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Open Mathematics

formerly Central European Journal of Mathematics

Editor-in-Chief: Gianazza, Ugo / Vespri, Vincenzo


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Volume 16, Issue 1

Issues

Volume 13 (2015)

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

Juan de Dios Pérez
Published Online: 2018-03-20 | DOI: https://doi.org/10.1515/math-2018-0021

Abstract

From the notion of Jacobi type vector fields for a real hypersurface in complex quadric Qm 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 Qm = SOm+2/SOmSO2 is a compact Hermitian symmetric space of rank 2. It is also a complex hypersurface in complex projective space ℂPm+1, [1]. Qm is equipped with two geometric structures: a complex conjugation A and a Kähler structure J.

Real hypersurfaces M in Qm are immersed submanifolds of real codimension 1. The Kähler structure J of Qm 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 Qm whose Reeb flow is isometric are classified in [2]. They obtain tubes around the totally geodesic ℂPk in Qm 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 (,) satisfies 𝔏X = 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 is of Jacobi type if it satisfies

¯X¯XY+R¯(Y,X)X=0(1)

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

As on a real hypersurface M in Qm 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

Let M be a real hypersurface in Qm, m ≥ 3. If M is 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

Let M be a compact or Hopf real hypersurface in Qm, m ≥ 3. Then the structure vector field is of Jacobi type if and only if m is even, say m = 2k, and M is locally congruent to a tube around a totally geodesicPk in Qm.

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

2 The space Qm

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

The complex projective space ℂPm+1 is considered as the Hermitian symmetric space of the special unitary group SUm+2, namely ℂPm+1 = SUm+2/S(Um+1U1). The symbol o= [0, …, 0, 1] in ℂPm+1 is the fixed point of the action of the stabilizer S(Um+1U1). The action of the special orthogonal group SOm+2SUm+2 on ℂPm+1 is of cohomogeneity one. A totally geodesic projective space ℝPm+1 ⊂ ℂPm+1 is an orbit containing o. The second singular orbit of this action is the complex quadric Qm = SOm+2/SOmSO2. It is a homogeneous model wich interprets geometrically the complex quadric Qm as the Grassmann manifold G2+ (ℝm+2) of oriented 2-planes in ℝm+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 Qm to be greater than or equal to 3.

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

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

The shape operator A of Qm with respect to the unit normal vector is given by

Az¯w=w¯

for all wT[z]Qm. Then A is a complex conjugation restricted to T[z]Qm. Thus T[z]Qm is decomposed into

T[z]Qm=V(Az¯)JV(Az¯)

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

The Gauss equation of Qm in ℂPm+1 yields that the Riemannian curvature tensor of Qm is given by

R¯(X,Y)Z=g(Y,Z)Xg(X,Z)Y+g(JY,Z)JXg(JX,Z)JY2g(JX,Y)JZ+g(AY,Z)AXg(AX,Z)AY+g(JAY,Z)JAXg(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 Qm there is a complex conjugation A ∈ 𝔄 and orthonormal vectors X, YV(A) such that

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

for some t ∈ [0, π4].

3 Real hypersurfaces in Qm

Let M be a real hypersurface in Qm, that is, a submanifold of Qm 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 ϕ X 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)SXg(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)Z1sin(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 ≤ tπ4. Therefore g(AN, ξ) = 0.

Let XT[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)Xg(X,Z)Y+g(ϕY,Z)ϕXg(ϕX,Z)ϕY2g(ϕX,Y)ϕZ+g(AY,Z)(AX)Tg(AX,Z)(AY)T+g(JAY,Z)(JAX)Tg(JAX,Z)(JAY)T+g(SY,Z)SXg(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)=(2m1)X3η(X)ξ+η(Bξ)BX+ρ(X)ϕBξ+η(BX)Bξ+(traceS)SXS2X(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(, ξ) is the Reeb function.

