields \(g(g(\phi AX,\) \(AY)\xi -\eta (AY)\phi AX-k\eta (X)\phi AY-g(\phi AX,Y)A\xi +\eta (Y)A\phi AX+k\eta (X)A\phi Y,\xi )=0\), for any X, Y tangent to M. Therefore
for any X, Y tangent to M.
Let us suppose that M is Hopf, that is, \(A\xi =\alpha \xi \). Then (3.1) gives \(g(A\phi AX,Y)-\alpha g(\phi AX,Y)=0\), for any X, Y tangent to M. Thus \(A\phi AX=\alpha \phi AX\), for any X tangent to M. If we choose \(X \in \mathbb {D}\) such that \(AX=\lambda X\), from Theorem 2.1 we should have \(A\phi X=\mu \phi X\), \(\mu =\frac{\alpha \lambda +2}{2\lambda -\alpha }\). Then \(\lambda \mu =\lambda \alpha \) and either \(\lambda =0\) or \(\mu =\alpha \).
If we suppose that in \(\mathbb {D}\) there exists a principal curvature \(\lambda \ne 0\), \(\mu =\alpha \) yields \(\alpha \lambda +2=2\alpha \lambda -\alpha ^2\). That is, \(\alpha \lambda =\alpha ^2+2\). This implies that \(\alpha \ne 0\) and then \(\lambda =\frac{\alpha ^2+2}{\alpha }\). As \(\mu =\alpha \), we also have \(\lambda \ne \mu \) and all the principal curvatures are constant. Therefore \(A\phi \ne \phi A\) and M cannot be of type (A). If there is not a vector field \(Y \in \mathbb {D}\) such that \(AY=0\), the unique principal curvatures on \(\mathbb {D}\) are \(\alpha \) and \(\frac{\alpha ^2+2}{\alpha }\). Looking at Takagi’s list, this is impossible.
Therefore, the unique principal curvature in \(\mathbb {D}\) is \(\lambda =0\). But then, \(\mu =-\frac{2}{\alpha }\) must be equal to 0 too, which is also impossible.
Then we must suppose that M is non Hopf. So we can write \(A\xi =\alpha \xi +\beta U\), where U is a unit vector field in \(\mathbb {D}\) and \(\beta \) is a function on M that does not vanish at least on a neighborhood of a point \(p \in M\). We will make all the calculations on such a neighborhood.
If we take \(Y=\xi \) in (3.1) we get \(2g(A\phi A\xi ,X)=0\) for any X tangent to M. That is, \(\beta g(A\phi U,X)=0\) for any X tangent to M, which yields
Taking \(X=\xi \) in (3.1) we obtain \(g(A\phi A\xi ,Y)-\alpha g(\phi A\xi ,Y)+kg(A\phi Y,\xi )=0\), for any Y tangent to M. Then, from (3.2), \(-\alpha \beta g(\phi U,Y)-k\beta g(\phi U,Y)=0\). As \(\beta \ne 0\), if we take \(Y=\phi U\) we have
If now we take \(Y=\phi U\) in (3.1) it follows \(-\alpha g(AX,U)-k\eta (X)g(AU,\xi )=0\), for any X tangent to M. That is, \(-\alpha g(AU,X)-k\beta \eta (X)=0\). From (3.3) we get \(kg(AU,X)-k\beta \eta (X)=0\), for any X tangent to M. Thus
From (3.2) and (3.4) we have that \({\mathbb {D}}_U=\{ X \in \mathbb {D}| g(X,U)=g(X,\phi U)=0 \}\) is A-invariant. Take \(X,Y \in {\mathbb {D}}_U\) in (3.1). Then \(g(A\phi AX,Y)-\alpha g(\phi AX,Y)=0\). From (3.3) this yields
for any \(X \in {\mathbb {D}}_U\). We can also write the equation above (3.5) as \(-g(A\phi AY,X)+\alpha g(A\phi Y,X)=0\), for any \(X,Y \in {\mathbb {D}}_U\). From (3.3) we obtain
for any \(X \in {\mathbb {D}}_U\). Adding (3.5) and (3.6) we have \(k(\phi A-A\phi )X=0\) for any \(X \in {\mathbb {D}}_U\) and, as \(k \ne 0\), we get
for any \(X \in {\mathbb {D}}_U\). Therefore, if \(X \in {\mathbb {D}}_U\) satisfies \(AX=\lambda X\), we obtain \(A\phi X=\lambda \phi X\). Moreover, from (3.5) it follows \(\lambda ^2+k\lambda =0\) and either \(\lambda =0\) or \(\lambda =-k\).
