J. Korean Math. Soc. 58 (2021), No. 2, pp. 473–486 https://doi.org/10.4134/JKMS.j200134
pISSN: 0304-9914 / eISSN: 2234-3008
KILLING STRUCTURE JACOBI OPERATOR OF A REAL HYPERSURFACE IN A COMPLEX PROJECTIVE SPACE
Juan de Dios P´erez
Abstract. We prove non-existence of real hypersurfaces with Killing structure Jacobi operator in complex projective spaces. We also classify real hypersurfaces in complex projective spaces whose structure Jacobi operator is Killing with respect to the k-th generalized Tanaka-Webster connection.
1. Introduction
Let CPm, m ≥ 2, be a complex projective space endowed with the metric g of constant holomorphic sectional curvature 4. Let M be a connected real hypersurface of CPm without boundary. Let ∇ be the Levi-Civita connection on M and J the complex structure of CPm. Take a locally defined unit normal vector field N on M and denote by ξ = −J N . This is a tangent vector field to M called the structure vector field on M . On M there exists an almost contact metric structure (φ, ξ, η, g) induced by the Kaehlerian structure of CPm, where φ is the tangent component of J and η is an one-form given by η(X) = g(X, ξ) for any X tangent to M . The classification of homogeneous real hypersurfaces in CPm was obtained by Takagi, see [6], [13], [14], [15]. His classification contains 6 types of real hypersurfaces. Among them we find type (A1) real hypersurfaces that are geodesic hyperspheres of radius r, 0 < r < π2 and type (A2) real hypersurfaces that are tubes of radius r, 0 < r < π2, over totally geodesic complex projective spaces CPn, 0 < n < m − 1. We will call both types of real hypersurfaces type (A) real hypersurfaces.
Ruled real hypersurfaces can be described as follows: Take a regular curve γ in CPm with tangent vector field X. At each point of γ there is a unique CPm−1 cutting γ so as to be orthogonal not only to X but also to J X. The union of these hyperplanes is called a ruled real hypersurface. It will be an
Received March 16, 2020; Accepted July 21, 2020.
2010 Mathematics Subject Classification. 53C15, 53B25.
Key words and phrases. g-Tanaka-Webster connection, complex projective space, real hypersurface, Killing structure Jacobi operator.
This work was supported by MINECO-FEDER Project MTM 2016-78807-C2-1-P.
c
2021 Korean Mathematical Society 473
embedded hypersurface locally, although globally it will in general have self- intersections and singularities. Equivalently, a ruled real hypersurface satisfies that the maximal holomorphic distribution on M , D, given at any point by the vectors orthogonal to ξ, is integrable with integral manifolds CPm−1, or g(AD, D) = 0. For examples of ruled real hypersurfaces see [7] or [9].
The Tanaka-Webster connection, [16], [18], is the canonical affine connection defined on a non-degenerate, pseudo-Hermitian CR-manifold. As a generaliza- tion of this connection, Tanno, [17], defined the generalized Tanaka-Webster connection for contact metric manifolds by
(1.1) ∇ˆXY = ∇XY + (∇Xη)(Y )ξ − η(Y )∇Xξ − η(X)φY.
Using the naturally extended affine connection of Tanno’s generalized Tanaka-Webster connection, Cho defined the g-Tanaka-Webster connection
∇ˆ(k) for a real hypersurface M in CPm given, see [4], [5], by (1.2) ∇ˆ(k)X Y = ∇XY + g(φAX, Y )ξ − η(Y )φAX − kη(X)φY
for any X,Y tangent to M where k is a non-zero real number. 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 φA + Aφ = 2kφ, the g-Tanaka-Webster connection coincides with the Tanaka-Webster connection.
Here we can consider the tensor field of type (1,2) given by the difference of both connections F(k)(X, Y ) = g(φAX, Y )ξ − η(Y )φAX − kη(X)φY , for any X, Y tangent to M , see [8] Proposition 7.10, pages 234–235. We will call this tensor the k-th Cho tensor on M . Associated to it, for any X tangent to M and any nonnull real number k we can consider the tensor field of type (1,1) FX(k), given by FX(k)Y = F(k)(X, Y ) for any Y ∈ T M . This operator will be called the k-th Cho operator corresponding to X. The torsion of the connection ˆ∇(k) is given by ˆT(k)(X, Y ) = FX(k)Y − FY(k)X for any X, Y tangent to M .
