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c World Scientific Publishing Company 3

DOI:10.1142/S0129167X1350050X 4

REAL HYPERSURFACES WITH ISOMETRIC REEB FLOW

5

IN COMPLEX QUADRICS

6

JURGEN BERNDT 7

Department of Mathematics 8

King’s College London 9

London WC2R 2LS, UK 10

jurgen.berndt@kcl.ac.uk 11

YOUNG JIN SUH 12

Department of Mathematics 13

College of Natural Sciences 14

Kyungpook National University 15

Daegu 702-701, Republic of Korea 16

yjsuh@knu.ac.kr 17

Received 18

Accepted 19

Published 20

We classify real hypersurfaces with isometric Reeb flow in the complex quadricsQ^m= 21

SO_m+2/SO_mSO₂,m≥3. We show thatmis even, saym= 2k, and any such hyper- 22

surface is an open part of a tube around ak-dimensional complex projective spaceCP^k 23

which is embedded canonically inQ^2kas a totally geodesic complex submanifold. As a 24

consequence, we get the non-existence of real hypersurfaces with isometric Reeb flow in 25

odd-dimensional complex quadricsQ^2k+1,k≥1. To our knowledge the odd-dimensional 26

complex quadrics are the first examples of homogeneous K¨ahler manifolds which do not 27

admit a real hypersurface with isometric Reeb flow.

28

Keywords: Real hypersurface; Reeb flow; complex quadric.

29

Mathematics Subject Classification 2010: Primary: 53C40; Secondary: 53C55, 53D15 30

1. Introduction

31

Let M be a real hypersurface in a K¨ahler manifold ¯M. The complex structure

32

J on ¯M induces locally an almost contact metric structure (φ, ξ, η, g) on M. In

33

the context of contact geometry, the unit vector fieldξ is often referred to as the

34

Reeb vector field onM and its flow is known as the Reeb flow. The Reeb flow has

been of significant interest in recent years, for example in relation to the Wein-

1

stein Conjecture. We are interested in the Reeb flow in the context of Riemannian

2

geometry, namely in the classification of real hypersurfaces with isometric Reeb

3

flow in homogeneous K¨ahler manifolds.

4

For the complex projective space CP^m a full classification was obtained by

5

Okumura in [4]. He proved that the Reeb flow on a real hypersurface in CP^m =

6

SU_m+1/S(U_mU₁) is isometric if and only ifM is an open part of a tube around a

7

totally geodesicCP^k ⊂CP^mfor somek∈ {0, . . . , m−1}. For the complex 2-plane

8

Grassmannian G₂(C^m+2) = SU_m+2/S(U_mU₂) the classification was obtained by

9

the authors in [2]. The Reeb flow on a real hypersurface inG₂(C^m+2) is isometric

10

if and only if M is an open part of a tube around a totally geodesicG₂(C^m+1)⊂

11

G₂(C^m+2). In this paper, we investigate this problem for the complex quadricQ^m=

12

SO_m+2/SO_mSO₂. In view of the previous two results a natural expectation is that

13

the classification involves at least the totally geodesic Q^m−1 ⊂ Q^m. Surprisingly,

14

this is not the case. Our main result states:

15

Theorem 1.1. Let M be a real hypersurface of the complex quadric Q^m, m ≥3.

16

The Reeb flow onM is isometric if and only if m is even, say m= 2k, and M is

17

an open part of a tube around a totally geodesicCP^k⊂Q^2k.

18

Every tube around a totally geodesicCP^k ⊂Q^2k is a homogeneous hypersur-

19

face. In fact, the closed subgroupU_k+1 ofSO_2k+2 acts onQ^2k with cohomogeneity

20

one. The two singular orbits are totally geodesic CP^k ⊂ Q^2k and the principal

21

orbits are the tubes around any of these two singular orbits. So as a corollary,

22

we get:

23

Corollary 1.2. Let M be a connected complete real hypersurface in the complex

24

quadricQ^2k, k≥2. If the Reeb flow on M is isometric, thenM is a homogeneous

25

hypersurface of Q^2k.

26

It is remarkable that in this situation the existence of a particular one-parameter

27

group of isometries implies transitivity of the isometry group. As another interesting

28

consequence we get:

29

Corollary 1.3. There are no real hypersurfaces with isometric Reeb flow in the

30

odd-dimensional complex quadric Q^2k+1, k≥1.

31

To our knowledge the odd-dimensional complex quadrics are the first examples

32

of homogeneous K¨ahler manifolds which do not admit a real hypersurface with

33

isometric Reeb flow.

34

Our paper is organized as follows. In Sec. 2, we present basic material about the

35

complex quadricQ^m, including its Riemannian curvature tensor and a description

36

of its singular tangent vectors. Apart from the complex structureJ there is another

37

distinguished geometric structure onQ^m, namely a parallel rank two vector bundle

38

Awhich contains anS¹-bundle of real structures on the tangent spaces ofQ^m. This

39

geometric structure determines a maximalA-invariant subbundleQof the tangent

1

bundleTM of a real hypersurfaceM inQ^m. In Sec. 3, we investigate the geometry

2

of this subbundleQ. In Sec. 4, we describe the canonical embedding ofCP^k into

3

Q^2k as a totally geodesic complex submanifold and investigate the geometry of the

4

tubes aroundCP^k. We will show that the Reeb flow on these tubes is isometric. In

5

Sec. 5, we determine some geometric consequences of the Codazzi equation for real

6

hypersurfaces inQ^m.

