On the structure vector field of a real hypersurface in complex quadric

Juan de Dios Pérez

doi:10.1515/math-2018-0021

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On the structure vector field of a real hypersurface in complex quadric

Juan de Dios Pérez

Published/Copyright: March 20, 2018

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$Open Mathematics$

From the journal Open Mathematics Volume 16 Issue 1

Abstract

From the notion of Jacobi type vector fields for a real hypersurface in complex quadric Q^m we prove that if the structure vector field is of Jacobi type it is Killing when the real hypersurface is either Hopf or compact. In such cases we classify real hypersurfaces whose structure vector field is of Jacobi type.

Keywords: Real hypersurfaces; Complex quadric; Jacobi type vector field; Structure vector field; Killing vector field

MSC 2010: 53C40; 53C15

1 Introduction

The complex quadric Q^m = SO_m+2/SO_mSO₂ is a compact Hermitian symmetric space of rank 2. It is also a complex hypersurface in complex projective space ℂP^m+1, [1]. Q^m is equipped with two geometric structures: a complex conjugation A and a Kähler structure J.

Real hypersurfaces M in Q^m are immersed submanifolds of real codimension 1. The Kähler structure J of Q^m induces on M an almost contact metric structure (ϕ, ξ, η, g), where ϕ is the structure tensor field, ξ is the structure (or Reeb) vector field, η is a 1-form and g is the the induced Riemannian metric on M.

Real hypersurfaces M in Q^m whose Reeb flow is isometric are classified in [2]. They obtain tubes around the totally geodesic ℂP^k in Q^m when m = 2k. The condition of isometric Reeb flow is equivalent to the commuting condition of the shape operator S with the structure tensor field ϕ of M.

It is known that a Killing vector field X on a Riemannian manifold (M̄,ḡ) satisfies 𝔏_Xḡ = 0, where 𝔏 denotes the Lie derivative. Killing vector fields are a powerful tool in studying the geometry of a Riemannian manifold. A Killing vector field is a Jacobi vector field along any geodesic. However the converse is not true: the position vector on the euclidean space ℝⁿ is a Jacobi field along any geodesic of ℝⁿ but it is not Killing. Studying when the structure vector field of a complex projective space is Killing, Deshmukh, [3], introduced the notion of Jacobi type vector fields on a Riemannian manifold. A vector field Y on M̄ is of Jacobi type if it satisfies

∇¯X∇¯XY+R¯(Y,X)X=0(1)

for any vector field X tangent to M̄, where ∇̄ denotes the Levi-Civita connection on M̄ and R̄ its Riemannian curvature tensor. Naturally any Jacobi type vector field on M̄ is a Jacobi vector field along any geodesic of M̄.

As on a real hypersurface M in Q^m we have a special vector field, the structure one ξ, it is interesting to see if it is Killing when it is of Jacobi type. In this sense we will prove the following

Theorem 1.1

LetMbe a real hypersurface inQ^m, m ≥ 3. IfMis either compact or Hopf and the structure vector field is of Jacobi type, it is a Killing vector field.

By this Theorem and the classification of real hypersurfaces with geodesic Reeb flow we obtain

Corollary 1.2

LetMbe a compact or Hopf real hypersurface inQ^m, m ≥ 3. Then the structure vector field is of Jacobi type if and only ifmis even, saym = 2k, andMis locally congruent to a tube around a totally geodesic ℂP^k inQ^m.

Similar results for real hypersurfaces of complex two-plane Grassmannians were obtained in [4].

2 The space Q^m

For the study of Riemannian geometry of Q^m see [1]. All the notations we will use since now are the ones in [2].

The complex projective space ℂP^m+1 is considered as the Hermitian symmetric space of the special unitary group SU_m+2, namely ℂP^m+1 = SU_m+2/S(U_m+1U₁). The symbol o= [0, …, 0, 1] in ℂP^m+1 is the fixed point of the action of the stabilizer S(U_m+1U₁). The action of the special orthogonal group SO_m+2 ⊂ SU_m+2 on ℂP^m+1 is of cohomogeneity one. A totally geodesic projective space ℝP^m+1 ⊂ ℂP^m+1 is an orbit containing o. The second singular orbit of this action is the complex quadric Q^m = SO_m+2/SO_mSO₂. It is a homogeneous model wich interprets geometrically the complex quadric Q^m as the Grassmann manifold G2+ (ℝ^m+2) of oriented 2-planes in ℝ^m+2. For m = 1 the complex quadric is isometric to a sphere S² of constant curvature. For m = 2 the complex quadric Q² is isometric to the Riemannian product of two 2-spheres with constant curvature. Therefore we assume the dimension of the complex quadric Q^m to be greater than or equal to 3.

