Geodesics and Circles on Real Hypersurfaces of Type A and B in a Complex Space Form

Hyang Sook Kim, Gil Sang Lee and Yong-Soo Pyo

Abstract

We denote by M_{n}(c) a complex space form with the metric of constant holomorphic sectional curvature 4c and M a real hypersurface in M_{n}(c). We will give characterizations of homogeneous real hypersurfaces of type A and B by observing the shape of geodesics and circles on M as curves in M_{n}(c).

Mathematics Subject Classification: 53C40 Key words: complex spce form, homogeneous real hypersurface of type A and B, shape operator, geodesics, cirle.

1 Introduction

We denote by M_{n}(c) a complete and simply connected complex n-dimensional Kählerian manifold of constant holomorphic sectional curvature 4c, which is called a complex space form. Such an M_{n}(c) is bi-holomorphically isometric to a complex projective space P_{n}C, a complex Euclidean space C^{n} or a complex hyperbolic space H_{n}C, according as c > 0, c = 0 or c < 0.

In this paper, we consider a real hypersurface M in M_{n}(c), c ≠ 0. Typical examples of M in P_{n}C are the six model spaces of type A_{1}, A_{2}, B, C, D and E (cf. Theorem A in §2), and the ones of M in H_{n}C are the four model spaces of type A_{0}, A_{1}, A_{2} and B (cf. Theorem B in §2), which are all given as orbits under certain Lie subgroups of the group consisting of all isometries of P_{n}C or H_{n}C. Denote by (ϕ, ξ, η, g) the almost contact metric structure of M induced from the almost complex structure of M_{n}(c) and A the shape operator of M. Eigenvalues and einvectors of A are called principal curvatures and principal vectors, respectively.

Many differential geometers have studied M from various points of view. For example, Berndt [1] and Takagi [13] investigated the homogeneity of M. Kimura [7] proved that if all principal curvatures are constant and ξ is principal vector, then M in P_{n}C is congruent to one of model spaces. Moreover, it is very interesting to characterize homogeneous real hypersurfaces of M_{n}(c). There are many characterizations of homogeneous ones of type A since these examples have a lot of beautiful geometric properties, where type A means type A_{1} or A_{2} in P_{n}C and type A_{0}, A_{1} or A_{2} in H_{n}C. Okumura [11] and Montiel-Romero [10] proved the fact in P_{n}C and H_{n}C, respectively that M satisfies Aϕ = ϕA if and only if M is locally congruent to type A. Also Maeda [9] gave a characterization of type A in P_{n}C (cf. Theorem E in §2). However, until now there are few results about characterizations of type B. Kim, Pyo and Ki-Nakagawa [5] characterized a real hypersurface of type B in M_{n}(c) (cf. Theorem F in §2).

Balkan Journal of Geometry and Its Applications, Vol.2, No.2, 1997, pp. 79-89 \begin{matrix}c \\ ◯ \end{matrix}Balkan Society of Geometers, Geometry Balkan Press

Recently, Maeda-Ogiue [8] investigated a geodesic hypersphere (i.e. type A_{1} in P_{n}C) by observing the shape of geodesics on M as curves in P_{n}C. Motivated by this result, we are interested in characterizing M of type A in M_{n}(c) by observing geodesics on M, and we will investigate circles on M of type B in M_{n}(c).

The purpose of this paper is to give characterizations of homogeneous real hypersurfaces of type A and B by studing geodesics and circles on M as curves in M_{n}(c).

2 Preliminaries

We begin with recalling the basic properties of a real hypersurface M of a complex space form M_{n}(c). Let N be a unit normal vector field on M. The Riemannian connections ˜∇ in M_{n}(c) and ∇ in M are related by

(2.1)  ˜∇_{X}Y = ∇_{X}Y + g(AX, Y)N

and

(2.2)  ˜∇_{X}N = −AX,

where g denotes the induced Riemannian metric on M. Let J the almost complex structure of M_{n}(c). For a vector field X on M, the images of X and N under the transformation J can be represented as