Let us suppose that there exists \(Y \in {\mathbb {D}}_U\) such that \(AY=-kY\) and \(A\phi Y=-k\phi Y\). The Codazzi equation yields \((\nabla _YA)\phi Y-(\nabla _{\phi Y}A)Y=-2\xi \). Therefore, \(-k\nabla _Y\phi Y-A\nabla _Y\phi Y+k\nabla _{\phi Y}Y+A\nabla _{\phi Y}Y=-2\xi \). Its scalar product with \(\xi \) gives \(kg(\phi Y,\phi AY)-g(\nabla _Y\phi Y,-k\xi +\beta U)-kg(Y,\phi A\phi Y)+g(\nabla _{\phi Y}Y,-k\xi +\beta U)=-2\). This yields \(\beta g([\phi Y,Y ],U)-k^2-kg(\phi Y,\phi AY)\) \(+kg(A\phi Y,\phi Y)+kg(Y,\phi A\phi Y)=-2\). Thus
Its scalar product with U implies \(-kg(\nabla _Y\phi Y,U)-g(\nabla _Y\phi Y,\beta \xi )+kg(\nabla _{\phi Y}Y,U)+g(\nabla _{\phi Y}Y,\) \(\beta \xi )=0\). That is, \(kg([\phi Y,Y ],U)+\beta g(\phi Y,\phi AY)-\beta g(Y,\phi A\phi Y)=0\). Then
From (3.7) and (3.8) \(\beta =-\frac{1}{\beta }\) would give \(\beta ^2=-1\), which is impossible.
We conclude that the unique principal curvature in \({\mathbb {D}}_U\) is 0 and M is ruled. The converse is straightforward and we finish the proof of Theorem 1.1.
In order to prove Theorem 1.2 let us suppose that \(g(A_F^{(k)}(X,Y),Z)=0\) for any X, Y tangent to M, \(Z \in \mathbb {D}\). This implies
for any X, Y tangent to M, \(Z \in {\mathbb {D}}\).
Let us suppose that M is Hopf and \(A\xi =\alpha \xi \). Taking \(X=\xi \) in (3.9) we get \(-kg(\phi AY,Z)+kg(A\phi Y, Z)=0\) for any Y tangent to M, \(Z \in {\mathbb {D}}\). As \(k \ne 0\), this means that \((A\phi -\phi A)X=0\) for any \(X \in \mathbb {D}\). From Theorem 2.2, M must be locally congruent to a real hypersurface of type (A).
If M is non Hopf we will write \(A\xi =\alpha \xi +\beta U\) with the same conditions as in the proof of Theorem 1.1. Taking \(X=Y=\xi \) in (3.9) we have \(-\alpha \beta g(\phi U,Z)-k\beta g(\phi U,Z)+\beta g(A\phi U,Z)=0\) for any \(Z \in \mathbb {D}\). This gives, bearing in mind that \(\beta \ne 0\),
If in (3.9) we put \(Y=\xi \) we get \(-\alpha g(\phi AX,Z)-k\beta \eta (X)g(\phi U,Z)+g(A\phi AX,Z)=0\), for any X tangent to M, \(Z \in \mathbb {D}\). If \(Z=\phi U\), we obtain \(-\alpha g(AU,X)-k\beta \eta (X)+(\alpha +k)g(AU,X)=0\), for any X tangent to M. This implies \(kAU=k\beta \xi \). As \(k \ne 0\) we obtain
If we take \(Y=\xi \), \(X=\phi U\) in (3.9) we have \(\alpha (\alpha +k)g(U,Z)-(\alpha +k)g(AU,Z)=0\), for any \(Z \in \mathbb {D}\). From (3.11) we get \(\alpha (\alpha +k)g(U,Z)=0\) for any \(Z \in \mathbb {D}\). Taking \(Z=U\) we obtain \(\alpha (\alpha +k)=0\).
