Mem. Fac. Sci. Eng. Shimane Univ.
Series B: Mathematical Science
38 (2005), pp. 1–5

REAL HYPERSURFACES OF COMPLEX SPACE FORMS WITHSYMMETRIC RICCI ∗-TENSOR
TATSUYOSHI HAMADA AND JUN-ICHI INOGUCHI
Communicated by Sadahiro Maeda(Received: September 7, 2004)

Abstract. Real hypersurfaces M ’s in non-flat complex space forms such thatthe symmetric part of the Ricci ∗-tensor of M is a constant multiple of the metricare classified.

1. Introduction
This note is a continuation of our previous paper [1].
Let (M, φ, ξ, η, g) be an almost contact metric manifold with Ricci tensor S. TheRicci ∗-tensor S ∗ is defined by1
S ∗ (X, Y ) = trace(Z 7→ R(X, φY )φZ), X, Y ∈ T M.
2An almost contact metric manifold is said to be ∗-Einstein if S ∗ is a constantmultiple of the metric g on the holomorphic distribution T ◦ M .
It should be remarked that Ricci ∗-tensor is not symmetric, in general. Thus thecondition “∗-Einstein” automatically requires a symmetric property of the Ricci∗-tensor.
On real hypersurfaces in almost Hermitian manifolds, almost contact structures
are naturally induced from the almost Hermitian structure of the ambient space.
In our previous paper [1], the first named author investigated real hypersurfaces in
non-flat complex space forms in terms of Ricci ∗-tensor. In particular, he classified ∗-Einstein real hypersurfaces in non-flat complex space forms whose structure
vector fields are principal.
The purpose of present note is to generalize the classification result of [1]. We
shall weaken the assumption “∗-Einstein” to “the symmetric part of S ∗ is a constantmultiple of g on T ◦ M ”. More precisely, we shall prove the following two results.
Theorem 1.1. Let M be a connected real hypersurface of Pn (C) of constant holomorphic sectional curvature 4c > 0. Assume that the symmetric part SymS ∗ ofKey words and phrases. real hypersurface, complex space form, Ricci ∗-tensor, ∗-Einstein.
12TATSUYOSHI HAMADA AND JUN-ICHI INOGUCHI

Ricci ∗-tensor of M is a constant multiple of the induced metric over the holomorphic distribution and the structure vector field ξ is a principal curvature vector.
Then M is an open subset of one of the following:√(i) a geodesic hypersphere of radius r (0 < r < π/(2 c)),(ii) a tube
√ over a totally geodesic complex projective space Pk (C) of radiusπ/(4 c), where 0 < k < n − 1,√(iii) a tube over a complex quadric Qn−1 of radius r (0 < r < π/(4 c)).
Theorem 1.2. Let M be a connected real hypersurface of Hn (C) of constant holomorphic sectional curvature 4c < 0. Assume that the symmetric part SymS ∗ ofRicci ∗-tensor of M is a constant multiple of the induced metric over the holomorphic distribution and the structure vector field ξ is a principal curvature vector.
Then M is an open subset of one of the following:(i) a geodesic hypersphere of radius r (0 < r < ∞),(ii) a tube over a totally geodesic complex hyperbolic hyperplane of radius r(0 < r < ∞),(iii) a tube over a totally real hyperbolic space H n (R) of radius r (0 < r < ∞),(iv) a horosphere.
2. Preliminaries
A complex n-dimensional Kähler manifold of constant holomorphic sectional curfn (4c). A complete
vature 4c is called a complex space form, which is denoted by M
and simply connected complex space form is a complex projective space Pn (C), acomplex Euclidean space Cn or a complex hyperbolic space Hn (C), according asc > 0, c = 0 or c < 0. Let M be a real hypersurface of a non-flat complex space
fn (4c).
form Mfn (4c). Then the Riemannain
Take a local unit normal vector filed N of M in M
˜ of Mfn (4c) and ∇ of M are related by the following Gauss formulaconnections ∇and Weingarten formula:e X Y = ∇X Y + g(AX, Y )N, X, Y ∈ X(M ),∇e X N = −AX, X ∈ X(M ).
∇Here g is the Riemannian metric of M induced by the Kähler metric G of thefn (4c). The (1, 1)-tensor field A is called the shape operator of M
ambient space M
derived from N .
An eigenvector X of the shape operator A is called a principal curvature vector.
The corresponding eigenvalue λ of A is called a principal curvature. As is wellknown, the Kähler structure (J, G) of the ambient space induces an almost contact
metric structure (φ, ξ, η, g) on M . In fact, the structure vector field ξ and its dual
1-form η are defined by
η(X) = g(ξ, X) = G(JX, N ), X ∈ T M.
The (1, 1)-tensor field φ is defined byg(φX, Y ) = G(JX, Y ), X, Y ∈ T M.

REAL HYPERSURFACES OF COMPLEX SPACE FORMS

3

One can easily check that this structure (φ, ξ, η, g) is an almost contact structure
on M , that is, it satisfies
φ2 X = −X + η(X)ξ,(1)η(ξ) = 1,φξ = 0.