Moreover, the complex quadric Q^m is the complex hypersurface in ℂP^m+1 defined by the equation z12 + … + zm+22 = 0, where z_i, i = 1, …, m + 2, are homogeneous coordinates on ℂP^m+1. The Kähler structure of complex projective space induces canonically a Kähler structure (J, g) on Q^m, where g is the Riemannian metric induced by the Fubini-Study metric of ℂP^m+1.

A point [z] in ℂP^m+1 is the complex span of z, that is [z] = {λz/ λ ∈ ℂ}, where z is a nonzero vector of ℂ^{m + 2}. For each [z] in ℂP^m+1 the tangent space T_[z]ℂP^m+1 can be identified canonically with the orthogonal complement of [z] ⊕ [z̄] in ℂ^{m + 2}.

The shape operator A_z̄ of Q^m with respect to the unit normal vector z̄ is given by

Az¯w=w¯

for all w ∈ T_[z]Q^m. Then A_z̄ is a complex conjugation restricted to T_[z]Q^m. Thus T_[z]Q^m is decomposed into

T[z]Qm=V(Az¯)⊕JV(Az¯)

where V(A_z̄) is the (+1)-eigenspace of A_z̄ and JV(A_z̄) is the (-1)-eigenspace of A_z̄. Geometrically, it means that A_z̄ defines a real structure on the complex vector space T_[z]Q^m. The set of all shape operators A_λz̄ defines a parallel rank 2 subbundle 𝔄 of the endomorphism bundle End(TQ^m) which consists of all the real structures of the tangent space of Q^m. For any A ∈ 𝔄, A² = I and AJ = −JA.

The Gauss equation of Q^m in ℂP^m+1 yields that the Riemannian curvature tensor R̄ of Q^m is given by

R¯(X,Y)Z=g(Y,Z)X−g(X,Z)Y+g(JY,Z)JX−g(JX,Z)JY−2g(JX,Y)JZ+g(AY,Z)AX−g(AX,Z)AY+g(JAY,Z)JAX−g(JAX,Z)JAY(2)

where J is the complex structure and A is a real structure in 𝔄.

For every vector field W tangent to Q^m there is a complex conjugation A ∈ 𝔄 and orthonormal vectors X, Y ∈ V(A) such that

W=cos(t)X+sin(t)JY

for some t ∈ [0, π4].

3 Real hypersurfaces in Q^m

Let M be a real hypersurface in Q^m, that is, a submanifold of Q^m with real codimension one. The induced Riemannian metric on M will also be denoted by g, and ∇ denotes the Riemannian connection of (M, g). Let N be a unit normal vector field of M and S the shape operator of M with respect to N. For any X tangent to M we write

JX=ϕX+η(X)N

where ϕX denotes the tangential component of JX and η(X)N its normal component. The structure vector field (or Reeb vector field) ξ is defined by ξ = −JN. The 1-form η is given by η(X) = g(X, ξ) for any vector field X tangent to M. Therefore, on M we have an almost contact metric structure (ϕ, ξ, η, g). Thus,

ϕ2X=−X+η(X)ξ,η(ξ)=1,g(ϕX,ϕY)=g(X,Y)−η(X)η(Y),ϕξ=0(3)

for all tangent vector fields X, Y on M. Moreover, the parallelism of J yields

(∇Xϕ)Y=η(Y)SX−g(SX,Y)ξ(4)

and

∇Xξ=ϕSX(5)

for any X, Y tangent to M.

At each point [z] ∈ M we choose a real structure A ∈ 𝔄_[z] such that

N[z]=cos(t)Z1+sin(t)JZ2AN[z]=cos(t)Z1−sin(t)JZ2ξ[z]=−cos(t)JZ1+sin(t)Z2Aξ[z]=cos(t)JZ1+sin(t)Z2(6)

where Z₁, Z₂ are orthonormal vectors in V(A) and 0 ≤ t ≤ π4. Therefore g(AN, ξ) = 0.