Let us suppose that \(\alpha =-k\). Then (3.10) and (3.11) imply \(A\xi =-k\xi +\beta U\), \(AU=\beta \xi \), \(A\phi U=0\). If we introduce \(X,Y \in {\mathbb {D}}_U\) in (3.9) we have \(-g(\phi AX,Y)g(A\xi ,Z)=0\), for any \(Z \in \mathbb {D}\). If \(Z=U\) we get \(g(\phi AX,Y)=0\) for any \(X,Y \in {\mathbb {D}}_U\). Now, if we take \(\phi Y\) instead of Y it follows \(g(AX,Y)=0\) for any \(X,Y \in {\mathbb {D}}_U\) and
for any \(X \in {\mathbb {D}}_U\). From (3.10), (3.11), (3.12) and the fact that \(\alpha =-k\), M should be ruled. But taking \(X=\xi \), \(Y=U\) in (3.10) we have \(k\beta g(\phi U,Z)=0\) for any \(Z \in \mathbb {D}\), which is impossible.
Suppose then that \(\alpha =0\). Therefore, \(A\xi =\beta U\), \(AU=\beta \xi \) and \(A\phi U=k\phi U\). Take \(X=\xi \), \(Y \in {\mathbb {D}}_U\) in (3.9). Then \(-kg(\phi AY,Z)+kg(A\phi Y,Z)=0\), for any \(Y \in {\mathbb {D}}_U\), \(Z \in \mathbb {D}\). This yields \(A\phi Y=\phi AY\) for any \(Y \in {\mathbb {D}}_U\). As \({\mathbb {D}}_U\) is A-invariant, if \(Y \in {\mathbb {D}}_U\) satisfies \(AY=\lambda Y\), \(A\phi Y=\lambda \phi Y\). If we take \(Y=\xi \), \(X \in {\mathbb {D}}_U\) in (3.9) we obtain \(g(A\phi AX,Z)=0\) for any \(X \in {\mathbb {D}}_U\), \(Z \in \mathbb {D}\). Therefore, \(A\phi AX=0\) for any \(X \in {\mathbb {D}}_U\). That is, if \(Y \in {\mathbb {D}}_U\) satisfies \(AY=\lambda Y\) we obtain \(\lambda =0\). Therefore \(AZ=0\) for any \(Z \in {\mathbb {D}}_U\). For such a Z Codazzi equation gives \((\nabla _ZA)\xi -(\nabla _{\xi }A)Z=-\phi Z\). Then \(\nabla _Z(\beta U)-A\phi AZ+A\nabla _{\xi }Z=-\phi Z\). This implies \(Z(\beta )U+\beta \nabla _ZU+A\nabla _{\xi }Z=-\phi Z\) and its scalar product with U implies \(Z(\beta )-\beta g(Z,\phi A\xi )=0\). We have proved that
for any \(Z \in {\mathbb {D}}_U\).
On the other hand, \((\nabla _UA)\xi -(\nabla _{\xi }A)U=-\phi U\) implies \(U(\beta )U+\beta \nabla _UU-\xi (\beta )\xi -\beta \phi A\xi +A\nabla _{\xi }U=-\phi U\). Its scalar product with \(\xi \) gives \(-\beta g(U,\phi AU)-\xi (\beta )+\beta g(\nabla _ {\xi }U,U)=0\). That is,
and its scalar product with U yields \(U(\beta )-\beta ^2g(U,\phi U)=0\). Thus
Also \((\nabla _{\phi U}A)\xi -(\nabla _{\xi }A)\phi U=U\) yields \((\phi U)(\beta )U+\beta \nabla _{\phi U}U+kAU-k\nabla _{\xi }\phi U+A\nabla _{\xi }\phi U=U\). Its scalar product with \(\xi \) implies \(3k\beta +\beta g(\nabla _{\xi }\phi U,U)=0\). Then
and its scalar product with U gives \((\phi U)(\beta )-kg(\nabla _{\xi }\phi U,U)-\beta g(A\xi ,U)=1\). Therefore, from (3.16),
From (3.13), (3.14), (3.15) and (3.17) we obtain \(grad(\beta )=\gamma \phi U\), where \(\gamma =-3k^2+\beta ^2+1\). As \(g(\nabla _Xgrad(\beta ),Y)=g(\nabla _Ygrad(\beta ),X)\), for any X, Y tangent to M, we get \(X(\gamma )g(\phi U,Y)+\gamma g(\nabla _X\phi U,Y)=Y(\gamma )g(\phi U,X)+\gamma g(\nabla _Y\phi U,X)\). If \(X=\xi \) we obtain \(\gamma g(\nabla _{\xi }\phi U,Y)=\gamma g(\nabla _Y\phi U,\xi )=-\gamma g(U,AY)\) for any Y tangent to M. If now \(Y=U\) it follows \(\gamma g(\nabla _{\xi }\phi U,U)=0\). From (3.16) we get \(-3k\gamma =0\). Thus \(\gamma =0\) and \(\beta \) is constant.