The Jacobi operator RX with respect to a unit vector field X is defined by RX= R(·, X)X, where R is the curvature tensor field on M . Then we see that RX is a self-adjoint endomorphism of the tangent space. It is related to Jacobi vector fields, which are solutions of the second-order differential equation (the Jacobi equation) ∇γ˙(∇γ˙Y )+R(Y, ˙γ) ˙γ = 0 along a geodesic γ in M . The Jacobi operator with respect to the structure vector field ξ, Rξ, is called the structure Jacobi operator on M .
Let L denote the Lie derivative of a Riemannian manifold ( ¯M , ¯g) with Levi- Civita connection ¯∇. Generalizing the notion of Killing vector field (a vector field on ¯M is Killing if LXg = 0), Blair, [1], introduced the notion of Killing¯ tensor along a geodesic γ of ¯M . A tensor of type (1,1) K on ¯M is called Killing along γ if (∇γ˙K) ˙γ = 0. We will say K is Killing if it is Killing along any geodesic of ¯M . Therefore K is Killing if ( ¯∇XK)X = 0 for any vector field X tangent to ¯M . Equivalently
( ¯∇XK)Y + ( ¯∇YK)X = 0
for any vector fields X, Y tangent to ¯M .
The purpose of the present paper is to study real hypersurfaces M in CPm whose structure Jacobi operator is Killing. We will prove the following:
Theorem 1. There does not exist any real hypersurface in CPm, m ≥ 3, with Killing structure Jacobi operator.
We will say that the structure Jacobi operator of M is Killing with respect to the k-th generalized Tanaka-Webster connection if ( ˆ∇(k)X Rξ)X = 0 for any vector field X tangent to M . Equivalently
( ˆ∇(k)X Rξ)Y + ( ˆ∇(k)Y Rξ)X = 0
for any vector fields X, Y tangent to M . We prove the following:
Theorem 2. Let M be a real hypersurface in CPm, m ≥ 2. Let k be a nonnull constant. Then the structure Jacobi operator of M is Killing with respect to the k-th generalized Tanaka-Webster connection if and only if M is locally congruent to either a tube of radius π4 over a complex submanifold of CPm or to a type (A) real hypersurface with radius r 6= π4.
2. Preliminaries
Throughout this paper, all manifolds, vector fields, etc., will be considered of class C∞ unless otherwise stated. Let M be a connected real hypersurface in CPm, 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) the Kaehlerian structure of CPm.
For any vector field X tangent to M we write J X = φX + η(X)N , and
−J N = ξ. Then (φ, ξ, η, g) is an almost contact metric structure on M , see [2].
That is, we have
(2.1) φ2X = −X + η(X)ξ, η(ξ) = 1, g(φX, φY ) = g(X, Y ) − η(X)η(Y ) for any tangent vectors X, Y to M . From (2.1) we obtain
(2.2) φξ = 0, η(X) = g(X, ξ).
From the parallelism of J we get
(2.3) (∇Xφ)Y = η(Y )AX − g(AX, Y )ξ and
(2.4) ∇Xξ = φAX
for any X, Y tangent to M , where A denotes the shape operator of the im- mersion. As the ambient space has holomorphic sectional curvature 4, the equations of Gauss and Codazzi are given, respectively, by
(2.5) R(X, Y )Z = g(Y, Z)X − g(X, Z)Y + g(φY, Z)φX − g(φX, Z)φY
− 2g(φX, Y )φZ + g(AY, Z)AX − g(AX, Z)AY,
and
(2.6) (∇XA)Y − (∇YA)X = η(X)φY − η(Y )φX − 2g(φX, Y )ξ
for any tangent vectors X, Y, Z to M , where R is the curvature tensor of M . We will call the maximal holomorphic distribution D on M to the following one: at any p ∈ M , D(p) = {X ∈ TpM | g(X, ξ) = 0}. We will say that M is Hopf if ξ is principal, that is, Aξ = αξ for a certain function α on M .
From the above formulas we have that the structure Jacobi operator on M is given by
(2.7) Rξ(X) = X − η(X)ξ + g(Aξ, ξ)AX − g(AX, ξ)Aξ for any X tangent to M . Therefore its covariant derivative is given by
(2.8)
(∇XRξ)Y = − g(Y, φAX)ξ − η(Y )φAX + g(∇XAξ, ξ)AY
+ g(Aξ, φAX)AY + g(Aξ, ξ)(∇XA)Y − g(Y, ∇XAξ)Aξ
− g(AY, ξ)∇XAξ.