4 Proof of Theorem 1.1

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

Take an orthonormal basis {e1, …, e2m−1} of vector fields tangent to M. As ξ is of Jacobi type, i=12m1eieiξ + Ric(ξ) = 0. That is,

i=12m1eiϕSei+Ric(ξ)=0(11)

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

Ric(ξ)=2(m2)ξ+2g(Aξ,ξ)Bξ+(traceS)SξS2ξ.(12)

From (11) and (12) we get i=12m1eiϕ Sei + 2(m − 2)ξ + 2g(, ξ) + (trace S)S2ξ = 0. Taking its scalar product with ξ and bearing in mind that g(, ξ) = g(, ξ) we obtain

Lemma 4.1

Let M be a real hypersurface in Qm, m ≥ 3, such that ξ is of Jacobi type. Then

traceS2+2(m2)+2g(Aξ,ξ)2+(traceS)η(Sξ)=0

Now we compute

ϕSSϕ2=i=12m1g((ϕSSϕ)ei,(ϕSSϕ)ei)=i,j=12m1g((ϕSSϕ)ei,ej)g((ϕSSϕ)ei,ej)=2i,j=12m1g(ϕSej,ei)g(ϕSej,ei)+2i,j=12m1g(ϕSej,ei)g(ϕSei,ej)=2j=12m1g(ϕSej,Sϕej)+2j=12m1g(ϕSej,ϕSej)=2traceS22g(S2ξ,ξ)2j=12m1g(ejξ,Sϕej)(13)

where we have used (4).

Take now U = ∇ξξ = ϕ . Then we have

div(U)=i=12m1g(eiU,ei)=i=12m1g(eiϕSξ,ei)=i=12m1g((eiϕ)Sξ,ei)+i=12m1g(ϕeiSξ,ei)=i=12m1g(η(Sξ)Seig(Sei,Sξ)ξ,ei)i=12m1g(eiSξ,ϕei),(14)

that is

Lemma 4.2

Let M be a real hypersurface in Qm, m ≥ 3, and U = ϕ . Then

div(U)=(traceS)η(Sξ)η(S2ξ)i=12m1g(eiSξ,ϕei).

From (13) and Lemma 4.2 we obtain

div(U)12ϕSSϕ2=traceS2+η(Sξ)(traceS)i=12m1g((eiS)ξ,ϕei).(15)

Then

i=12m1(g((ξS)ei,ϕei)=i=12m1g(ϕ(ξS)ei,ei)=trace(ϕ(ξS))=trace((ξS)ϕ)=i=12m1g((ξS)ϕei,ei)=i=12m1g((ξS)ei,ϕei).(16)

Thus we conclude

i=12m1g((ξS)ei,ϕei)=0.(17)

Bearing in mind (17) Codazzi equation yields

i=12m1g((eiS)ξ,ϕei)=i=12m1g(ϕei,ϕei)+i=12m1g(ei,AN)g(Aξ,ϕei)+i=12m1g(ei,Aξ)g(JAξ,ϕei)i=12m1g(ξ,Aξ)g(JAei,ϕei)=2(m2)g(AN,N)2+g(ξ,Aξ)g(AN,N)g(ξ,Aξ)(traceA).(18)

From (6) g(AN, N) = cos(2t) = −g(, ξ). Moreover, as {e1, …e2m−1, N} is an orthonormal basis of vectors tangent to Qm at any point of M, {Je1, …, Je2m−1, JN} is also an orthonormal basis. Then trace A = i=12m1 g(AJei, Jei) + g(AJN, JN) = −i=12m1 g(JAei, Jei) − g(JAN, AN) = −i=12m1 g(Aei, ei) − g(AN, N) = −trace A. Thus trace A = 0 and (18) becomes

i=12m1g((eiS)ξ,ϕei)=2(m2)2g(Aξ,ξ)2.(19)

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

div(U)=12ϕSSϕ2.(20)

Now if M is Hopf, U = 0 and then ϕ SSϕ = 0.

If M is compact, 12 Mϕ SSϕ2dV = 0. Thus again ϕ SSϕ = 0.

In both cases as (𝔏ξ g)(X, Y) = g((ϕ SSϕ)X, Y), for any X, YTM, we conclude 𝔏ξ g = 0 and ξ is Killing, obtaining our Theorem.

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

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.

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About the article

Received: 2017-09-28

Accepted: 2018-01-25

Published Online: 2018-03-20


Citation Information: Open Mathematics, Volume 16, Issue 1, Pages 185–189, ISSN (Online) 2391-5455, DOI: https://doi.org/10.1515/math-2018-0021.

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© 2018 de Dios Pérez, published by De Gruyter. This work is licensed under the Creative Commons Attribution-NonCommercial-NoDerivatives 4.0 License. BY-NC-ND 4.0

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