Then \((\nabla _{\phi U}A)U-(\nabla _UA)\phi U=2\xi \) yields \(\beta \phi A\phi U-A\nabla _{\phi U}U-k\nabla _U\phi U+A\nabla _U\phi U=2\xi \). Its scalar product with \(\xi \) gives \(kg(U,AU)+\beta g(\nabla _U\phi U,U)=2\). Therefore,
and its scalar product with U implies \(-\beta k+\beta g(U,\phi A\phi U)-kg(\nabla _U\phi U,U)=0\). That is, \(-2\beta k=kg(\nabla _U\phi U,U)\). Then
From (3.18) and (3.19) we have \(-\beta ^2=1\), which is impossible and this finishes the proof of Theorem 1.2.
4 Proofs of Theorems 1.3 and 1.4
If we suppose that \(g(A_T^{(k)}(X,Y),\xi )=0\) for any X, Y tangent to M we obtain
for any X, Y tangent to M.
Let us suppose that M is Hopf, \(A\xi =\alpha \xi \), and take \(X,Y \in \mathbb {D}\) in (4.1). Then \(g(\phi AX,AY)-g(\phi A^2Y,X)-\alpha g(\phi AX,Y)+\alpha g(\phi AY,X)=0\). Therefore, we obtain \(A\phi AX+A^2\phi X-\alpha \phi AX-\alpha A\phi X=0\) for any \(X \in \mathbb {D}\). If \(X \in \mathbb {D}\) satisfies \(AX=\lambda X\), from Theorem 2.1 we know that \(A\phi X=\mu \phi X\). Thus \(\lambda \mu +\mu ^2-\alpha \lambda -\alpha \mu =0\). That is, \((\lambda +\mu )\mu -(\lambda +\mu )\alpha =0\), or \((\lambda +\mu )(\mu -\alpha )=0\). If \(\lambda +\mu =0\), as \(\mu =\frac{\alpha \lambda +2}{2\lambda -\alpha }\), we obtain \(2\lambda ^2+2=0\), which is impossible. Therefore \(\mu =\alpha \) and then, as in the proof of Theorem 1.1, this case is not possible.
Therefore M must be non Hopf. We continue writing \(A\xi =\alpha \xi +\beta U\) as in Sect. 3.
Taking \(X=\xi \) in (4.1) we obtain \(\beta g(\phi U,AY)-\alpha \beta g(\phi U,Y)+kg(\phi Y,A\xi )-g(\phi AY,A\xi )=0\), for any Y tangent to M. This gives \(\beta A\phi U-\alpha \beta \phi U-k\beta \phi U+\beta A\phi U=0\). Then \(2A\phi U-(\alpha +k)\phi U=0\) and
If now we take \(Y=\xi \) in (4.1) it follows \(2\,g(\phi AX,A\xi )-g(\phi A^2\xi ,X)+\alpha g(\phi A\xi ,X)-kg(\phi X,A\xi )=0\), for any X tangent to M. Then \(-2\beta g(A\phi U,X)-g(\phi A(\alpha \xi +\beta U),X)+\alpha \beta g(\phi U,X)+k\beta g(\phi U,X)=0\), for any X tangent to M. From (4.2) we get \(-\alpha g(\phi A\xi ,X)-\beta g(\phi AU,X)=0\), for any X tangent to M. Therefore, \(-\alpha \beta \phi U-\beta \phi AU=0\), or \(\phi AU=-\alpha \phi U\). Applying \(\phi \) we obtain
Take \(X=\phi U\) in (4.1). Then \(g(\phi A\phi U,AY)-g(A^2Y,U)-\alpha g(\phi A\phi U,Y)+\eta (Y)g(\phi A\phi U,A\xi )\) \(+\alpha g(AY,U)+k\eta (Y)g(U,A\xi )=0\), for any Y tangent to M. From (4.2) we get \(-(\frac{\alpha +k}{2})\) \(g(AU,Y)-g(A^2U,Y)+\alpha (\frac{\alpha +k}{2})g(U,Y)-\beta (\frac{\alpha +k}{2})\eta (Y)+\alpha g(AU,Y)+k\beta \eta (Y)=0\), for any Y tangent to M. Therefore, \((\alpha -(\frac{\alpha +k}{2}))AU-A^2U+\alpha (\frac{\alpha +k}{2})U+\beta (k-(\frac{\alpha +k}{2}))\xi =0\). This and (4.3) yield