7

Finally, in Sec. 6, we present the proof of Theorem1.1. The first step is to prove

8

that the normal bundle of a real hypersurfaceM with isometric Reeb flow consists

9

of a particular type of singular tangent vectors of Q^m, the so-called A-isotropic

10

vectors. The next step is to show that Q is invariant under the shape operator

11

of M. Putting all this information into the Codazzi equation then allows us to

12

compute explicitly the principal curvatures and principal curvatures spaces of M.

13

A particular consequence of this is thatm is even, saym= 2k. Using Jacobi field

14

theory, we then show thatM has a smooth focal manifold at a constant distance

15

which is embedded inQ^2k as a totally geodesic complex submanifold of complex

16

dimension k. Using Klein’s [3] classification of totally geodesic submanifolds in

17

complex quadrics we will then show that this focal manifold is a totally geodesic

18

CP^k ⊂Q^2k.

19

2. The Complex Quadric

20

The homogeneous quadratic equationz₁²+· · ·+z_m+2² = 0 onC^m+2defines a complex

21

hypersurfaceQ^m in the (m+ 1)-dimensional complex projective space CP^m+1 =

22

SU_m+2/S(U_m+1U₁). The hypersurfaceQ^mis known as them-dimensional complex

23

quadric. The complex structureJ onCP^m+1naturally induces a complex structure

24

on Q^m which we will denote by J as well. We equip Q^m with the Riemannian

25

metricg which is induced from the Fubini–Study metric onCP^m+1with constant

26

holomorphic sectional curvature 4. The 1-dimensional quadricQ¹is isometric to the

27

round 2-sphereS². Form≥2 the triple (Q^m, J, g) is a Hermitian symmetric space

28

of rank two and its maximal sectional curvature is equal to 4. The 2-dimensional

29

quadricQ²is isometric to the Riemannian productS²×S². We will assumem≥3

30

for the main part of this paper.

31

For a nonzero vector z ∈C^m+1 we denote by [z] the complex span of z, that

32

is, [z] ={λz| λ∈C}. Note that by definition [z] is a point in CP^m+1. As usual,

33

for each [z] ∈ CP^m+1 we identify T_[z]CP^m+1 with the orthogonal complement

34

C^m+2[z] of [z] in C^m+2. For [z] ∈ Q^m the tangent space T_[z]Q^m can then be

35

identified canonically with the orthogonal complementC^m+2([z]⊕[¯z]) of [z]⊕[¯z]

36

in C^m+2. Note that ¯z ∈ ν_[z]Q^m is a unit normal vector of Q^m in CP^m+1 at the

37

point [z].

38

We denote byA_¯zthe shape operator ofQ^minCP^m+1with respect to ¯z. Then we haveA_¯zw=wfor allw∈T_[z]Q^m, that is,A_z¯is just complex conjugation restricted

toT_[z]Q^m. The shape operatorA_z¯is an antilinear involution on the complex vector spaceT_[z]Q^mand

T_[z]Q^m=V(A_¯z)⊕JV(A_z¯),

where V(A_z¯) = R^m+2 ∩T_[z]Q^m is the (+1)-eigenspace and JV(A_¯z) = iR^m+2∩

1

T_[z]Q^m is the (−1)-eigenspace of A_z¯. Geometrically this means that the shape

2

operator A_z¯ defines a real structure on the complex vector space T_[z]Q^m. Recall

3

that a real structure on a complex vector space V is by definition an antilinear

4

involution A : V →V. Since the normal space ν_[z]Q^m of Q^m in CP^m+1 at [z] is

5

a complex subspace ofT_[z]CP^m+1 of complex dimension one, every normal vector

6

in ν_[z]Q^mcan be written as λ¯z with someλ∈C. The shape operatorsA_λ¯z ofQ^m

7

define a rank two vector subbundle A of the endomorphism bundle End(TQ^m).

8

Since the second fundamental form of the embeddingQ^m⊂CP^m+1is parallel (see

9

e.g. [5]), Ais a parallel subbundle of End(TQ^m). For λ∈ S¹ ⊂C, we again get

10

a real structureA_λ¯z onT_[z]Q^mand we haveV(A_λ¯z) = λV(A_¯z). We thus have an

11

S¹-subbundle ofAconsisting of real structures on the tangent spaces ofQ^m.

12

The Gauss equation for the complex hypersurfaceQ^m⊂CP^m+1implies that the Riemannian curvature tensorRofQ^mcan be expressed in terms of the Riemannian metricg, the complex structureJ and a generic real structureAin A:

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.

Note that the complex structureJ anti-commutes with each endomorphismA∈A,

13

that is,AJ =−JA.

