DIFFERENTIAL GEOMETRY OF REAL HYPERSURFACES INHERMITIAN SYMMETRIC SPACES WITH RANK 2
JÜRGEN BERNDT AND YOUNG JIN SUH

Abstract. In this talk, ﬁrst we introduce the classiﬁcation of homogeneous
hypersurfaces in some Hermitian symmetric spaces of rank 1 or rank 2. In particular, we give a full expression of the geometric structures for hypersurfaces
in complex two-plane Grassmannians G2 (Cm+2 ) or in complex hyperbolic twoplane Grassmannians G∗2 (Cm+2 ).
Next by using the isometric Reeb ﬂow we give a complete classiﬁcation
for hypersurfaces M in complex two-plane Grassmannians G2 (Cm+2 ), complex hyperbolic two-plane Grassmannians G∗2 (Cm+2 ), complex quadric Qn
and its noncompact dual Qn∗ . Moreover, we give a classiﬁcation of contact
hypersurfaces with constant mean curvature in the complex quadric Qn =o /SO SO for n ≥ 3
SOn+2 /SOn SO2 and its noncompact dual Qn∗ = SOn,2n2

1. Compact Hermitian Symmetric Space with rank 2
The study of real hypersurfaces in non-ﬂat complex space forms or quaternionicspace forms which belong to HSSP with rank 1 of compact type in section 1 isa classical topic in diﬀerential geometry. For instance, there have been many investigations for homogeneous hypersurfaces of type A1 , A2 , B, C, D and E incomplex projective space CP m . They are completely classiﬁed by Cecil and Ryan[9], Kimura [13] and Takagi [30]. Here, explicitly, we mention that A1 : Geodesic
hyperspheres, A2 : a tube around a totally geodesic complex projective spaces CP k ,B : a tube around a complex quadric Qm−1 and can be viewed as a tube around areal projective space RP m , C : a tube around the Segre embedding of CP 1 × CP k
into CP 2k+1 for some k≥2, D : a tube around the Plücker embedding into CP 9 of
the complex Grassmannian manifold G2 (C5 ) of complex 2-planes in C5 and E : atube around the half spin embedding into CP 15 of the Hermitian symmetric spaceSO(10)/U (5).
Now let us study hypersurfaces in complex two-plane Grassmanians G2 (Cm+2 )which is a kind of HSSP with rank two of compact type. The ambient space
G2 (Cm+2 ) is known to be the unique compact irreducible Riemannian symmetricspace equipped with both a Kähler structure J and a quaternionic Kähler structure
J not containing J.
12000 Mathematics Subject Classiﬁcation. Primary 53C40; Secondary 53C15.
Key words and phrases : Hermitian symmetric spaces, Real hypersurfaces, Noncompact complex two-plane Grassmannians, Isometric Reeb ﬂow, Totally geodesic, Horosphere at inﬁnity,
Contact hypersurfaces.
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This work was supported by grant Proj. No. NRF-2012-R1A2A2A-01043023 from National
Research Foundation of Korea.
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JÜRGEN BERNDT AND YOUNG JIN SUH