Let X ∈ T_[z]M. Then AX is decomposed into

AX=BX+ρ(X)N(7)

where BX is the tangential component of AX and ρ(X)N is its normal component, with ρ(X) = g(AX, N). As seen above ρ(ξ) = 0.

From (2) the curvature tensor R of M is given by

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)T−g(AX,Z)(AY)T+g(JAY,Z)(JAX)T−g(JAX,Z)(JAY)T+g(SY,Z)SX−g(SX,Z)SY(8)

for any X, Y, Z tangent to M, where (.)^T denotes the tangential component of the correspondent vector field. From (8) the Ricci tensor of M is given (see [5]) by

Ric(X)=(2m−1)X−3η(X)ξ+η(Bξ)BX+−ρ(X)ϕBξ+η(BX)Bξ+(traceS)SX−S2X(9)

for any X tangent to M. Moreover, the Codazzi equation is given by

g((∇XS)Y−(∇YS)X,Z)=η(X)g(ϕY,Z)−η(Y)g(ϕX,Z)−2g(ϕX,Y)η(Z)+g(X,AN)g(AY,Z)−g(Y,AN)g(AX,Z)+g(X,Aξ)g(JAY,Z)−g(Y,Aξ)g(JAX,Z)(10)

for any X, Y, Z tangent to M.

The real hypersurface M is called Hopf if the Reeb vector field is an eigenvector of the shape operator S, that is

Sξ=αξ

where α = g(Sξ, ξ) is the Reeb function.

4 Proof of Theorem 1.1

Let us suppose that ξ is of Jacobi type. Then ∇_X∇_Xξ + R(ξ, X)X = 0 for any X tangent to M.

Take an orthonormal basis {e₁, …, e_2m−1} of vector fields tangent to M. As ξ is of Jacobi type, ∑i=12m−1 ∇_{e_i}∇_{e_i}ξ + Ric(ξ) = 0. That is,

∑i=12m−1∇eiϕSei+Ric(ξ)=0(11)

From (9)Ric(ξ) = 2(m − 2)ξ + η(Bξ)Bξ −ρ(ξ)ϕBξ + η(Bξ)Bξ + (trace S)Sξ − S²ξ. As ρ(ξ) = 0 and η(Bξ) = g(Aξ, ξ) we obtain

Ric(ξ)=2(m−2)ξ+2g(Aξ,ξ)Bξ+(traceS)Sξ−S2ξ.(12)

From (11) and (12) we get ∑i=12m−1 ∇_{e_i}ϕSe_i + 2(m − 2)ξ + 2g(Aξ, ξ)Bξ + (trace S)Sξ −S²ξ = 0. Taking its scalar product with ξ and bearing in mind that g(Bξ, ξ) = g(Aξ, ξ) we obtain

Lemma 4.1

LetMbe a real hypersurface inQ^m, m ≥ 3, such thatξis of Jacobi type. Then

−traceS2+2(m−2)+2g(Aξ,ξ)2+(traceS)η(Sξ)=0

Now we compute

∥ϕS−Sϕ∥2=∑i=12m−1g((ϕS−Sϕ)ei,(ϕS−Sϕ)ei)=∑i,j=12m−1g((ϕS−Sϕ)ei,ej)g((ϕS−Sϕ)ei,ej)=2∑i,j=12m−1g(ϕSej,ei)g(ϕSej,ei)+2∑i,j=12m−1g(ϕSej,ei)g(ϕSei,ej)=−2∑j=12m−1g(ϕSej,Sϕej)+2∑j=12m−1g(ϕSej,ϕSej)=2traceS2−2g(S2ξ,ξ)−2∑j=12m−1g(∇ejξ,Sϕej)(13)

where we have used (4).