14

A nonzero tangent vectorW ∈T_[z]Q^mis called singular if it is tangent to more

15

than one maximal flat in Q^m. There are two types of singular tangent vectors for

16

the complex quadricQ^m:

17

(i) If there exists a real structureA∈A_[z]such thatW ∈V(A), thenW is singular.

18

Such a singular tangent vector is calledA-principal.

19

(ii) If there exist a real structure A∈A_[z] and orthonormal vectorsX, Y ∈V(A)

20

such thatW/W= (X+J Y)/√

2, thenW is singular. Such a singular tangent

21

vector is calledA-isotropic.

22

Basic complex linear algebra shows that for every unit tangent vectorW ∈T_[z]Q^m there exist a real structure A ∈ A_[z] and orthonormal vectorsX, Y ∈ V(A) such that

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

for some t∈[0, π/4]. The singular tangent vectors correspond to the valuest= 0

23

andt=π/4.

24

3. The Maximal A-invariant Subbundle Q ofTM

1

LetM be a real hypersurface in Q^m and denote by (φ, ξ, η, g) the induced almost

2

contact metric structure onM and by∇the induced Riemannian connection onM.

3

Note thatξ=−J N, whereN is a (local) unit normal vector field ofM. The vector

4

fieldξis known as the Reeb vector field ofM. If the integral curves ofξare geodesics

5

inM, the hypersurfaceM is called a Hopf hypersurface. The integral curves ofξare

6

geodesics inM if and only ifξis a principal curvature vector ofM everywhere. The

7

tangent bundle TM ofM splits orthogonally intoTM =C ⊕ F, whereC= ker(η)

8

is the maximal complex subbundle ofTM andF =Rξ. The structure tensor field

9

φ restricted to C coincides with the complex structure J restricted to C, and we

10

haveφξ = 0. We denote byνM the normal bundle ofM.

11

We first introduce some notations. For a fixed real structure A ∈ A_[z] and X ∈T_[z]M we decomposeAX into its tangential and normal component, that is,

AX =BX+ρ(X)N, whereBX is the tangential component ofAX and

ρ(X) =g(AX, N) =g(X,AN) =g(X,AJξ) =g(JX, Aξ).

SinceJX =φX+η(X)N and Aξ=Bξ+ρ(ξ)N, we also have ρ(X) =g(φX, Bξ) +η(X)ρ(ξ) =η(BφX) +η(X)ρ(ξ).

We also define

δ=g(N,AN) =g(JN,JAN) =−g(JN,AJN) =−g(ξ, Aξ).

At each point [z]∈M, we define

Q_[z]={X∈T_[z]M|AX∈T_[z]M for allA∈A_[z]}, which is the maximalA_[z]-invariant subspace ofT_[z]M.

12

Lemma 3.1. The following statements are equivalent:

13

(i) The normal vectorN_[z] is A-principal.

14

(ii) Q_[z]=C_[z].

15

(iii) There exists a real structure A∈A_[z] such thatAN_[z] ∈Cν_[z]M.

16

Proof. We first assume that N_[z] is A-principal. By definition, this means that

17

there exists a real structureA∈A_[z] such thatN_[z]∈V(A), that is,AN_[z]=N_[z].

18

Then we have Aξ_[z] = −AJN_[z] = JAN_[z] = J N_[z] = −ξ_[z]. It follows that A

19

restricted to the planeCν_[z]M is the orthogonal reflection in the lineν_[z]M. Since

20

all endomorphisms inA_[z]differ just by scalar multiplication with a complex number

21

we see thatCν_[z]M is invariant underA_[z]. This implies thatC_[z]=T_[z]Q^mCν_[z]M

22

is invariant underA_[z], and henceQ_[z]=C_[z].

23

Conversely, assume thatQ_[z] =C_[z]. ThenCν_[z]M =T_[z]Q^m C_[z] is invariant

24

under A_[z]. Since the real dimension of Cν_[z]M is 2 and all real structures inA_[z]

25

differ just by scalar multiplication with a unit complex number, there exists a

1

real structure A∈A_[z] which fixesN_[z]. ThusN_[z] is A-principal. This proves the

2

equivalence of (i) and (ii).

3

The equivalence of (ii) and (iii) follows from the fact that all real structures in

4

A_[z] differ just by scalar multiplication with a unit complex number.

5

Assume now that the normal vectorN_[z] is notA-principal. Then there exists a real structureA∈A_[z] such that

N_[z]= cos(t)Z₁+ sin(t)JZ₂

for some orthonormal vectorsZ₁, Z₂∈V(A) and 0< t≤ ^π₄. This implies N_[z] = cos(t)Z₁+ sin(t)JZ₂,

AN_[z] = cos(t)Z₁−sin(t)JZ₂, ξ_[z] = sin(t)Z₂−cos(t)JZ₁, Aξ_[z] = sin(t)Z₂+ cos(t)JZ₁

and thereforeQ_[z]=T_[z]Q^m([Z₁]⊕[Z₂]) is strictly contained inC_[z]. Moreover, we have

Aξ_[z]=Bξ_[z] and ρ(ξ_[z]) = 0.