On the other hand, Cecil and Ryan [9] proved that any tube M around a complex
submanifold in complex projective space CP m are characterized by the invariancyof Aξ = αξ, where the Reeb vector ξ is deﬁned by ξ = −JN for a Kähler structure
J and a unit normal N to hypersurfaces M in Pm (C). Moreover, the corresponding
geometrical feature for hypersurfaces in QP m is the invariance of the distributionD⊥ = Span {ξ1 , ξ2 , ξ3 } by the shape operator, where ξi = −Ji N , Ji ∈J. In fact every
tube around a quaternionic submanifold QP m satisﬁes such kind of geometricalfeature (See [14], [16], [17]).
From such a view point, we considered two natural geometric conditions for real
hypersurfaces in G2 (Cm+2 ) that means that the maximal complex subbundle C anda maximal quaternionic subbundle Q of T M are both invariant under the shapeoperator of M , where the maximal complex subbundle C of the tangent bundle
T M of M is deﬁned by C = {X∈T M |JX∈T M }, and the maximal quaternionicsubbundle Q of T M is deﬁned by Q = {X∈T M |JX∈T M } respectively. By using
such conditions and the result in Alekseevskii [1], Berndt and Suh [3] proved thefollowing
Let M be a connected real hypersurface in G2 (Cm+2 ), m ≥ 3.
Theorem 1.1.
Then the maximal complex subbundle C and a maximal quaternionic subbundle Q
of T M are both invariant under the shape operator of M if and only if(A) M is an open part of a tube around a totally geodesic G2 (Cm+1 ) in G2 (Cm+2 ),or(B) m is even, say m = 2n, and M is an open part of a tube around a totally
geodesic HP n in G2 (Cm+2 ).
When the Reeb ﬂow on M in G2 (Cm+2 ) is isometric, we say that the Reeb
vector ﬁeld ξ on M is Killing. Moreover, the Reeb vector ﬁeld ξ is said to be Hopf
if it is invariant by the shape operator A. The 1-dimensional foliation of M by the
integral manifolds of the Reeb vector ﬁeld ξ is said to be a Hopf foliation of M . We
say that M is a Hopf hypersurface in G2 (Cm+2 ) if and only if the Hopf foliation of
M is totally geodesic.
By using Theorem 1, in a paper due to Berndt and Suh [4] we have given acomplete classiﬁcation of real hypersurfaces in G2 (Cm+2 ) with isometric Reeb ﬂow
as follows:
Theorem 1.2.
Let M be a connected real hypersurface in G2 (Cm+2 ), m ≥ 3.
Then the Reeb flow on M is isometric if and only if M is an open part of a tube
around a totally geodesic G2 (Cm+1 ) in G2 (Cm+2 ).
2. Complex hyperbolic two-plane Grassmannian SU2,m /S(U2 ·Um )Now let us consider for the case that the Riemannian manifold M̄ becomes a
Riemannian symmetric spaces of non compact type with rank 1 or rank 2. As someexamples of non compact type with rank 1 we say a real hyperbolic space RH n =SO0 (1, n)/SO(n), a complex hyperbolic space CH n = SU (1, n)/S(U (1)×U (n)), a
quaternionic hyperbolic space HH n = Sp(1, n)/Sp(1)×Sp(n), and a Caley projective plane OP 2 = F4 /Spin(9). The study of homogeneous hypersurfaces in such aHERMITIAN SYMMETRIC SPACES WITH RANK 2

3symmetric spaces of noncompact type with rank 1 was investigated in Berndt [5],Berndt and Tamaru [8].
In this section we consider a hypersurface in HSSP of noncompact type withrank 2. Among some examples of noncompact type with rank 2 given in section
1 we focus on a dual complex two-plane Grassmannian SU (2, m)/S(U (2)×U (m)).
The Riemannian symmetric space SU (2, m)/S(U (2)×U (m)) is a connected, simply
connected, irreducible Riemannian symmetric space of noncompact type with rank
2.
Let G = SU2,m and K = S(U2 ·Um ), and denote by g and k the corresponding
Lie algebra of the Lie group G and K respectively. Let B be the Killing form ofg and denote by p the orthogonal complement of k in g with respect to B. The
resulting decomposition g = k ⊕ p is a Cartan decomposition of g. The Cartan
involution θ ∈ Aut(g) on su2,m is given by θ(A) = I2,m AI2,m , where()−I2 02,m
I2,m =
0m,2 ImI2 and Im denote the identity (2 × 2)-matrix and (m × m)-matrix respectively.
Then < X, Y >= −B(X, θY ) becomes a positive deﬁnite Ad(K)-invariant innerproduct on g. Its restriction to p induces a metric g on SU2,m /S(U2 ·Um ), which isalso known as the Killing metric on SU2,m /S(U2 ·Um ). Throughout this paper weconsider SU2,m /S(U2 ·Um ) together with this particular Riemannian metric g.
The Lie algebra k decomposes orthogonally into k = su2 ⊕ sum ⊕ u1 , where u1
is the one-dimensional center of k. The adjoint action of su2 on p induces thequaternionic Kähler structure J on SU2,m /S(U2 ·Um ), and the adjoint action of
( mi
)
I2
02,m
Z = m+2
∈ u1
−2i
0m,2
m+2 Iminduces the Kähler structure J on SU2,m /S(U2 ·Um ). By construction, J commuteswith each almost Hermitian structure Jν in J for ν = 1, 2, 3. Recall that a canonicallocal basis J1 , J2 , J3 of a quaternionic Kähler structure J consists of three almost
Hermitian structures J1 , J2 , J3 in J such that Jν Jν+1 = Jν+2 = −Jν+1 Jν , where
the index ν is to be taken modulo 3. The tensor ﬁeld JJν , which is locally deﬁned
on SU2,m /S(U2 ·Um ), is selfadjoint and satisﬁes (JJν )2 = I and tr(JJν ) = 0,
where I is the identity transformation. For a nonzero tangent vector X we deﬁne
RX = {λX|λ ∈ R}, CX = RX ⊕ RJX, and HX = RX ⊕ JX.
Then by the argument asserted in section 1, we note that any homogeneous
hypersurfaces in G∗2 (Cm+2 ) becomes a tube around one singular orbit. By virtueof this fact and using geometric tools given in Helgason [11], [12], Eberlein [10], Berndt and Suh [5] proved a characterization of homogeneous hypersurfaces inG∗2 (Cm+2 ) as follows:
Theorem 2.1. Let M be a connected real hypersurface in the complex hyperbolic
two-plane Grassmanniaan SU (2, m)/S(U (2)×U (m)), m ≥ 2, with constant principal curvatures. Then the maximal complex subbundle C and a maximal quaternionicsubbundle Q of T M are both invariant under the shape operator of M if and only
if M is congruent to an open part of one of the following hypersurfaces:

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JÜRGEN BERNDT AND YOUNG JIN SUH(A) a tube around a totally geodesic SU (2, m − 1)/S(U (2)×U (m − 1)) in
SU (2, m)/S(U (2)×U (m)),(B) a tube around a totally geodesic quaternionic hyperbolic space HH n in
SU (2, 2)/S(U (2)×U (m)), m = 2n,(C) a horosphere in SU (2, m)/S(U (2)×U (m)) whose center at infinity is singular.
In this section we give a classiﬁcation of all real hypersurfaces with isometric
Reeb flow in complex hyperbolic two-plane Grassmann manifold SU2,m /S(U2 ·Um )as follows (see Suh [26]):
Theorem 2.2. Let M be a connected orientable real hypersurface in the complex
hyperbolic two-plane Grassmannian SU2,m /S(U2 ·Um ), m ≥ 3. Then the Reeb flowon M is isometric if and only if M is an open part of a tube around some totally
geodesic SU2,m−1 /S(U2 ·Um−1 ) in SU2,m /S(U2 ·Um ) or a horosphere whose centerat infinity is singular.
A tube around SU2,m−1 /S(U2 ·Um−1 ) in SU2,m /S(U2 ·Um ) is a principal orbitof the isometric action of the maximal compact subgroup SU1,m+1 of SUm+2 ,and the orbits of the Reeb flow corresponding to the orbits of the action of U1 .
The action of SU1,m+1 has two kinds of singular orbits. One is a totally geodesic
SU2,m−1 /S(U2 ·Um−1 ) in SU2,m /S(U2 ·Um ) and the other is a totally geodesic CH m
in SU2,m /S(U2 ·Um ).
A remarkable consequence of our Main Theorem is that a connected complete real
hypersurface in SU2,m /S(U2 ·Um ), m ≥ 3 with isometric Reeb flow is homogeneous.
This was also true in complex two-plane Grassmannians G2 (Cm+2 ), which could
be identiﬁed with symmetric space of compact type SUm+2 /S(U2 ·Um ), as follows
from the classiﬁcation. It would be interesting to understand the actual reason forit (See [3], [4], [17], and [22]).
3. Isometric Reeb Flow in Complex Quadric Qm
2= 0 on Cm+2 deﬁnes
The homogeneous quadratic equation z12 + . . . + zm+2ma complex hypersurface Q in the (m + 1)-dimensional complex projective space
CP m+1 = SUm+2 /S(Um+1 U1 ). The hypersurface Qm is known as the m-dimensional
complex quadric. The complex structure J on CP m+1 naturally induces a complex structure on Qm which we will denote by J as well. We equip Qm with the
Riemannian metric g which is induced from the Fubini Study metric on CP m+1
with constant holomorphic sectional curvature 4. The 1-dimensional quadric Q1 isisometric to the round 2-sphere S 2 . For m ≥ 2 the triple (Qm , J, g) is a Hermitiansymmetric space of rank two and its maximal sectional curvature is equal to 4. The
2-dimensional quadric Q2 is isometric to the Riemannian product S 2 × S 2 . We willassume m ≥ 3 for the main part of this paper.
For a nonzero vector z ∈ Cm+1 we denote by [z] the complex span of z, that is,[z] = {λz | λ ∈ C}. Note that by deﬁnition [z] is a point in CP m+1 . As usual,for each [z] ∈ CP m+1 we identify T[z] CP m+1 with the orthogonal complement
Cm+2 ⊖ [z] of [z] in Cm+2 . For [z] ∈ Qm the tangent space T[z] Qm can then be
identiﬁed canonically with the orthogonal complement Cm+2 ⊖ ([z] ⊕ [z̄]) of [z] ⊕ [z̄]HERMITIAN SYMMETRIC SPACES WITH RANK 2