In the sequel we need the following results:
Theorem 2.1 ([11]). Let M be a real hypersurface of CPm, m ≥ 2. Then the following are equivalent:
(1) M is locally congruent to either a geodesic hypersphere or a tube of radius r, 0 < r < π2, over a totally geodesic CPn, 0 < n < m − 1.
(2) φA = Aφ.
Theorem 2.2 ([10]). If ξ is a principal curvature vector with corresponding principal curvature α, it is constant and if X ∈ D is principal with principal curvature λ, then φX is principal with principal curvature 2λ−ααλ+2.
3. Proof of Theorem 1
Suppose Rξ is Killing and M is Hopf with Aξ = αξ. If X ∈ D, then Rξ(X) = X + αAX. As (∇XRξ)ξ + (∇ξRξ)X = 0 we get, bearing in mind that α is constant, −Rξ(φAX) + ∇ξ(X + αAX) − Rξ(∇ξX) = 0. That is,
−Rξ(φAX) + α∇ξAX − αA∇ξX = 0. Thus we have Rξ(φAX) = α(∇ξA)X for any X ∈ D.
Suppose that X ∈ D satisfies AX = λX. From Theorem 2.2 AφX = µφX, where µ = αλ+22λ−α. Therefore λφX + λαAφX = α(∇ξA)X. That is,
(3.1) (λ + λαµ)φX = α(∇ξA)X.
As also Rξ(φAφX) = α(∇ξA)φX, we obtain
(3.2) − (µ + λαµ)X = α(∇ξA)φX.
Taking the scalar product of (3.1) and φX we have
λ + λαµ = αg((∇ξA)X, φX) = αg((∇ξA)φX, X) = −µ − λαµ.
Thus λ + µ = −2αλµ. Bearing in mind the value of µ it follows λ + αλ+22λ−α =
−2αλαλ+22λ−α. Therefore
(3.3) (1 + α2)λ2+ 2αλ + 1 = 0.
But this equation with unknown λ does not admit any real root, which is impossible.
That means our real hypersurface is non Hopf. Then, we write Aξ = αξ+βU , where U is a unit vector field in D and β a function on M non-vanishing on a certain neighborhood of a point p. From now on the calculations will be made on such a neighborhood. We will call DU = {X ∈ D/g(X, U ) = g(X, φU ) = 0}.
As (∇ξRξ)ξ = 0, we have −Rξ(φAξ) = −βRξ(φU ) = 0. As β 6= 0 we get Rξ(φU ) = 0 and this yields
(3.4) α 6= 0
and
(3.5) AφU = −1
αφU.
From (2.8) our condition (∇XRξ)Y + (∇YRξ)X = 0 becomes
(3.6)
− g(Y, φAX)ξ − η(Y )φAX + g(∇XAξ, ξ)AY + g(Aξ, φAX)AY + α(∇XA)Y − g(Y, ∇XAξ)Aξ − g(AY, ξ)∇XAξ − g(X, φAY )ξ
− η(X)φAY + g(∇YAξ, ξ)AX + g(Aξ, φAY )AX + α(∇YA)X
− g(X, ∇YAξ)Aξ − g(AX, ξ)∇YAξ = 0
for any X, Y tangent to M . From (3.6) and Codazzi equation we obtain
(3.7)
2α(∇XA)Y
= α{η(X)φY − η(Y )φX − 2g(φX, Y )ξ}
+ g(Y, φAX)ξ + η(Y )φAX − g(∇XAξ, ξ)AY − g(Aξ, φAX)AY + g(Y, ∇XAξ)Aξ + g(AY, ξ)∇XAξ + g(X, φAY )ξ + η(X)φAY
− g(∇YAξ, ξ)AX − g(Aξ, φAY )AX + g(X, ∇YAξ)Aξ + g(AX, ξ)∇YAξ.