We have

g(Bξ_[z]+δξ_[z], N_[z]) = 0, g(Bξ_[z]+δξ_[z], ξ_[z]) = 0, g(Bξ_[z]+δξ_[z], Bξ_[z]+δξ_[z]) = sin²(2t) and therefore

U_[z]= 1

sin(2t)(Bξ_[z]+δξ_[z]) is a unit vector in C_[z] and

C_[z]=Q_[z]⊕[U_[z]] (orthogonal sum).

IfN_[z] is notA-principal at [z], thenN is notA-principal in an open neighborhood

6

of [z], and thereforeU is a well-defined unit vector field on that open neighborhood.

7

We summarize this in the following lemma.

8

Lemma 3.2. Assume thatN_[z] is notA-principal at[z]. Then there exists an open

9

neighborhood of [z] in M and a section A inA on that neighborhood consisting of

10

real structures such that

11

(i) Aξ=Bξ andρ(ξ) = 0,

12

(ii) U = (Bξ+δξ)/Bξ+δξ is a unit vector field tangent toC,

13

(iii) C=Q ⊕[U].

14

In the following, we will always assume thatAis chosen in line with Lemma3.2

1

in neighborhoods where the normal vector fieldN is notA-principal.

2

4. Tubes Around the Totally Geodesic CP^k ⊂Q^2k

3

We assume thatmis even, saym= 2k. The map

CP^k →Q^2k⊂CP^2k+1,[z₁, . . . , z_k+1]→[z₁, . . . , z_k+1, iz₁, . . . , iz_k+1]

provides an embedding ofCP^kintoQ^2kas a totally geodesic complex submanifold.

4

We define a complex structurej onC^2k+2 by

j(z₁, . . . , z_k+1, z_k+2, . . . , z_2k+2) = (−z_k+2, . . . ,−z_2k+2, z₁, . . . , z_k+1).

Note thatij =ji. We can then identifyC^2k+2 with C^k+1⊕jC^k+1 and get T_[z]CP^k ={X+jiX|X ∈C^k+1[z]}={X+ijX|X ∈V(A_z¯)}.

Note that the complex structureionC^2k+2 corresponds to the complex structure J onT_[z]Q^2k via the obvious identifications. For the normal spaceν_[z]CP^k ofCP^k at [z] we have

ν_[z]CP^k=A_z¯(T_[z]CP^k) ={X−ijX|X ∈V(A_z¯)}.

It is easy to see that both the tangent bundle and the normal bundle ofCP^k consist

5

ofA-isotropic singular tangent vectors ofQ^2k.

6

We will now calculate the principal curvatures and principal curvature spaces of the tube with radius 0 < r < π/2 around CP^k in Q^2k. Let N be a unit nor- mal vector of CP^k in Q^2k at [z] ∈ CP^k. Since N is A-isotropic, the four vectors N,JN,AN,JAN are pairwise orthonormal and the normal Jacobi operatorR_N is given by

R_NZ=R(Z, N)N =Z−g(Z, N)N+ 3g(Z,JN)JN

−g(Z,AN)AN −g(Z,JAN)JAN.

This implies readily that R_N has the three eigenvalues 0,1,4 with corresponding

7

eigenspacesRN⊕[AN],T_[z]Q^2k([N]⊕[AN]) andRJN. Since [N]⊂ν_[z]CP^k and

8

[AN]⊂T_[z]CP^k, we conclude that both T_[z]CP^k and ν_[z]CP^k are invariant under

9

R_N.

10

To calculate the principal curvatures of the tube of radius 0< r < π/2 around CP^k we use the Jacobi field method as described in [1, Sec. 8.2]. Let γ be the geodesic in Q^2k with γ(0) = [z] and ˙γ(0) = N and denote by γ^⊥ the parallel subbundle ofTQ^2k alongγdefined byγ_γ(t)^⊥ =T_[γ(t)]Q^2kRγ(t). Moreover, define˙ theγ^⊥-valued tensor fieldR^⊥_γ alongγ by R^⊥_γ(t)X =R(X,γ(t)) ˙˙ γ(t). Now consider the End(γ^⊥)-valued differential equation

Y+R^⊥_γ ◦Y = 0.

LetD be the unique solution of this differential equation with initial values D(0) =

I 0 0 0

, D(0) = 0 0

0 I

, where the decomposition of the matrices is with respect to γ_[z]^⊥ =T_[z]CP^k⊕(ν_[z]CP^kRN)

and I denotes the identity transformation on the corresponding space. Then the shape operatorS(r) of the tube of radius 0< r < π/2 aroundCP^kwith respect to

−γ(r) is given by˙

S(r) =D(r)◦D⁻¹(r).