If now we take \(Y=\phi U\) in (4.1) we have \(g(\phi AX,A\phi U)-g(\phi A^2\phi U,X)-\alpha g(AX,U)-k\eta (X)g(U,A\xi )+\alpha g(\phi A\phi U,X)-\beta g(\phi A\phi U,U)\eta (X)=0\), for any X tangent to M. This yields \(((\frac{\alpha +k}{2})-\alpha )AU+(\frac{\alpha +k}{2})((\frac{\alpha +k}{2})-\alpha )U+\beta ((\frac{\alpha +k}{2})-\alpha )\xi =0\), that is, \((\frac{k-\alpha }{2})AU+(\frac{k+\alpha }{2})(\frac{k-\alpha }{2})U-\beta (\frac{k-\alpha }{2})\xi =0\). If \(\alpha =k\), from (4.4), \(\beta =0\), which is impossible. Therefore, \(k \ne \alpha \) and we get
From (4.3) and (4.5), \(\alpha =\frac{\alpha +k}{2}\), and then, \(\alpha =k\), that we have seen that is impossible, finishing the proof of Theorem 1.3.
Suppose now that \(g(A_T^{(k)}(X,Y),Z)=0\) for any X, Y tangent to M, \(Z \in \mathbb {D}\). This implies
for any X, Y tangent to M, \(Z \in \mathbb {D}\).
Let us suppose that M is Hopf with \(A\xi =\alpha \xi \). Take \(X=\xi \), \(Y \in \mathbb {D}\) in (4.6). Then we get \(-kg(\phi AY,Z)+g(\phi A^2Y,Z)-kg(\phi Y,AZ)-g(\phi AY,AZ)=0\), for any \(Y,Z \in \mathbb {D}\). Therefore,
for any \(Y \in \mathbb {D}\). If we interchange Y and Z we also obtain
for any \(Y \in \mathbb {D}\). If such a Y satisfies \(AY=\lambda Y\), from (4.7) and Theorem 2.1 we obtain
where \(\mu =\frac{\alpha \lambda +2}{2\lambda -\alpha }\). From (4.8) we also get
From (4.9) and (4.10) either \(\lambda =\mu \) for any principal curvature in \(\mathbb {D}\), and in this case, from Theorem 2.2, M is locally congruent to a real hypersurface of type (A) or there exists \(\lambda \) such that \(\mu \ne \lambda \). Then \(\lambda =\mu =k\), which is impossible.
Suppose now that M is non Hopf and write \(A\xi \) as before. Take \(X=Y=\xi \) in (4.6). Then \(-\alpha g(\phi A\xi ,Z)-kg(\phi A\xi ,Z)+g(\phi A^2\xi ,Z)+g(\phi A\xi ,AZ)-g(\phi A\xi ,AZ)=0\). Therefore, \(-(\alpha +k)\beta g(\phi U,Z)+g(\phi A(\alpha \xi +\beta U),Z)=0\), for any \(Z \in \mathbb {D}\). This yields \(-k\beta g(\phi U,Z)+\beta g(\phi AU,Z)=0\), for any \(Z \in \mathbb {D}\). Then \(\phi AU=k\phi U\), and applying \(\phi \) we get
Take now \(X=\xi \), \(Y=\phi U\) in (4.6). We obtain \(-kg(\phi A\phi U,Z)+g(\phi A^2\phi U,Z)-g(A\xi ,U)g(A\xi ,Z)-kg(AU,Z)-g(\phi A\phi U,AZ)=0\), for any \(Z \in \mathbb {D}\). If we take \(Z=U\) we get \(kg(A\phi U,\phi U)-g(A^2\phi U,\phi U)-\beta ^2-k^2+g(A\phi U,\phi AU)=0\). That is,
If we take \(X=U\), \(Y=\phi U\) in (4.6) we have \(-g(\phi AU,\phi U)g(A\xi ,Z)+g(\phi A\phi U,U)g(A\xi ,Z)=0\), for any \(Z \in \mathbb {D}\). From (4.11) it follows \(-kg(A\xi ,Z)-g(A\phi U,\phi U)g(A\xi ,Z)=0\), for any \(Z \in \mathbb {D}\). If \(Z=U\) we get \(g(A\phi U,\phi U)=-k\), and from (4.12) \(g(A\phi U,A\phi U)+\beta ^2+3k^2=0\), which is impossible, finishing the proof of Theorem 1.4.