In particular, if Y = ξ we get
(3.8)
2α(∇XA)ξ = − αφX + φAX − g(Aξ, φAX)Aξ + α∇XAξ + g(X, φAξ)ξ + η(X)φAξ − g(∇ξAξ, ξ)AX + g(X, ∇ξAξ)Aξ + g(AX, ξ)∇ξAξ
for any X tangent to M . Bearing in mind (3.7) and (3.8) and that g((∇XA)ξ, Y ) = g((∇XA)Y, ξ)
we have
(3.9)
− αg(φX, Y ) + g(X, φAY ) − g(∇YAξ, ξ)g(AX, ξ)
− g(Aξ, φAY )g(AX, ξ) + αg(X, ∇YAξ) + g(AX, ξ)g(∇YAξ, ξ)
= g(X, φAξ)η(Y )
+ η(X)g(φAξ, Y ) − g(∇ξAξ, ξ)g(AX, Y ) + g(X, ∇ξAξ)g(Aξ, Y ) + g(AX, ξ)g(∇ξAξ, Y )
for any X, Y tangent to M .
From (3.8), bearing in mind (3.5) it follows (3.10) α∇UAξ = 2αAφAU − αφU + φAU
− g(∇ξAξ, ξ)AU + g(U, ∇ξAξ)Aξ + β∇ξAξ and taking X ∈ DU, Y = U in (3.9) we obtain
(3.11) g(X, φAU ) + αg(X, ∇UAξ) = −g(∇ξAξ, ξ)g(AX, U ) + βg(X, ∇ξAξ) for any X ∈ DU.
Introducing (3.11) in (3.10) we have
(3.12) g(φAU, X) + αg(AφAU, X) = 0
for any X ∈ DU. Therefore φAU + αAφAU ∈ Span{ξ, U, φU }. As g(φAU + αAφAU, ξ) = 0 from (3.5), g(φAU + αAφAU, U ) = αg(AU, φAU ) = 0 and g(φAU + αAφAU, φU ) = g(AU, U ) + αg(AφU, φAU ) = g(AU, U ) − g(φU, φAU )
= 0, we get
(3.13) φAU + αAφAU = 0.
Taking X = U , Y ∈ DU in (3.9) we obtain
(3.14) αg(U, ∇YAξ) = −g(∇ξAξ, ξ)g(AU, Y ) + βg(∇ξAξ, Y ) for any Y ∈ DU. From (3.8) it follows
(3.15) α∇YAξ = 2αAφAY − αφY + φAY − g(∇ξAξ, ξ)AY + g(Y, ∇ξAξ)Aξ.
Introducing (3.15) in (3.14) we have 2αg(AφAY, U ) − g(∇ξAξ, ξ)g(AY, U ) + βg(Y, ∇ξAξ) = −g(∇ξAξ, ξ)g(AU, Y ) + βg(∇ξAξ, Y ). Therefore, as α 6= 0, for any Y ∈ DU, g(AφAY, U ) = 0. That is, AφAU ∈ Span{ξ, U, φU }. As g(AφAU, ξ) = βg(φAU, U ) = 0, g(AφAU, U ) = g(φAU, AU ) = 0 and g(AφAU , φU ) = g(φAU, AφU ) = −α1g(φAU , φU ) = −α1g(AU, U ) we conclude
(3.16) AφAU = −1
αg(AU, U )φU.
From (3.13) and (3.16) we get φAU = g(U, AU )φU . This yields
(3.17) AU = βξ + g(AU, U )U
and this means that DU is A-invariant.
Take X, Y ∈ DU in (3.9). We have
(3.18) − αg(φX, Y ) + g(X, φAY ) + αg(X, ∇YAξ) = −g(∇ξAξ, ξ)g(AX, Y ).
Moreover, taking Y ∈ DU in (3.8) we obtain
(3.19) α∇YAξ = 2αAφAY − αφY + φAY − g(∇ξAξ, ξ)AY + g(Y, ∇ξAξ)Aξ.
From (3.18) and (3.19) we get
(3.20) 2g(X, φAY ) + 2αg(AφAY, X) = 0
for any X, Y ∈ DU. Then φAY + αAφAY = 0 for any Y ∈ DU. Suppose that Y ∈ DU is unit and satisfies AY = λY . Then λφY + λαAφY = 0. Thus, either λ = 0 or AφY = −α1φY . Interchanging X and Y in (3.20) we have AφY + αAφAY = 0. Therefore AφY = φAY . If AY = 0, AφY = 0 and if AφY = −α1φY = φAY , then AY = −α1Y . Therefore the unique principal curvatures appearing on DU are 0 and −α1.
Taking X = φU in (3.8) we obtain
(3.21)
α∇φUAξ = 2αAφAφU + αU + φAφU + βg(AφU, φU )Aξ + βξ + 1
αg(∇ξAξ, ξ)φU + g(φU, ∇ξAξ)Aξ.