JX = ϕX + η(X)N , JN = −ξ,

where ϕ defines a skew-symmetric transformation on the tangent bundle T M of M, while η and ξ denote a 1-form and a vector field on M, respectively. Moreover, it is seen that g(ξ, X) = η(X). By the properties of the almost complex structure J, the set (ϕ, ξ, η, g) of tensors satisfies

ϕ^{2} = −I + η ⊗ ξ , ϕξ = 0 , η(ϕX) = 0 , η(ξ) = 1

and

g(ϕX, ϕY) = g(X, Y) − η(X)η(Y),

where I denotes the identity transformation. Accordingly, this set (ϕ, ξ, η, g) defines the almost contact metric structure on M. Furthermore, the covariant derivatives of the structure tensors are given by

(2.3)  (∇_{X}ϕ)Y = η(Y)AX − g(AX, Y)ξ,

(2.4)  ∇_{X}ξ = ϕAX.

Since the ambient space is of constant holomorphic sectional curvature 4c, the equation of Codazzi is given as follows

(2.5)  (∇_{X}A)Y − (∇_{Y} A)X = c{η(X)ϕY − η(Y)ϕX − 2g(ϕX, Y)ξ}.

It is well-known that there exist no totally umbilical real hypersurfaces in M_{n}(c). So, a real hypersurface M of M_{n}(c) is said to be totally η-umbilical if its shape operator A satisfies

AX = aX + bη(X)ξ

for some smooth functions a and b on M.

In the following, we use the same terminology and notations as the above unless otherwise stated. Now we quote the following in order to prove our results. Theorem A ([13]). Let M be a homogeneous real hypersurface of P_{n}C. Then M is a tube of radius r over one of the following Kähler submanifolds:

(A_{1} ) a hyperplane P_{n−1}C, where 0 < r < \frac{π}{2} ,

(A_{2} ) a totally geodesic P_{k}C (1 ≤ k ≤ n − 2), where 0 < r < \frac{π}{2} ,

(B ) a complex quadratic Q_{n−1}, where 0 < r < \frac{π}{4} ,

(C ) P_{1}C × P_{(n−1)/2}C, where 0 < r < \frac{π}{4} and n(≥ 5) is odd,

(D ) a complex Grassmann G_{2,5}C, where 0 < r < \frac{π}{4} and n = 9,

(E ) a Hermitian symmetric space SO(10)/U(5),

where 0 < r < \frac{π}{4} and n = 15 . Theorem B ([1]). Let M be a real hypersurface of H_{n}C. Then M has constant principal curvatures and ξ is principal if and only if M is locally congruent to one of the following:

(A_{0}) a horosphere in H_{n}C,

(A_{1}) a geodesic hypersphere H_{0}C or a tube over a hyperplane H_{n−1}C, (A_{2}) a tube over a totally geodesic H_{k}C (1 ≤ k ≤ n − 2), (B) a tube over a totally real hyperbolic space H_{n}R.

Theorem C ([10], [11]). Let M be a real hypersurface of M_{n}(c). Then M satisfies Aϕ = ϕA if and only if M is locally congruent to one of type A_{1} and A_{2} when c > 0, and of type A_{0}, A_{1} and A_{2} when c < 0. Theorem D ([10], [14]). Let M be a real hypersurface of M_{n}(c). Then M is totally η-umbilical if and only if M is locally congruent to one of type A_{1} when c > 0, and of type A_{0} and A_{1} when c < 0. Theorem E ([3], [9]. Let M be a real hypersurface of M_{n}(c). Then the following are equivalent:

(1) M is locally congruent to one of type A,

(2) (∇_{X}A)Y = −c{g(ϕX, Y)ξ + η(Y)ϕX} for any vector fields X and Y on M.

Theorem F ([5]). Let M be a real hypersurface of M_{n}(c). Then the following are equivalent:

(1) M is locally congruent to type B,

(2) (∇_{X}A)Y = k{2η(X)(Aϕ − ϕA)Y + η(Y)(Aϕ − 3ϕA)X +g((Aϕ − 3ϕA)X, Y)ξ} for any vector fields X and Y on M and k ∈ R. Proposition A ([4], [9]). Let M be a real hypersurface of M_{n}(c), c ≠ 0. If ξ is principal, then the corresponding principal curvature α is locally constant.