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in Cm+2 . Note that z̄ ∈ ν[z] Qm is a unit normal vector of Qm in CP m+1 at the
point [z].
We denote by Az̄ the shape operator of Qm in CP m+1 with respect to z̄. Then we
have Az̄ w = w for all w ∈ T[z] Qm , that is, Az̄ is just complex conjugation restricted
to T[z] Qm . The shape operator Az̄ is an antilinear involution on the complex vector
space T[z] Qm andT[z] Qm = V (Az̄ ) ⊕ JV (Az̄ ),where V (Az̄ ) = Rm+2 ∩ T[z] Qm is the (+1)-eigenspace and JV (Az̄ ) = iRm+2 ∩T[z] Qm is the (−1)-eigenspace of Az̄ . Geometrically this means that the shapeoperator Az̄ deﬁnes a real structure on the complex vector space T[z] Qm . Recall
that a real structure on a complex vector space V is by deﬁnition an antilinear
involution A : V → V . Since the normal space ν[z] Qm of Qm in CP m+1 at [z] isa complex subspace of T[z] CP m+1 of complex dimension one, every normal vectorin ν[z] Qm can be written as λz̄ with some λ ∈ C. The shape operators Aλz̄ of
Qm deﬁne a rank two vector subbundle A of the endomorphism bundle End(T Qm ).
Since the second fundamental form of the embedding Qm ⊂ CP m+1 is parallel (seee.g. [29]), A is a parallel subbundle of End(T Qm ). For λ ∈ S 1 ⊂ C we again geta real structure Aλz̄ on T[z] Qm and we have V (Aλz̄ ) = λV (Az̄ ). We thus have an
S 1 -subbundle of A consisting of real structures on the tangent spaces of Qm .
The Gauss equation for the complex hypersurface Qm ⊂ CP m+1 implies that the
Riemannian curvature tensor R of Qm can be expressed in terms of the Riemannian
metric g, the complex structure J and a generic real structure A in 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 structure J anti-commutes with each endomorphism A ∈ A,
that is, AJ = −JA.
A nonzero tangent vector W ∈ T[z] Qm is called singular if it is tangent to morethan one maximal ﬂat in Qm . There are two types of singular tangent vectors forthe complex quadric Qm :1. If there exists a real structure A ∈ A[z] such that W ∈ V (A), then W issingular. Such a singular tangent vector is called A-principal.
2. If there exist a real structure A ∈ A[z] and orthonormal vectors X, Y ∈√V (A) such that W/||W || = (X + JY )/ 2, then W is singular. Such a
singular tangent vector is called A-isotropic.
Basic complex linear algebra shows that for every unit tangent vector W ∈ T[z] Qm
there exist a real structure A ∈ A[z] and orthonormal vectors X, Y ∈ V (A) suchthat
W = cos(t)X + sin(t)JY
for some t ∈ [0, π/4]. The singular tangent vectors correspond to the values t = 0
and t = π/4.
Let M be a real hypersurface in a Kähler manifold M̄ . The complex structure
J on M̄ induces locally an almost contact metric structure (ϕ, ξ, η, g) on M . In
the context of contact geometry, the unit vector ﬁeld ξ is often referred to as theReeb vector ﬁeld on M and its ﬂow is known as the Reeb ﬂow. The Reeb ﬂow has