And taking X = U , Y = φU in (3.9) it follows
(3.22) − α +1 − β2
α + αg(U, ∇φUAξ) = βg(∇ξAξ, φU ).
From (3.21) and (3.22) we have
(3.23) g(AU, U ) = β2− 1
α .
From Proposition 3.3 in [12], there must be a principal curvature on DU not equal to −1α.
Codazzi equation applied to X ∈ DU and φX such that AX = AφX = 0 yields β2= 1. Our Theorem follows from Propositions 3.1 and 3.2 in [12].
4. Proof of Theorem 2
If we suppose that ( ˆ∇(k)X Rξ)Y + ( ˆ∇(k)Y Rξ)X = 0 for any X, Y tangent to M , developing it we get
(4.1)
(∇XRξ)Y + (∇YRξ)X
= − g(φAX, Rξ(Y ))ξ + kη(X)φRξ(Y ) − η(Y )Rξ(φAX)
− kη(X)Rξ(φY ) − g(φAY, Rξ(X))ξ + kη(Y )φRξ(X)
− η(X)Rξ(φAY ) − kη(Y )Rξ(φX).
From (2.8) it follows
(4.2)
− g(φAX, Y )ξ − η(Y )φAX + g(∇XAξ, ξ)AY + g(Aξ, φAX)AY + α(∇XA)Y − g(Y, ∇XAξ)Aξ − g(AY, ξ)∇XAξ − g(X, φAY )ξ
− η(X)φAY + g(∇YAξ, ξ)AX + g(Aξ, φAY )AX + α(∇YA)X
− g(X, ∇YAξ)Aξ − g(AX, ξ)∇YAξ
= − g(φAX, Rξ(Y ))ξ + kη(Y )φRξ(Y ) − η(Y )Rξ(φAX)
− kη(X)Rξ(φY ) − g(φAY, Rξ(X))ξ + kη(Y )φRξ(X)
− η(X)Rξ(φAY ) − kη(Y )Rξ(φX) for any X, Y tangent to M .
Let us suppose M is Hopf with Aξ = αξ. If at a point p ∈ M α 6= 0, there exists an open neighborhood of p on which α does not vanish. All the following computations are made on this neighborhood.
From (4.2) it follows
(4.3)
α(∇XA)Y + α(∇YA)X
= g(Y, φAX)ξ + η(Y )φAX + g(Y, ∇XAξ)Aξ + g(AY, ξ)∇XAξ + g(X, φAY )ξ + η(X)φAY + g(X, ∇YAξ)Aξ + g(AX, ξ)∇YAξ
− g(φAX, Rξ(Y ))ξ + kη(X)φRξ(Y ) − η(Y )Rξ(φAX)
− kη(X)Rξ(φY ) − g(φAY, Rξ(X))ξ + kη(Y )φRξ(X)
− η(X)Rξ(φAY ) − kη(Y )Rξ(φX)
for any X, Y tangent to M . From Codazzi equation we have
(4.4)
2α(∇XA)Y
= αη(X)φY − αη(Y )φX − 2αg(φX, Y )ξ + g(Y, φAX)ξ + η(Y )φAX + g(Y, ∇XAξ)Aξ + g(AY, ξ)∇XAξ + g(X, φAY )ξ + η(X)φAY + g(X, ∇YAξ)Aξ
+ g(AX, ξ)∇YAξ − g(φAX, Rξ(Y ))ξ + kη(X)φRξ(Y )
− η(Y )Rξ(φAX) − kη(X)Rξ(φY ) − g(φAY, Rξ(X))ξ + kη(Y )φRξ(X) − η(X)Rξ(φAY ) − kη(Y )Rξ(φX) for any X, Y tangent to M
In particular, if Y = ξ we get
(4.5) 2α(∇XA)ξ = − αφX + φAX + α∇XAξ + g(X, ∇ξAξ)Aξ
− Rξ(φAX) + kφRξ(X) − kRξ(φX)
for any X tangent to M . From (4.4), (4.5) and the fact that g((∇XA)Y, ξ) = g(Y, (∇XA)ξ) we have
(4.6)
− αg(φX, Y ) + g(AY, ξ)g(∇XAξ, ξ) + g(X, φAY ) + αg(X, ∇YAξ) − g(φAY, Rξ(X))
= g(X, ∇ξAξ)g(Aξ, Y ) + αg(∇XAξ, Y ) + kg(φRξ(X), Y )
− kg(Rξ(φX), Y )
for any X, Y tangent to M . Taking X, Y ∈ D in (4.6) we obtain
(4.7) − αg(φX, Y ) + g(X, φAY ) + α2g(X, φAY ) − g(φAY, Rξ(X))
= α2g(φAX, Y ) + kg(φRξ(X), Y ) − kg(Rξ(φX), Y ), where we have applied that α is constant.