If we decompose γ_[z]^⊥ further into

γ_[z]^⊥ = (T_[z]CP^k[AN])⊕[AN]⊕(ν_[z]CP^k[N])⊕RJ N, we get by explicit computation that

S(r) =







−tan(r) 0 0 0

0 0 0 0

0 0 cot(r) 0

0 0 0 2 cot(2r)







with respect to that decomposition. Therefore, the tube of radius 0 < r < π/2

1

aroundCP^k has four distinct constant principal curvatures−tan(r), 0, cot(r) and

2

2 cot(2r) (unless m= 2 in which case there are only two distinct constant princi-

3

pal curvatures 0 and 2 cot(2r)). The corresponding principal curvature spaces are

4

[AN],T_[z]CP^k[AN],ν_[z]CP^k[N] andRJN, respectively, where we identify the

5

subspaces obtained by parallel translation along γ from [z] toγ(r). Note that the

6

parallel translate of [AN] corresponds toC Q, the parallel translate of [N] cor-

7

responds toCνM, and the parallel translate ofRJN corresponds to F. Moreover,

8

we haveA(T_[z]CP^k[AN]) =ν_[z]CP^k[N].

9

SinceJN is a principal curvature vector we also conclude that every tube around CP^kis a Hopf hypersurface. We also see that all principal curvature spaces orthog- onal to RJN areJ-invariant. Thus, if φ denotes the structure tensor field on the tube which is induced byJ, we getSφ=φS. Since the complex structure onQ^m is parallel, we have

g(∇Xξ, Y) +g(X,∇Yξ) =g((Sφ−φS)X, Y)

for allX, Y ∈TM. Asξis a Killing vector field if and only if∇ξis a skew-symmetric

10

tensor field, we see that the Reeb flow onM is isometric if and only ifSφ=φS.

11

We summarize the previous discussion in the following proposition.

12

Proposition 4.1. Let M be the tube of radius 0 < r < π/2 around the totally

13

geodesic CP^k inQ^2k, k≥2. Then the following statements hold:

14

(i) M is a Hopf hypersurface.

15

Table 1. Principal curvatures ofM. Principal curvature Eigenspace Multiplicity

2 cot(2r) F 1

0 C Q 2

−tan(r) TCP^k(C Q) 2k−2

cot(r) νCP^kCνM 2k−2

(ii) The tangent bundle TM and the normal bundleνM ofM consist ofA-isotropic

1

singular tangent vectors of Q^2k.

2

(iii) M has four distinct constant principal curvatures. Their values and corre-

3

sponding principal curvature spaces and multiplicities are given in Table 1.

4

The real structure A determined by the A-isotropic unit normal vector at [z]

5

maps T_[z]CP^k(C_[z] Q_[z])ontoν_[z]CP^kCν_[z]M, and vice versa.

6

(iv) The shape operator S of M and the structure tensor field φ of M commute

7

with each other,that is, Sφ=φS.

8

(v) The Reeb flow onM is an isometric flow.

9

5. The Codazzi Equation and Some Consequences

10

From the explicit expression of the Riemannian curvature tensor of the complex quadric Q^m we can easily derive the Codazzi equation for a real hypersurface M ⊂Q^m:

g((∇_XS)Y −(∇_YS)X, Z)

=η(X)g(φY, Z)−η(Y)g(φX, Z)−2η(Z)g(φX, Y) +ρ(X)g(BY, Z)−ρ(Y)g(BX, Z)

−η(BX)g(BY, φZ)−η(BX)ρ(Y)η(Z) +η(BY)g(BX, φZ) +η(BY)ρ(X)η(Z).

Now, we assume thatM is a Hopf hypersurface. Then, we have Sξ=αξ

with the smooth function α = g(Sξ, ξ) on M. Inserting Z = ξ into the Codazzi equation leads to

g((∇XS)Y −(∇YS)X, ξ) =−2g(φX, Y) + 2ρ(X)η(BY)−2ρ(Y)η(BX).

On the other hand, we have

g((∇_XS)Y −(∇_YS)X, ξ) =g((∇_XS)ξ, Y)−g((∇_YS)ξ, X)

=dα(X)η(Y)−dα(Y)η(X)

+αg((Sφ+φS)X, Y)−2g(SφSX, Y).

Comparing the previous two equations and puttingX =ξyields dα(Y) =dα(ξ)η(Y)−2δρ(Y).

Reinserting this into the previous equation yields

g((∇XS)Y −(∇YS)X, ξ) = 2δη(X)ρ(Y)−2δρ(X)η(Y)

+αg((φS+Sφ)X, Y)−2g(SφSX, Y).

Altogether this implies

0 = 2g(SφSX, Y)−αg((φS+Sφ)X, Y)−2g(φX, Y)

+ 2δρ(X)η(Y) + 2ρ(X)η(BY)−2ρ(Y)η(BX)−2δη(X)ρ(Y)

=g((2SφS−α(φS+Sφ)−2φ)X, Y) + 2ρ(X)η(BY+δY)−2ρ(Y)η(BX+δX)

=g((2SφS−α(φS+Sφ)−2φ)X, Y)

+ 2ρ(X)g(Y, Bξ+δξ)−2g(X, Bξ+δξ)ρ(Y).

If AN =N we haveρ= 0, otherwise we can use Lemma 3.2to calculate ρ(Y) =

1

g(Y,AN) =g(Y,AJξ) =−g(Y,JAξ) =−g(Y,JBξ) =−g(Y, φBξ). Thus, we have

2

proved.