Here we consider the case where the structure vector ξ is principal, namely, Aξ = αξ. It follows from (2.5) that

(2.6)  2AϕA = 2cϕ + α(Aϕ + ϕA)

and hence, if AX = λX for any vector field X orthogonal to ξ, then we get

(2λ − α)AϕX = (αλ + 2c)ϕX.

Accordingly, it turns out that in the case where α^{2} + c ≠ 0, ϕX is also principal vector with principal curvature μ = (αλ + 2c)/(2λ − α), that is, we obtain

(2.7)  \begin{matrix}AϕX = μϕX, \\ 2λ − α ≠ 0, μ = (αλ + 2c)/(2λ − α). \end{matrix}

Finally, we recall the definition of helices in Riemannian geometry.

A smooth curve γ = γ(s) in a Riemannian manifold parametrized by its arc length s is called a helix of proper order d if there exists an orthonormal frame {V_{1} = γ̇, …, V_{d}} along γ and positive constants k_{1}, …, k_{d−1} which satisfy

∇γ̇_{i}V_{j}(s) = −k_{j−1}V_{j−1}(s) + k_{j}V_{j+1}(s), j = 1, …, d,

where V_{0} = V_{d+1} = 0. The constants k_{j} (1 ≤ j ≤ d − 1) and the orthonormal frame {V_{1}, …, V_{d}} are called the curvatures and the Frenet frame of γ, respectively. And a smooth curve is called a helix of order d if it is a helix of proper order r (≤ d).

Note that a helix of order 1 is nothing but a geodesic, and a helix of order 2 is called a circle. That is, a smooth curve γ = γ(s) in a Riemannian manifold parametrized by its arc length s is called a circle if there exists a field Y = Y (s) of unit vectors along γ which satisfies ∇γ̇ γ̇ = kY and ∇γ̇Y = −kγ̇ for some positive constant k which is called the curvature of γ. Moreover, for an arbitrary point x, an arbitrary orthonormal pair (u, v) of vectors at x and an arbitrary positive number k, there exists a unique circle γ = γ(s) with γ(0) = x, γ̇(0) = u and Y (0) = v.

3 Real hypersurfaces of type A

We denote by M_{n}(c) a complex space form with the metric of constant holomorphic sectional curvature 4c and M a real hypersurface in M_{n}(c), c ≠ 0. In this section, we are concerned with homogeneous real hypersurfaces of type A. Then, according to Takagi’s classification theorem [13] and Berndt’s one [1], the principal curvatures and their multiplicities of type A in M_{n}(c) are given as follows:

In the case c > 0,

(i) type A_{1} has two distinct constant principal curvatures α = 2 cot 2r with multiplicity 1 and λ = cot r with multiplicity 2n − 2,

(ii) type A_{2} has three distinct constant principal curvatures α = 2 cot 2r with multiplicity 1, λ = − tan r with multiplicity 2k and μ = cot r with multiplicity 2(n − k − 1), where 1 ≤ k ≤ n − 1.

In the case c < 0,

(i) type A_{0} has two distinct constant principal curvatures α = 2 with multiplicity 1 and λ = 1 with multiplicity 2n − 2,

(ii) type A_{1} has two distinct constant principal curvatures α = 2 coth (2r) with multiplicity 1 and λ = tanh(r) if 0 < λ < 1 or λ = coth(r) if λ > 1 with multiplicity 2n − 2,

(iii) type A_{2} has three distinct constant principal curvatures α = 2 coth(2r) with multiplicity 1, λ = tanh(r) with multiplicity 2k and μ = coth(r) with multiplicity 2(n − k − 1), where 1 ≤ k ≤ n − 1.

The following discussion in the case c > 0 is partially indebted to Maeda and Ogiue [8]:

First of all, we prove the following Lemma 3.1 Let M be a real hypersurface of type A in M_{n}(c), c ≠ 0. Take orthonormal vectors (v_{1}, v_{2}, …, v_{2n−2}) orthogonal to ξ at an arbitrary point p of M in such a way that (v_{1}, v_{2}, …, v_{2k}) (resp. (v_{2k+1}, v_{2k+2}, …, v_{2n−2})) are principal vectors with principal curvature λ (resp. μ). Then (v_{1}, v_{2}, …, v_{2n−2}) satisfy the following:

(1) All geodesics γ_{i} on M with γ_{i}(0) = p and γ̇_{i}(0) = v_{i} (1 ≤ i ≤ 2k) are circles of the curvature λ in M_{n}(c).