As for any X ∈ D, Rξ(X) = X + αAX, (4.7) yields
(4.8) − φX + (k − α)AφX − (k + α)φAX + AφAX = 0 for any X ∈ D. But interchanging X and Y we obtain
(4.9) φX − (k − α)φAX + (k + α)AφX − AφAX = 0.
From (4.8) and (4.9) we get 2kAφX −2kφAX = 0 for any X ∈ D. This yields Aφ = φA and from Theorem 2.1, M is locally congruent to a real hypersurface of type (A). The converse is immediate.
Suppose now that α = 0 on M . From (4.2) we get
(4.10)
− g(Y, φAX)ξ − η(Y )φAX − g(X, φAY )ξ − η(X)φAY
= − g(φAX, Rξ(Y ))ξ + kη(X)φRξ(Y ) − η(Y )Rξ(φAX)
− kη(X)Rξ(φY ) − g(φAY, Rξ(X))ξ
+ kη(Y )φRξ(X) − η(X)Rξ(φAY ) − kη(Y )Rξ(φX).
In this case Rξ(X) = X − η(X)ξ and, for any X ∈ D, Rξ(X) = X. If X, Y ∈ D, (4.10) gives −g(Y, φAX)ξ − g(X, φAY )ξ = −g(φAX, Y )ξ − g(φAY, X)ξ. If X = ξ and Y is tangent to M we obtain −φAY = −φAY , proving that any real hypersurface such that Aξ = 0 satisfies our condition. The results follows from [3].
In the following we suppose M is non Hopf and as in the previous section we write Aξ = αξ + βU for a unit U ∈ D. Let p be a point of M such that α 6= 0. This is true on a neighborhood of such a point and we will make the calculations on that neighborhood.
Bearing in mind (4.2) and Codazzi equation α(∇XA)Y − α(∇YA)X = αη(X)φY − αη(Y )φX − 2αg(φX, Y )ξ we obtain
(4.11)
2α(∇XA)Y
= g(Y, φAX)ξ + η(Y )φAX − g(∇XAξ, ξ)AY − g(Aξ, φAX)AY + g(Y, ∇XAξ)Aξ + g(AY, ξ)∇XAξ + g(X, φAY )ξ + η(X)φAY
− g(∇YAξ, ξ)AX − g(Aξ, φAY )AX + g(X, ∇YAξ)Aξ
− g(AX, ξ)∇YAξ − g(φAX, Rξ(Y ))ξ + kη(X)φRξ(Y )
− η(Y )Rξ(φAX) − kη(X)Rξ(φY ) − g(φAY, Rξ(X))ξ + kη(Y )φRξ(X) − η(X)Rξ(φAY ) − kη(Y )Rξ(φX) + αη(X)φY − αη(Y )φX − 2αg(φX, Y )ξ
for any X, Y tangent to M . In particular, for Y = ξ, (4.11) yields
(4.12)
2α(∇XA)ξ
= φAX − g(Aξ, φAX)Aξ + α∇XAξ + g(X, φAξ)ξ + η(X)φAξ − g(∇ξAξ, ξ)AX + g(X, ∇ξAξ)Aξ
− g(AX, ξ)∇ξAξ − Rξ(φAX) − g(φAξ, Rξ(X))ξ + kφRξ(X)
− η(X)Rξ(φAξ) − kRξ(φX) − αφX
for any X tangent to M . From (4.11) and (4.12) we have
(4.13)
− g(∇XAξ, ξ)g(AY, ξ) + g(X, φAY ) − g(Aξ, φAY )g(AX, ξ) + αg(X, ∇YAξ) − g(AX, ξ)g(∇YAξ, ξ) − g(φAY, Rξ(X))
= η(X)g(φAξ, Y ) − g(∇ξAξ, ξ)g(AX, Y ) − g(φAξ, Rξ(X))η(Y ) + kg(φRξ(X), Y ) − η(X)g(Rξ(φAξ), Y ) − kg(Rξ(φX), Y ) for any X, Y tangent to M . For X = ξ, (4.13) gives −αg(Aξ, φAY ) = g(φAξ, Y ) − g(Rξ(φAξ), Y ) for any Y tangent to M . That is,
−αβg(U, φAY ) = βg(φU, Y ) − βg(Rξ(φU ), Y )
= βg(φU, Y ) − βg(φU, Y ) − αβg(AφU, Y )
= αβg(U, φAY ).