3

Lemma 5.1. LetM be a Hopf hypersurface inQ^m, m≥3. Then we have (2SφS−α(φS+Sφ)−2φ)X =−2ρ(X)(Bξ+δξ)−2g(X, Bξ+δξ)φBξ.

IfN isA-principal we can choose a real structureA ∈Asuch that AN =N. Then, we haveρ= 0 andφBξ=−φξ= 0, and therefore

2SφS−α(φS+Sφ) = 2φ.

IfN is notA-principal we can choose a real structureA∈Aas in Lemma3.2and get

ρ(X)(Bξ+δξ) +g(X, Bξ+δξ)φBξ

=−g(X, φ(Bξ+δξ))(Bξ+δξ) +g(X, Bξ+δξ)φ(Bξ+δξ)

=Bξ+δξ²(g(X, U)φU)−g(X, φU)U)

= sin²(2t)(g(X, U)φU−g(X, φU)U),

which is equal to 0 onQ and equal to sin²(2t)φX on C Q. Altogether, we have

4

proved:

5

Lemma 5.2. LetM be a Hopf hypersurface inQ^m, m≥3. Then the tensor field 2SφS−α(φS+Sφ)

leaves QandC Q invariant and we have

2SφS−α(φS+Sφ) = 2φon Q and

2SφS−α(φS+Sφ) = 2δ²φonC Q.

We will now prove that the principal curvature α of a Hopf hypersurface is constant if the normal vectors areA-isotropic. Assume thatN isA-isotropic every- where. Then we haveδ= 0 and we get

Y α=dα(ξ)η(Y)

for allY ∈TM. Since grad^Mα=dα(ξ)ξ, we can compute the Hessian hess^Mαby (hess^Mα)(X, Y) =g(∇_Xgrad^Mα, Y)

=d(dα(ξ))(X)η(Y) +dα(ξ)g(φSX, Y).

As hess^Mαis a symmetric bilinear form, the previous equation implies dα(ξ)g((Sφ+φS)X, Y) = 0

for all vector fieldsX, Y onM which are tangential toC.

1

Now, let us assume that Sφ+φS = 0. For every principal curvature vector X ∈ C such that SX = λX this implies SφX = −φSX = −λφX. We assume X= 1 and putY =φX. Using Lemma5.1, we get

1≤λ²+ 1 =ρ(X)η(BφX)−ρ(φX)η(BX)

=g(X, AN)²+g(X, Aξ)²=X_CQ²≤1,

where X_CQ denotes the orthogonal projection of X onto C Q. This implies

2

X_CQ²= 1 for all principal curvature vectorsX ∈ C withX= 1. This is only

3

possible if C =C Q, or equivalently, ifQ= 0. Since m≥3 this is not possible.

4

Hence, we must have Sφ+φS = 0 everywhere and therefore dα(ξ) = 0, which

5

implies grad^Mα= 0. SinceM is connected this implies that αis constant. Thus,

6

we have proved:

7

Lemma 5.3. Let M be a Hopf hypersurface inQ^m, m≥3,such that the normal

8

vector fieldN isA-isotropic everywhere. Thenαis constant.

9

6. Proof of Theorem 1.1

10

Let M be a real hypersurface in Q^m, m ≥ 3, with isometric Reeb flow. As we

11

have seen above, this geometric condition is equivalent to the algebraic condition

12

Sφ =φS. Applying this equation to ξ gives 0 = Sφξ = φSξ, which implies that

13

Sξ=αξ withα=g(Sξ, ξ). Therefore, any real hypersurface in Q^m with isometric

14

Reeb flow is a Hopf hypersurface.

15

Differentiating the equationSφ−φS= 0 gives

0 = (∇XS)φY +S(∇Xφ)Y −(∇Xφ)SY −φ(∇XS)Y

= (∇XS)φY +S(η(Y)SX−g(SX, Y)ξ)

−(η(SY)SX−g(SX,SY)ξ)−φ(∇XS)Y

= (∇_XS)φY +η(Y)S²X−αg(SX, Y)ξ

−η(SY)SX+g(SX,SY)ξ−φ(∇_XS)Y.

If we define

Ψ(X, Y, Z) =g((∇XS)Y, φZ) +g((∇XS)Z, φY), the previous equation implies

Ψ(X, Y, Z) =αη(Z)g(SX, Y)−η(Z)g(SX,SY) +η(SY)g(SX, Z)−η(Y)g(S²X, Z).

Evaluating Ψ(X, Y, Z) + Ψ(Y, Z, X)−Ψ(Z, X, Y) leads to

2g((∇_XS)Y, φZ) = Ω(X, Y, Z)−Ω(Y, Z, X) + Ω(Z, X, Y) + 2αη(Z)g(SX, Y)−2η(Z)g(S²X, Y), where

Ω(X, Y, Z) =g((∇_XS)Y −(∇_YS)X, φZ).