5 Proofs of Theorems 1.5 and 1.6
If we suppose that \(A_F^{(k)}(X,Y)=\omega (X)AY\) for any \(X,Y \in \mathbb {D}\) we get
for any \(X,Y \in \mathbb {D}\).
Let us suppose that M is Hopf and that \(A\xi =\alpha \xi \). Then (5.1) becomes
for any \(X,Y \in \mathbb {D}\). The scalar product of (5.2) and \(\xi \) gives \(g(\phi AX,AY)-\alpha g(\phi AX,Y)=0\), for any \(X,Y \in \mathbb {D}\). Therefore, we have
for any \(X \in \mathbb {D}\), and interchanging X and Y we also get
for any \(X \in \mathbb {D}\). From (5.3) and (5.4) it follows \(\alpha (\phi A-A\phi )X=0\) for any \(X \in \mathbb {D}\). Let us suppose that \(\alpha =0\). Then, from (5.3) we obtain \(A\phi AX=0\) for any \(X \in \mathbb {D}\) and if we suppose that \(AX=\lambda X\), from Theorem 2.1, \(\lambda (\frac{2}{2\lambda })=0\), which is impossible. Therefore, \(\phi A-A\phi =0\), and from Theorem 2.2, M must be locally congruent to a real hypersurface of type (A). In this case (5.3) gives \(A^2\phi X-\alpha A\phi X=0\) for any \(X \in \mathbb {D}\) and also \(\phi A^2X-\alpha \phi AX=0\). Thus \(\mu (\mu -\alpha )=\lambda (\lambda -\alpha )=0\). We have now that either \(\mu =0\) or \(\mu =\alpha \) and, at the same time, either \(\lambda =0\) or \(\lambda =\alpha \). These four possibilities give contradictions and M must be non Hopf.
As in previous sections we write \(A\xi =\alpha \xi +\beta U\). Then (5.1) looks like
for any \(X,Y \in \mathbb {D}\). Taking \(Y=U\) in (5.5) we get \(g(\phi AX,AU)\xi -\beta \phi AX-g(\phi AX,U)A\xi =\omega (X)AU\). Its scalar product with U yields \(-2\beta g(\phi AX,U)=\omega (X)g(AU,U)\) for any \(X \in \mathbb {D}\). If, in particular, \(X=U\) we obtain
Taking the scalar product of (5.5) and \(\phi U\) we have \(-\beta g(U,Y)g(AX,U)=\omega (X)g(AY,\phi U)\), for any \(X,Y \in \mathbb {D}\). If \(X=Y=U\) it follows
The linear system given by (5.6) and (5.7) satisfies \((\omega (U))^2+2\beta ^2 \ne 0\), and therefore
Taking \(X \in {\mathbb {D}}_U\) in (5.5) and its scalar product with \(\phi U\) we obtain \(-\beta g(U,Y)g(AU,X)=\omega (X)g(AY,\phi U)\) for any \(X \in {\mathbb {D}}_U\), \(Y \in \mathbb {D}\). Bearing in mind (5.8), if \(Y=U\) and \(X \in {\mathbb {D}}_U\) we get \(-\beta g(AU,X)=0\), for any \(X \in {\mathbb {D}}_U\). As \(\beta \ne 0\), it follows
for any \(X \in {\mathbb {D}}_U\). Now (5.8) and (5.9) yield
The scalar product of (5.5) and U gives \(\beta g(U,Y)g(A\phi U,X)-\beta g(\phi AX,Y)=\omega (X)g(AY,U)\) \(=0\), for any \(X,Y \in \mathbb {D}\). Taking \(Y=U\) we have \(2\beta g(A\phi U,X)=0\), for any \(X \in \mathbb {D}\). Thus
Take now \(X,Y \in {\mathbb {D}}_U\) in (5.5). Then, \(g(\phi AX,AY)\xi -g(\phi AX,Y)A\xi =\omega (X)AY\), and its scalar product with U yields \(\beta g(AX,\phi Y)=0\), for any \(X,Y \in {\mathbb {D}}_U\). Therefore, \(AX=0\), for any \(X \in {\mathbb {D}}_U\). This, (5.10) and (5.11) imply that M is locally congruent to a ruled real hypersurface, finishing the proof of Theorem 1.5.