Therefore, as αβ 6= 0, g(AφU, Y ) = 0 for any Y tangent to M . That is,
(4.14) AφU = 0.
From (4.14) we obtain Rξ(φU ) = φU and (4.12) becomes
(4.15)
α∇XAξ = 2αAφAX + φAX − g(∇ξAξ, ξ)AX
+ g(X, ∇ξAξ)Aξ − g(AX, ξ)∇ξAξ − Rξ(φAX) + kφRξ(X) − kRξ(φX) − αφX
for any X tangent to M . If X = ξ it follows α∇ξAξ = −α∇ξAξ. As α 6= 0 we have
(4.16) ∇ξAξ = 0
and (4.15) becomes
(4.17) α∇XAξ = 2αAφAX + φAX − Rξ(φAX) + kφRξ(X) − kRξ(φX) − αφX for any X tangent to M .
From (4.13) we obtain
(4.18)
αg(∇XAξ, ξ)g(AY, ξ) − αg(X, φAY ) − α2g(X, ∇YAξ) + αg(AX, ξ)g(∇YAξ, ξ) + αg(φAY, Rξ(X))
= αg(φAξ, Rξ(X))η(Y ) − kαg(φRξ(X), Y ) + kαg(Rξ(φX), Y ).
Taking X = φU , Y = ξ in (4.18) we get αg(∇φUAξ, ξ) = 2β, but from (4.17) α∇φUAξ = kφRξ(φU ) + kRξ(U ) + U . Therefore αg(∇φUAξ, ξ) = 0, which yields β = 0, giving a contradiction.
Thus we must suppose α = 0. Let X ∈ D. Then ( ˆ∇(k)X Rξ)ξ +( ˆ∇(k)ξ Rξ)X = 0 gives ( ˆ∇(k)ξ Rξ)X = 0. That is, ˆ∇(k)ξ Rξ(X) − Rξ( ˆ∇(k)ξ X) = 0. This yields
(4.19)
(∇ξRξ)X = −βg(φU, Rξ(X))ξ + kφRξ(X) − kRξ(φX)
= −βg(φU, X)ξ − kβg(Aξ, X)φU + kg(Aξ, φX)Aξ
= −βg(φU, X)ξ − kβ2g(U, X)φU + kβ2g(U, φX)U for any X ∈ D.
In particular,
(∇ξRξ)φU = −βξ − kβ2U
= ∇ξφU − Rξ(∇ξφU )
= −g(φU, φAξ)ξ + β2g(U, ∇ξφU )U
= −βξ + β2g(U, ∇ξφU )U, that implies
(4.20) g(∇ξU, φU ) = k.
Taking X = U in (4.19) we have (∇ξRξ)U = −kβ2φU = ∇ξ(1 − β2)U − Rξ(∇ξU ) = ∇ξ(−β2U ) = −ξ(β2)U − β2∇ξU . Thus we obtain
(4.21) ξ(β) = 0.
If m ≥ 3 we define DU as in the previous section. Taking X ∈ DU in (4.19) we get (∇ξRξ)(X) = 0 = ∇ξX −Rξ(∇ξX) = g(Aξ, ∇ξX)Aξ = β2g(U, ∇ξX)U . This yields
(4.22) g(∇ξU, X) = 0
for any X ∈ DU. As g(∇ξU, U ) = g(∇ξU, ξ) = 0, from (4.20) and (4.22) we have
(4.23) ∇ξU = kφU.
Take X, Y ∈ D in (4.2). In our case we get
(4.24)
− g(Y, φAX)ξ − βg(Y, ∇X(βU ))U − βg(Y, U )∇X(βU )
− g(X, φAY )ξ − βg(X, ∇Y(βU ))U − βg(X, U )∇Y(βU )
= − g(φAX, Rξ(Y ))ξ − g(φAY, Rξ(X))ξ for any X, Y ∈ D.
The scalar product of (4.24) and U , bearing in mind β 6= 0, yields (4.25) 2X(β)g(Y, U ) + βg(Y, ∇XU ) + 2Y (β)g(X, U ) + βg(X, ∇YU ) = 0.