It follows that∇X ξ = φAX.
e and R be the Riemannian curvature tensors of M
fn (4c) and M , respectively.
Let Re of M
fn (4c), we have the following¿From the expression of the curvature tensor R
equations of Gauss and Codazzi:
R(X, Y )Z= c(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,(∇X A)Y − (∇Y A)X = c(η(X)φY − η(Y )φX − 2g(φX, Y )ξ).
To close this section, we recall the following two fundamental results (See e.g., [2]).
Lemma 2.1. If ξ is a principal curvature vector, then the corresponding principal
curvature α is locally constant.
Lemma 2.2. Assume that ξ is a principal curvature vector and the corresponding
principal curvature is α. If AX = λX for X ⊥ ξ, then we have (2λ − α)AφX =(αλ + 2c)φX.
We refer to the reader [2] about general theory of differential geometry of real
hypersurfaces in complex space forms.
3. ∗-Einstein real hypersurfaces
Let us denote by S ∗ the Ricci ∗-tensor of a real hypersurface M which is definedby1
S ∗ (X, Y ) = trace(Z 7→ R(X, φY )φZ).
2
Then the Gauss equation implies that(2)S ∗ (X, Y ) = 2cn(g(X, Y ) − η(X)η(Y )) − g(φAφAX, Y ),for all X, Y ∈ T M .
The Ricci ∗-operator Q∗ is the linear endomorphism field associated to S ∗ ;S ∗ (X, Y ) = g(Q∗ X, Y ), X, Y ∈ T M.
The trace ρ∗ of Q∗ is called the ∗-scalar curvature of M .
Let T ◦ M be a distribution defined by a subspace
Tx◦ M = {X ∈ Tx M : X ⊥ ξx }in the tangent space Tx M . The formulas (1) imply that the distribution T ◦ M isinvariant under φ. The distribution T ◦ M is called the holomorphic distribution of

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TATSUYOSHI HAMADA AND JUN-ICHI INOGUCHI

M . If the Ricci ∗-tensor is a constant multiple of the Riemannian metric for theholomorphic distribution, i.e.
ρ∗S ∗ (X, Y ) =g(X, Y )2(n − 1)for X, Y ∈ T ◦ M on M , then M is said to be a ∗-Einstein real hypersurface.
The first author proved the following results in [1].
Proposition 3.1. Let M be a connected ∗-Einstein real hypersurface of Pn (C) ofconstant holomorphic sectional curvature 4c > 0, whose structure vector field ξ isa principal curvature vector. Then M is an open subset of one of the following:√(i) a geodesic hypersphere of radius r (0 < r < π/(2 c)),(ii) a tube
√ over a totally geodesic complex projective space Pk (C) of radiusπ/(4 c), where 0 < k < n − 1,√(iii) a tube over a complex quadric Qn−1 of radius r (0 < r < π/(4 c)).
Proposition 3.2. Let M be a connected ∗-Einstein real hypersurface of Hn (C) ofconstant holomorphic sectional curvature 4c < 0, whose structure vector field ξ isa principal curvature vector. Then M is an open subset of one of the following:(i) a geodesic hypersphere of radius r (0 < r < ∞),(ii) a tube over a totally geodesic complex hyperbolic hyperplane of radius r(0 < r < ∞),(iii) a tube over a totally real hyperbolic space H n (R) of radius r (0 < r < ∞),(iv) a horosphere.
Now we take the symmetric part SymS ∗ and the alternate part AltS ∗ of Ricci∗-tensor S ∗ of M ;1
SymS ∗ (X, Y ) = (S ∗ (X, Y ) + S ∗ (Y, X)),
2
1AltS ∗ (X, Y ) = (S ∗ (X, Y ) − S ∗ (Y, X)),2
for any X, Y ∈ T M .
Using (2), we see that
(3)

(4)

SymS ∗ (X, Y ) = 2cn(g(X, Y ) − η(X)η(Y ))1− g((φAφA + AφAφ)X, Y ),21AltS ∗ (X, Y ) = g((AφAφ − φAφA)X, Y ).
2
4. Proof of main theorems

To prove our theorems, we need the following lemma.
fn (4c).
Lemma 4.1. Let M be a real hypersurface of a non-flat complex space form M
If ξ is a principal curvature vector, then the Ricci ∗-tensor of M is symmetric, i.e.
AltS ∗ = 0.

REAL HYPERSURFACES OF COMPLEX SPACE FORMS

5Proof. Let X be a unit principal curvature vector orthogonal to ξ with principalcurvature λ. From Lemma 2.2, the tangent vector φX is also a principal curvaturevector. By calculating (4), we get AltS ∗ (X, Y ) = 0, for any X ∈ T ◦ M and Y ∈TM.
On the other hand, by the assumption, we have φAφAξ = 0 and (1) shows
AφAφξ = 0. Thus, we get AltS ∗ (ξ, Y ) = 0 for any Y ∈ T M .
¤fn (4c) with c 6= 0 whose
Proof of theorems. Now let M be a real hypersurface in M∗◦SymS is a constant multiple of g over T M . Assume that the structure vector
field ξ is principal. Then Lemma 4.1 implies that S ∗ (X, Y ) = SymS ∗ (X, Y ) for
X, Y ∈ T M . Hence M is ∗-Einstein. This fact, together with Propositions 3.1 and3.2, yields the required results.
¤References
1. T. Hamada, Real hypersurfaces of complex space forms in terms of Ricci ∗-tensor, Tokyo J.
Math. 25 (2002), no. 2, 473–483.
2. R. Niebergall and P. J. Ryan, Real hypersurfaces in complex space forms, Tight and taut submanifolds, (T. E. Cecil and S. S. Chern, S.S. eds.), Math. Sci. Res. Inst. Publ., 32 Cambridge
Univ. Press, Cambridge, 1997, pp. 233–305.
Department of Applied Mathematics, Faculty of Science, Fukuoka University,Fukuoka, 814-0180, Japan
E-mail address: hamada@sm.fukuoka-u.ac.jp
Department of Mathematics Education, Faculty of Education, Ustunomiya University, Utsunomiya, 321-8505, Japan
E-mail address: inoguchi@cc.utsunomiya-u.ac.jp