If now \(A_T^{(k)}(X,Y)=\delta (X)AY\), for any \(X,Y \in \mathbb {D}\), we obtain
for any \(X,Y \in \mathbb {D}\).
If we suppose that M is Hopf, \(A\xi =\alpha \xi \), and take the scalar product of (5.12) and \(\xi \), we get
for any \(X,Y \in \mathbb {D}\). Then (5.13) yields
for any \(X \in \mathbb {D}\) and, interchanging X and Y,
for any \(X \in \mathbb {D}\). From (5.14) and (5.15) we have \(A^2\phi X-\phi A^2X=0\), for any \(X \in \mathbb {D}\). If we suppose that \(AX=\lambda X\), from Theorem 2.1, \(A\phi X=\mu \phi X\) and \(\mu ^2=\lambda ^2\). If \(-\lambda =\mu =\frac{\alpha \lambda +2}{2\lambda -\alpha }\), it yields \(\alpha \lambda -2\lambda ^2=\alpha \lambda +2\). Therefore, \(\lambda ^2+1=0\), which is impossible. Therefore \(\lambda =\mu \) and \(\phi A=A\phi \). In this case (5.14) becomes \(2A^2\phi X-2\alpha A\phi X=0\) and then \(\mu (\mu -\alpha )=0\). In the same way, (5.15) implies \(-2\phi A^2X+2\alpha \phi AX=0\) and \(\lambda (\alpha -\lambda )=0\). The four possibilities that we obtain imply contradictions and M must be non Hopf. Write as usual \(A\xi =\alpha \xi +\beta U\).
The scalar product of (5.12) and \(\phi U\) gives
for any \(X,Y \in \mathbb {D}\). Taking \(X=\phi U\) in (5.16) and \(Y=U\) we get \(-\beta g(A\phi U,U)=\delta (\phi U)g(AU,\phi U)\). Thus
If we put \(Y=\phi U\) in (5.16) we obtain
for any \(X \in \mathbb {D}\).
Take the scalar product of (5.12) and U. Then it follows
for any \(X,Y \in \mathbb {D}\). Taking \(Y=\phi U\) in (5.19) we obtain \(-\beta g(AX,U)+\beta g(\phi A\phi U,X)=\delta (X)g(A\phi U,U)\). If \(X=\phi U\) it follows \(-\beta g(A\phi U,U)+\beta g(A\phi U,U)=\delta (\phi U)g(A\phi U,U)\). That is,
Suppose that \(\delta (\phi U)=-\beta \). Then, from (5.18), \(g(A\phi U,\phi U)=0\) and from (5.20), \(g(A\phi U,U)=0\). If \(\delta (\phi U) \ne -\beta \), from (5.17), \(g(AU,\phi U)=0\). Thus we have proved that always
If we take \(X=Y=U\) in (5.19), bearing in mind (5.21), we obtain
If now we take \(Y=U\) in (5.16) we get \(-\beta g(AU,X)+k\beta g(U,X)=\delta (X)g(AU,\phi U)=0\), for any \(X \in \mathbb {D}\). This yields
and from (5.22) we also have \(\delta (U)=0\).