Taking X = Y = U in (4.25) we obtain
(4.26) U (β) = 0.
Taking X ∈ D and orthogonal to U , Y = U in (4.25) we have (4.27) 2X(β) + βg(X, ∇UU ) = 0
for any X ∈ D and orthogonal to U .
As Rξ(U ) = (1 − β2)U , if we develop ( ˆ∇(k)U Rξ)U = 0, bearing in mind (4.26) we obtain β2∇UU + β2g(φAU, U )ξ = 0. Therefore, g(∇UU, X) = 0 for any X ∈ D and orthogonal to U . From (4.27) it follows
(4.28) X(β) = 0
for any X ∈ D and orthogonal to U . From (4.21), (4.26) and (4.28), β is a constant. By Codazzi equation (∇ξA)U − (∇UA)ξ = φU . This yields
(4.29) ∇ξAU − kAφU − β∇UU + AφAU = φU.
Its scalar product with ξ gives
(4.30) g(∇ξAU, ξ) + 2βg(U, φAU ) = 0.
As g(AU, ξ) = β, a constant, g(∇ξAU, ξ) = −g(AU, φAξ) = βg(φAU, U ).
From (4.30) we obtain
(4.31) g(AU, φU ) = 0.
This and above results yield
(4.32) ∇UU = 0.
Let X ∈ DU (if m ≥ 3). Codazzi equation gives (∇XA)ξ − (∇ξA)X = −φX.
This yields β∇XU − AφAX − ∇ξAX + A∇ξX = −φX. Its scalar product with ξ implies 3βg(AφU, X)−βg(X, ∇ξU ) = 0. From (4.23) we have g(AφU, X) = 0 for any X ∈ DU. Now, from (4.30) we conclude that AφU = γφU for a certain function γ.
Once again, from Codazzi equation (∇φUA)ξ − (∇ξA)φU = U . This yields β∇φUU − AφAφU − ∇ξ(γφU ) + A∇ξφU = U . Its scalar product with ξ gives
−2βg(U, φAφU ) + γg(φU, φAξ) + βg(∇ξφU, U ) = 0. From (4.23) we have 3γ − k = 0. That is,
(4.33) AφU = k
3φU.
Moreover, (∇φUA)U − (∇UA)φU = 2ξ. That is, (4.34) ∇φUAU − A∇φUU −k
3∇UφU + A∇UφU = 2ξ.
Its scalar product with ξ implies g(∇φUAU, ξ) +k
3g(φU, φAU ) + βg(∇UφU, U ) = 2.
This gives g(∇φUAU, ξ) + k3g(U, AU ) = 2. As g(AU, ξ) = β is constant, g(∇φUAU, ξ) = −g(AU, φAφU ). Therefore g(φAU, AφU ) + k3g(U, AU ) = 2, which yields
(4.35) g(AU, U ) = 3
k.
We have ( ˆ∇(k)U Rξ)φU + ( ˆ∇(k)φURξ)U = 0. As Rξ(φU ) = φU and Rξ(U ) = (1 − β2)U , this yields ˆ∇(k)U φU − Rξ( ˆ∇(k)U φU ) + (1 − β2) ˆ∇(k)φUU − Rξ( ˆ∇(k)φUU ) = 0.
That is, β2g(U, ˆ∇(k)U φU )U − β2∇ˆ(k)φUU = 0. As g(U, ˆ∇(k)U φU ) = g(U, ∇UφU + g(φAU φU )ξ) = 0, we get ˆ∇(k)φUU = 0 = ∇φUU + g(φAφU, U )ξ = ∇φUU − k3ξ.
Therefore
(4.36) ∇φUU =k
3ξ.
The scalar product of (4.34) and U yields
−g(AU, ∇φUU ) −k
3β + g(∇UφU, AU ) = 0, where we have applied that g(AU, U ) = 3k is constant. Therefore
g(∇φUAU, U ) = −g(AU, ∇φUU ).
As (∇Uφ)U = −g(AU, U )ξ = −3kξ and ∇UφU = (∇Uφ)U + φ∇UU = (∇Uφ)U , from (4.32) and (4.35) we get 2k3β + 3kβ = 0. This implies 2k2+ 9 = 0, which is impossible, finishing the proof.
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Juan de Dios P´erez
Departamento de Geometria y Topologia Universidad de Granada
18071 Granada, Spain Email address: [email protected]