Take now U = ∇_ξξ = ϕSξ. Then we have

div(U)=∑i=12m−1g(∇eiU,ei)=∑i=12m−1g(∇eiϕSξ,ei)=∑i=12m−1g((∇eiϕ)Sξ,ei)+∑i=12m−1g(ϕ∇eiSξ,ei)=∑i=12m−1g(η(Sξ)Sei−g(Sei,Sξ)ξ,ei)−∑i=12m−1g(∇eiSξ,ϕei),(14)

that is

Lemma 4.2

LetMbe a real hypersurface inQ^m, m ≥ 3, andU = ϕSξ. Then

div(U)=(traceS)η(Sξ)−η(S2ξ)−∑i=12m−1g(∇eiSξ,ϕei).

From (13) and Lemma 4.2 we obtain

div(U)−12∥ϕS−Sϕ∥2=−traceS2+η(Sξ)(traceS)−∑i=12m−1g((∇eiS)ξ,ϕei).(15)

Then

∑i=12m−1(g((∇ξS)ei,ϕei)=−∑i=12m−1g(ϕ(∇ξS)ei,ei)=−trace(ϕ(∇ξS))=−trace((∇ξS)ϕ)=−∑i=12m−1g((∇ξS)ϕei,ei)=−∑i=12m−1g((∇ξS)ei,ϕei).(16)

Thus we conclude

∑i=12m−1g((∇ξS)ei,ϕei)=0.(17)

Bearing in mind (17) Codazzi equation yields

∑i=12m−1g((∇eiS)ξ,ϕei)=−∑i=12m−1g(ϕei,ϕei)+∑i=12m−1g(ei,AN)g(Aξ,ϕei)+∑i=12m−1g(ei,Aξ)g(JAξ,ϕei)−∑i=12m−1g(ξ,Aξ)g(JAei,ϕei)=2(m−2)−g(AN,N)2+g(ξ,Aξ)g(AN,N)−g(ξ,Aξ)(traceA).(18)

From (6)g(AN, N) = cos(2t) = −g(Aξ, ξ). Moreover, as {e₁, …e_2m−1, N} is an orthonormal basis of vectors tangent to Q^m at any point of M, {Je₁, …, Je_2m−1, JN} is also an orthonormal basis. Then traceA = ∑i=12m−1g(AJe_i, Je_i) + g(AJN, JN) = −∑i=12m−1g(JAe_i, Je_i) − g(JAN, AN) = −∑i=12m−1g(Ae_i, e_i) − g(AN, N) = −trace A. Thus trace A = 0 and (18) becomes

∑i=12m−1g((∇eiS)ξ,ϕei)=−2(m−2)−2g(Aξ,ξ)2.(19)

From this, Lemma 4.1, Lemma 4.2 and (17) we get

div(U)=12∥ϕS−Sϕ∥2.(20)

Now if M is Hopf, U = 0 and then ϕS − Sϕ = 0.

If M is compact, 12∫_M ∥ϕS − Sϕ∥²dV = 0. Thus again ϕS − Sϕ = 0.

In both cases as (𝔏_ξg)(X, Y) = g((ϕS − Sϕ)X, Y), for any X, Y ∈ TM, we conclude 𝔏_ξg = 0 and ξ is Killing, obtaining our Theorem.

As ϕ S = Sϕ we have, [2], that m = 2k and M must be locally congruent to a tube around a totally geodesic ℂP^k in Q^m.

Bearing in mind the expression of the shape operator S of such a real hypersurface, [2], it is immediate to see that its structure vector field is of Jacobi type and we conclude the proof of our Corollary.

Acknowledgement

Supported by MINECO-FEDER Project MTM 2013-47828-C2-1-P.

References

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Received: 2017-09-28

Accepted: 2018-01-25

Published Online: 2018-03-20

This work is licensed under the Creative Commons Attribution-NonCommercial-NoDerivatives 4.0 License.

Articles in the same Issue

https://doi.org/10.1515/math-2018-0021

Keywords for this article

Real hypersurfaces; Complex quadric; Jacobi type vector field; Structure vector field; Killing vector field

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On the structure vector field of a real hypersurface in complex quadric

Article

Abstract

1 Introduction

Theorem 1.1

Corollary 1.2

2 The space Qm

3 Real hypersurfaces in Qm

4 Proof of Theorem 1.1

Lemma 4.1

Lemma 4.2

Acknowledgement

References

Articles in the same Issue

Articles in the same Issue

Articles in the same Issue

2 The space Q^m

3 Real hypersurfaces in Q^m