If we take \(X=U\) in (5.19) we obtain \(-\beta g(\phi AU,Y)+\beta g(\phi AY,U)=0\) for any \(Y \in \mathbb {D}\). This yields \(A\phi U=-\phi AU\), and bearing in mind (5.23), we arrive at
From (5.23) and (5.24) we know that \({\mathbb {D}}_U\) is A-invariant. Taking \(Y=U\), \(X \in {\mathbb {D}}_U\) in (5.12) we have \(-\beta \phi AX+k\beta \phi X=\delta (X)AU\), for any \(X \in {\mathbb {D}}_U\). If we take its scalar product with \(\xi \) we get \(\beta \delta (X)=0\), for any \(X \in {\mathbb {D}}_U\). Thus \(\delta (X)=0\) for such an X, and the above equation implies \(\phi AX=k\phi X\) for any \(X \in {\mathbb {D}}_U\). If we apply \(\phi \) we obtain \(AX=kX\) for any \(X \in {\mathbb {D}}_U\). For such a vector field \(AX=kX\), \(A\phi X=k\phi X\). Codazzi equation implies \((\nabla _XA)\phi X-(\nabla _{\phi X}A)X=-2\xi \). Therefore, \(k\nabla _X\phi X-A\nabla _X\phi X-k\nabla _{\phi X}X+A\nabla _{\phi X}X=-2\xi \). Its scalar product with U gives \(-kg([\phi X,X ],U)-g(\nabla _X\phi X,\beta \xi +kU)+g(\nabla _{\phi X}X,\beta \xi +kU)=0\). This yields \(\beta g(\phi X,\phi AX)-\beta g(X,\phi A\phi X)=0\). Thus \(2k\beta =0\), which is impossible and finishes the proof of Theorem 1.6.
Suppose finally that M satisfies \(A_F^{(k)}(X,Y)=\omega (X)AY\) for any X, Y tangent to M, From Theorem 1.5M must be locally congruent to either a real hypersurface of type (A) or to a ruled real hypersurface. Moreover, M must satisfy
for any X, Y tangent to M. Suppose that M is a real hypersurface of type (A) and take \(X=\xi \) in (5.25). We get
for any Y tangent to M. As our real hypersurface satisfies \(A\phi =\phi A\), from (5.26) we have \(\omega (\xi )AY=0\) for any Y tangent to M. If \(\omega (\xi ) \ne 0\) we should have \(AY=0\) for any Y tangent to M. That is, M is totally geodesic, which is impossible. Therefore \(\omega (\xi )=0\).
Take then \(Y=\xi \) in (5.25). We obtain
for any X tangent to M. Consider \(X \in \mathbb {D}\) and take the scalar product of (5.27) and \(Z \in \mathbb {D}\). This gives \(-\alpha g(\phi AX,Z)+g(A\phi AX,Z)=0\), for any \(X,Z \in \mathbb {D}\). Therefore \(-\alpha \phi AX+A\phi AX=0\). As \(A\phi =\phi A\) we have \(-\alpha \phi AX+\phi A^2X=0\) for any \(X \in \mathbb {D}\) Suppose that \(AX=\lambda X\). Then \(-\alpha \lambda +\lambda ^2=0\), and the unique principal curvatures in \(\mathbb {D}\) are \(\alpha \) and 0. Thus M has, exactly, two distinct constant principal curvatures and looking at Takagi’s list M must be locally congruent to a geodesic hypersphere. But a geodesic hypersphere has not such principal curvatures.
If M is ruled and we take \(Y=\phi U\) in (5.25) it follows \(-g(\phi AX,\phi U)A\xi -k\eta (X)AU=\omega (X)A\phi U=0\). Then \(-g(AX,U)A\xi -k\eta (X)AU=0\), for any X tangent to M. Its scalar product with U implies \(-\beta g(AX,U)=0\) for any X tangent to M. If \(X=\xi \) we get \(\beta ^2=0\), which is impossible. Thus we have obtained the following
Corollary 5.1
There does not exist any real hypersurface M in \(\mathbb {C}P^m\), \(m \ge 3\), such that \(A_F^{(k)}(X,Y)=\omega (X)AY\), for a certain nonnull 1-form \(\omega \) on M, any X, Y tangent to M and a nonnull real number k.
Similarly, we have
Corollary 5.2
There does not exist any real hypersurface M in \(\mathbb {C}P^m\), \(m \ge 3\), such that \(A_T^{(k)}(X,Y)=\delta (X)AY\), for a certain nonnull 1-form \(\delta \) on M, any X, Y tangent to M and a nonnull real number k.
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First author is partially supported by project PID 2020-116126GB-I00 from MICINN. The authors want to thank the referees for valuable comments that have improved the paper.
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Pérez, J.d.D., Pérez-López, D. Some Conditions Concerning the Shape Operator of a Real Hypersurface in Complex Projective Space. Bull. Iran. Math. Soc. 49, 55 (2023). https://doi.org/10.1007/s41980-023-00797-1
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DOI: https://doi.org/10.1007/s41980-023-00797-1
Keywords
- kth Generalized Tanaka–Webster connection
- Complex projective space
- Real hypersurface
- Shape operator
- Lie derivative