The three Ω-terms can be evaluated using the Codazzi equation, which leads to 2g((∇XS)Y, φZ) =ρ(X)(g(AY, φZ)−g(AZ, φY))

+η(BX)(g(JAY, φZ)−g(JAZ, φY))

−ρ(Y)(g(AX, φZ) +g(AZ, φX))

−η(BY)(g(JAX, φZ) +g(JAZ, φX)) +ρ(Z)(g(AX, φY) +g(AY, φX)) +η(BZ)(g(JAX, φY) +g(JAY, φX)) + 2η(Z)g(φX, φY)−2η(Y)g(φX, φZ) + 2αη(Z)g(SX, Y)−2η(Z)g(S²X, Y).

ReplacingφZ byJ Z−η(Z)N, and similarly forX andY, one can easily calculate that

g(JAY, φZ)−g(JAZ, φY) =η(Y)η(BZ)−η(Z)η(BY), g(JAX, φZ) +g(JAZ, φX) = 2g(BX, Z)−η(X)η(BZ)

−η(Z)η(BX),

g(JAX, φY) +g(JAY, φX) = 2g(BX, Y)−η(X)η(BY)

−η(Y)η(BX).

Inserting this into the previous equation gives

2g((∇XS)Y, φZ) =ρ(X)(g(BY, φZ)−g(BZ, φY))

−ρ(Y)(g(BX, φZ) +g(BZ, φX)) +ρ(Z)(g(BX, φY) +g(BY, φX))

−2η(BY)g(BX, Z) + 2η(BZ)g(BX, Y) + 2η(Z)g(φX, φY)−2η(Y)g(φX, φZ) + 2αη(Z)g(SX, Y)−2η(Z)g(S²X, Y).

Since

g(BY, φZ)−g(BZ, φY) =η(Y)ρ(Z)−η(Z)ρ(Y), g(BX, φY) +g(BY, φX) = 2g(BX, φY) +η(Y)ρ(X)

−η(X)ρ(Y),

g(BX, φZ) +g(BZ, φX) = 2g(BX, φZ) +η(Z)ρ(X)

−η(X)ρ(Z), we get

g((∇XS)Y, φZ) =ρ(X)η(Y)ρ(Z)−ρ(X)ρ(Y)η(Z)

−ρ(Y)g(BX, φZ) +ρ(Z)g(BX, φY)

−η(BY)g(BX, Z) +η(BZ)g(BX, Y) +η(Z)g(φX, φY)−η(Y)g(φX, φZ) +αη(Z)g(SX, Y)−η(Z)g(S²X, Y).

ReplacingZ byφZ and usingφ²Z=−Z+η(Z)ξgives

−g((∇XS)Y, Z) +η(Z)g((∇XS)Y, ξ)

=ρ(X)η(Y)ρ(φZ) +ρ(Y)g(BX, Z)

−ρ(Y)η(Z)g(BX, ξ) +ρ(φZ)g(BX, φY)

−η(BY)g(BX, φZ) +η(BφZ)g(BX, Y) +η(Y)g(φX, Z).

Since

g((∇XS)Y, ξ) =dα(X)η(Y) +αg(SφX, Y)−g(S²φX, Y),

this implies

g((∇_XS)Y, Z) =dα(X)η(Y)η(Z) +η(Z)g((αSφ−S²φ)X, Y)

−ρ(X)η(Y)ρ(φZ)−ρ(Y)g(BX, Z) +ρ(Y)η(Z)g(BX, ξ)−ρ(φZ)g(BX, φY) +η(BY)g(BX, φZ)−η(BφZ)g(BX, Y)

−η(Y)g(φX, Z).

From this, we get an explicit expression for the covariant derivative of the shape operator,

(∇XS)Y ={dα(X)η(Y) +g((αSφ−S²φ)X, Y) +δη(Y)ρ(X) +δg(BX, φY) +η(BX)ρ(Y)}ξ

+{η(Y)ρ(X) +g(BX, φY)}Bξ+g(BX, Y)φBξ

−ρ(Y)BX −η(Y)φX−η(BY)φBX.

PuttingY =ξand X∈ C then leads to

αSφX−S²φX =δρ(X)ξ+ρ(X)Bξ+η(BX)φBξ−φX+δφBX. On the other hand, from Lemma5.1, we get

αSφX−S²φX =δρ(X)ξ+ρ(X)Bξ+η(BX)φBξ−φX.

Comparing the previous two equations leads to δφBX = 0

for allX∈ C. Let us first assume thatδ= 0. Then we haveφBX = 0 for allX∈ C, which impliesBX =η(BX)ξ for allX ∈ C, and therefore

AX =BX+ρ(X)N =η(BX)ξ+ρ(X)N

for all X ∈ C. This implies A(C) ⊂ [N], which gives a contradiction since A is

1

an isomorphism everywhere and the rank of C is equal to 2(m−1) and m ≥ 3.

2

Therefore, we must haveδ= 0, which means thatN isA-isotropic. We thus have

3

proved the following proposition.

4

Proposition 6.1. LetM be a real hypersurface inQ^m, m≥3,with isometric Reeb

5

flow. Then the normal vector fieldN isA-isotropic everywhere.

6

From Proposition6.1and Lemma5.3, we conclude that the principal curvature function αis constant. Since the normal vector field N is A-isotropic, the vector

fieldsN,ξ,Bξ andφBξ are pairwise orthonormal. This implies that C Q= [Bξ].