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JÜRGEN BERNDT AND YOUNG JIN SUH

been of signiﬁcant interest in recent years, for example in relation to the Weinstein
Conjecture. We are interested in the Reeb ﬂow in the context of Riemannian
geometry, namely in the classiﬁcation of real hypersurfaces with isometric Reeb
ﬂow in homogeneous Kähler manifolds.
For the complex projective space CP m a full classiﬁcation was obtained by Okumura in [19]. He proved that the Reeb ﬂow on a real hypersurface in CP m =SUm+1 /S(Um U1 ) is isometric if and only if M is an open part of a tube around a
totally geodesic CP k ⊂ CP m for some k ∈ {0, . . . , m − 1}. For the complex 2-plane
Grassmannian G2 (Cm+2 ) = SUm+2 /S(Um U2 ) the classiﬁcation was obtained by
the author in [26]. We have proved that the Reeb ﬂow on a real hypersurface in
G2 (Cm+2 ) is isometric if and only if M is an open part of a tube around a totally
geodesic G2 (Cm+1 ) ⊂ G2 (Cm+2 ). Finally, related to the isometric Reeb ﬂow, wegive a mention for our recent work due to Berndt and Suh [6]. In this lecture wewant to investigate this problem for the complex quadric Qm = SOm+2 /SOm SO2 .
In view of the previous two results a natural expectation is that the classiﬁcation
involves at least the totally geodesic Qm−1 ⊂ Qm . Surprisingly, this is not the case.
Our main result states:
Theorem 3.1. Let M be a real hypersurface of the complex quadric Qm , m ≥ 3.
The Reeb flow on M is isometric if and only if m is even, say m = 2k, and M isan open part of a tube around a totally geodesic CP k ⊂ Q2k .
Every tube around a totally geodesic CP k ⊂ Q2k is a homogeneous hypersurface.
In fact, the closed subgroup Uk+1 of SO2k+2 acts on Q2k with cohomogeneity one.
The two singular orbits are totally geodesic CP k ⊂ Q2k and the principal orbits
are the tubes around any of these two singular orbits. So as a corollary we get:Corollary 3.1. Let M be a connected complete real hypersurface in the complex
quadric Q2k , k ≥ 2. If the Reeb flow on M is isometric, then M is a homogeneous
hypersurface of Q2k .
It is remarkable that in this situation the existence of a particular one-parametergroup of isometries implies transitivity of the isometry group. As another interesting consequence we get:Corollary 3.2. There are no real hypersurfaces with isometric Reeb flow in the
odd-dimensional complex quadric Q2k+1 , k ≥ 1.
To our knowledge the odd-dimensional complex quadrics are the ﬁrst examples
of homogeneous Kähler manifolds which do not admit a real hypersurface withisometric Reeb ﬂow. Recently, as a further characterization of the tube around a
totally geodesic CP k we want to remark the following two theorems in [27] and [28]respectively.
When the shape operator S of M in Qm is parallel, that is, ∇ξ S = 0 along thedirection the structure vector ﬁeld ξ, we say that the shape operator is Reeb parallel.
Moreover, we say that the Reeb principal curvature is constant if the function α
deﬁned by α = g(Sξ, ξ) is constant. Motivated by theses results, we have given
a complete classiﬁcation of real hypersurfaces in complex quadric Qm with Reeb
parallel shape operator in [27] as follows:
Let M be a real hypersurface in complex quadric Qm , m≥3 with
Theorem 3.2.
Reeb parallel shape operator and non-vanishing constant Reeb curvature. ThenHERMITIAN SYMMETRIC SPACES WITH RANK 2

7m = 2k, and M is locally congruent to a tube over a totally geodesic complex
projective space CP k in Q2k .
The shape operator S is said to be Reeb invariant if the shape operator is Lieparallel along the Reeb direction, that is, Lξ S = 0. From such a view point, wewant to give another characterization of the tube around a totally geodesic CP k in[28] as follows:
Theorem 3.3.
Let M be a real hypersurface in complex quadric Qm , m≥3 withReeb parallel shape operator and non-vanishing constant Reeb curvature. Thenm = 2k, and M is locally congruent to a tube over a totally geodesic complex
projective space CP k in Q2k .