From Lemma5.2, we know that the tensor field 2SφS−α(φS+Sφ) = 2φ(S²−αS) leavesQandC Qinvariant and

φ(S²−αS) =φonQ and φ(S²−αS) = 0 onC Q. Sinceφis an isomorphism ofQand ofC Qthis implies

S²−αS=I onQ and S²−αS= 0 onC Q.

AsM is a Hopf hypersurface we haveS(C)⊂ C. LetX ∈ Cbe a principal curvature vector of M with corresponding principal curvature λ, that is, SX = λX. We decomposeX intoX =Y +Z withY ∈ Qand Z∈ C Q. Then, we get

(λ²−αλ)Y + (λ²−αλ)Z = (λ²−αλ)X = (S²−αS)X =Y.

If λ²−αλ = 0 we must have Y = 0 and therefore X ∈ C Q. If λ²−αλ = 0

1

we must have Z = 0 and therefore X ∈ Q. Altogether, this implies the following

2

proposition.

3

Proposition 6.2. Let M be a real hypersurface in Q^m, m ≥ 3, with isometric

4

Reeb flow. Then the distributions Q and C Q = [Bξ] are invariant under the

5

shape operatorS ofM.

6

Assume thatSX =λX withX ∈ C. From Lemma5.2, we getλ²−αλ−1 = 0

7

if X ∈ Q and λ²−αλ = 0 if X ∈ C Q. Recall that α is constant. We put

8

α = 2 cot(2r) with 0 < r < ^π₂ and define T_α = Rξ = F. Then the solutions of

9

x²−αx−1 = 0 are λ= cot(r) and µ= −tan(r). We denote by T_λ and T_µ the

10

subbundles of Q consisting of the corresponding principal curvature vectors. The

11

rank ofC Q is equal to 2 andC Qis bothS- andφ-invariant. Therefore, since

12

Sφ = φS, there is exactly one principal curvature κ on C Q which is equal to

13

either κ= 0 orκ=α. We defineT_κ =C Q. Note that, since Sφ=φS, we have

14

J T_ρ=φT_ρ=T_ρforρ∈ {λ, µ, κ}.

15

According to Lemma5.1, we have

(αSφ−S²φ)X =−φX+ρ(X)Bξ+g(X, Bξ)φBξ

=−φX+η(BφX)Bξ+η(BX)φBξ.

Inserting this into the above expression for the covariant derivative ofS and replac- ingρ(X) andρ(Y) byη(BφX) andη(BφY), respectively leads to

(∇XS)Y ={g((αSφ−S²φ)X, Y) +η(BX)η(BφY)}ξ

+{η(Y)η(BφX) +g(BX, φY)}Bξ+g(BX, Y)φBξ

−η(BφY)BX −η(Y)φX−η(BY)φBX

={η(BφX)η(BY)−g(φX, Y)}ξ

+{η(BφX)η(Y)−g(φBX, Y)}Bξ+g(BX, Y)φBξ

−η(BφY)BX −η(Y)φX−η(BY)φBX. This implies

g(BX, Y) =g((∇_XS)Y, φBξ).

LetX andY be sections inT_λ. Then the previous equation implies g(BX, Y) = (λ−κ)g(∇_XY, φBξ) = (κ−λ)g(∇_X(φBξ), Y)

= (κ−λ)g(φ∇_X(Bξ), Y).

SinceBξ=Aξ, the Gauss formula forM inQ^mgives

∇_X(Bξ) =∇_X(Aξ) = ¯∇_X(Aξ),

where ¯∇ is the Riemannian connection of Q^m. Since Ais a parallel subbundle of End(TQ^m), there existsA ∈Asuch that ¯∇XA=A, and we get

∇¯X(Aξ) =A∇¯Xξ+Aξ=A∇Xξ+Aξ=AφSX+Aξ=λAφX+Aξ.

Since A differs from A by a complex scalar we haveAξ ∈[Aξ] = [Bξ] and thus g(φAξ, Y) = 0. Altogether, we therefore get

g(BX, Y) = (κ−λ)λg(φAφX, Y) = (κ−λ)λg(AX, Y)

= (κ−λ)λg(BX, Y).

Assume thatg(BX, Y)= 0. Then we have (κ−λ)λ= 1. Recall thatκ∈ {0, α}. If κ= 0 we get−λ²= 1, and if κ=αwe get−1 = (α−λ)λ= 1, both of which gives a contradiction. Therefore, we must haveg(BX, Y) = 0 for all sectionsX andY in T_λ. The same argument can be repeated forT_µ. SinceQ=T_λ⊕T_µ andB(Q) =Q, we conclude

B(T_λ) =T_µ and B(T_µ) =T_λ.

Since B =A onT_λ and T_µ we can replace B byA here. As both T_λ and T_µ are

1

complex we see that Q and hence Q^m must have even complex dimension. We

2

summarize this in the following proposition.

3

Proposition 6.3. LetM be a real hypersurface inQ^m, m≥3,with isometric Reeb

4

flow. Then mis even, saym= 2k,and the real structureA maps T_λ ontoT_µ, and

5

vice versa.

6