4. Contact hypersurfaces in Complex Quadric Qn
This section is a recent work due to Berndt and Suh [7]. A contact manifold isa smooth (2n − 1)-dimensional manifold M together with a one-form η satisfyingη ∧ (dη)n−1 ̸= 0, n ≥ 2. The one-form η on a contact manifold is called a contact
form. The kernel of η deﬁnes the so-called contact distribution C in the tangent
bundle T M of M . Note that if η is a contact form on a smooth manifold M , thenρη is also a contact form on M for each smooth function ρ on M which is nonzero
everywhere. The origin of contact geometry can be traced back to Hamiltonianmechanics and geometric optics. The standard example of a contact manifold is R3
together with the contact form η = dz − y dx.
Another standard example is a round sphere in an even-dimensional Euclideannspace. Consider the sphere S 2n−1 (r) with radius·⟩∑nr ∈ R+ in C and denote by ⟨·,nthe inner product on C given by ⟨z, w⟩ = Re ν=1 zν w̄ν . By deﬁning ξz = − 1r iz
for z ∈ S 2n−1 (r) we obtain a unit tangent vector ﬁeld ξ on S 2n−1 (r). We denoteby η the dual one-form given by η(X) = ⟨X, ξ⟩ and by ω the Kähler form on Cn
given by ω(X, Y ) = ⟨iX, Y ⟩. A straightforward calculation shows that dη(X, Y ) =− 2r ω(X, Y ). Since the Kähler form ω has rank 2(n − 1) on the kernel of η it follows
that η∧(dη)n−1 ̸= 0. Thus S 2n−1 (r) is a contact manifold with contact form η. This
argument for the sphere motivates a natural generalization to Kähler manifolds.
Let (M̄ , J, g) be a Kähler manifold of complex dimension n and let M be aconnected oriented real hypersurface of M̄ . The Kähler structure on M̄ induces
an almost contact metric structure (ϕ, ξ, η, g) on M . The Riemannian metric on
M is the one induced from the Riemannian metric on M̄ , both denoted by g. The
orientation on M determines a unit normal vector ﬁeld N of M . The so-called Reeb
vector ﬁeld ξ on M is deﬁned by ξ = −JN and η is the dual one form on M , that is,
η(X) = g(X, ξ). The tensor ﬁeld ϕ on M is deﬁned by ϕX = JX − g(JX, N )N =JX − η(X)N , so that ϕX is just the tangential component of JX. The tensor ﬁeld
ϕ determines the fundamental 2-form ω on M by ω(X, Y ) = g(ϕX, Y ). M is said
to be a contact hypersurface if there exists an everywhere nonzero smooth function
ρ on M such that dη = 2ρω. It is clear that if dη = 2ρω holds then η ∧ (dη)n−1 ̸= 0,
that is, every contact hypersurface in a Kähler manifold is a contact manifold.
Contact hypersurfaces in complex space forms of complex dimension n ≥ 3 havebeen investigated and classiﬁed by Okumura [20] (for the complex Euclidean space

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JÜRGEN BERNDT AND YOUNG JIN SUH

Cn and the complex projective space CP n ) and Vernon [21] (for the complex hyperbolic space CH n ). In this paper we carry out a systematic study of contact hypersurfaces in Kähler manifolds. We will then apply our results to the complex quadrico
Qn = SOn+2 /SOn SO2 and its noncompact dual space Qn∗ = SOn,2/SOn SO2 toprove the following two classiﬁcations:
Theorem 4.1.
Let M be a connected orientable real hypersurface with constantomean curvature in the complex quadric Qn = SOn+2/SOn SO2 and n ≥ 3. Then
M is a contact hypersurface if and only if M is congruent to an open part of theπtube of radius 0 < r < 2√around a real form S n of Qn .
2
Theorem 4.2.
Let M be a connected orientable real hypersurface with constantomean curvature in the noncompact dual Qn∗ = SOn,2/SOn SO2 of the complex
quadric and n ≥ 3. Then M is a contact hypersurface if and only if M is congruent
to an open part of one of the following contact hypersurfaces in Qn∗ :(i) the tube of radius r ∈ R+ around the totally geodesic Q(n−1)∗ in Qn∗ ;(ii) a horosphere in Qn∗ whose center at infinity is determined by an A-principal
geodesic in Qn∗ ;(iii) the tube of radius r ∈ R+ around a real form RH n in Qn∗ .
The symbol A refers to a circle bundle of real structures on Qn∗ and the notionof A-principal will be explained later. Every contact hypersurface in a complex
space form has constant mean curvature. Our results on contact hypersurfaces inKähler manifolds suggest that it is natural to impose the condition of constantmean curvature in the more setting.
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Jürgen Berndt
King’s College London,
Department of Mathematics,
London WC2R 2LS, United Kingdom
E-mail address: jurgen.berndt@kcl.ac.uk

Young Jin Suh
Kyungpook National University,
Department of Mathematics,
Taegu 702-701, Korea
E-mail address: yjsuh@knu.ac.kr