(2) All geodesics γ_{i} on M with γ_{i}(0) = p and γ̇_{i}(0) = v_{i}

(2k + 1 ≤ i ≤ 2n − 2) are circles of the curvature μ in M_{n}(c).

Proof. Let γ_{i} = γ_{i}(s) (1 ≤ i ≤ 2n − 2) be geodesics on M with γ_{i}(0) = p and γ̇_{i}(0) = v_{i}. Then, taking account of (2.4) and Theorem C, we have

∇γ̇_{i}(g(γ̇_{i}, ξ)) = g(γ̇_{i}, ∇γ̇_{i}ξ) = g(γ̇_{i}, ϕAγ̇_{i}) = g(γ̇_{i}, Aϕγ̇_{i}) = −g(ϕAγ̇_{i}, γ̇_{i}) = 0.

This implies that each γ̇_{i} (1 ≤ i ≤ 2n−2) is perpendicular to ξ since g(γ̇_{i}(0), ξ) = g(v_{i}, ξ) = 0.

Thus, owing to Theorem E, we get

∇γ̇_{i} ∥ Aγ̇_{i} − λγ̇_{i} ∥^{2}= 2g((∇γ̇_{i}A)γ̇_{i}, Aγ̇_{i} − λγ̇_{i}) = 0,

where 1 ≤ i ≤ 2k. Since Aγ̇_{i}(0)−λγ̇_{i}(0) = Av_{i} −λv_{i} = 0, we obtain Aγ̇_{i} −λγ̇_{i} = 0 (1 ≤ i ≤ 2k). Here we note that k = n − 1 in type A_{1} when c > 0, and in type A_{0} and A_{1} when c < 0. Therefore, we see from (2.1) and (2.2) that

˜∇γ̇_{i} γ̇_{i} = g(Aγ̇_{i}, γ̇_{i})N = g(λγ̇_{i}, γ̇_{i})N = λN

and

˜∇γ̇_{i}N = −Aγ̇_{i} = −λγ̇_{i}.

This implies that γ_{i} (1 ≤ i ≤ 2k) are circles of the curvature λ and the Frenet frame {γ̇_{i}, N} in M_{n}(c).

Similarly in the case where M is of type A_{2}, we can show that γ_{i} (2k + 1 ≤ i ≤ 2n−2) are circles of the curvature μ and the Frenet frame {γ̇_{i}, N} in M_{n}(c). Theorem 3.2. Let M be a real hypersurface of M_{n}(c), c ≠ 0. Then M is locally congruent to one of type A_{1} when c > 0, and of type A_{0} and A_{1} when c < 0 if and only if there exist orthonormal vectors (v_{1}, v_{2}, …, v_{2n−2}) orthogonal to ξ at an arbitrary point p of M such that all geodesics γ_{i} = γ_{i}(s) on M through p in the direction v_{i} + v_{j} (1 ≤ i ≤ j ≤ 2n − 2) are circles in M_{n}(c). Proof. Let M be locally congruent to one of type A_{1} when c > 0, and of type A_{0} and A_{1} when c < 0. Then Lemma 3.1 shows that, for an arbitrary unit vector X⊥ξ at p ∈ M, a geodesic γ = γ(s) with γ(0) = p and γ̇(0) = X is a circle in M_{n}(c). Thus there exist orthonormal vectors (v_{1}, v_{2}, …, v_{2n−2}) orthogonal to ξ at an arbitrary point p of M such that all geodesics γ_{i} = γ_{i}(s) on M through p in the direction v_{i} (1 ≤ i ≤ 2n − 2) are circles in M_{n}(c).

Conversely, let γ_{i} = γ_{i}(s) (1 ≤ i ≤ 2n − 2) be geodesics on M with γ_{i}(0) = p and γ̇_{i}(0) = v_{i}. Then by such assumption that all geodesics γ_{i} = γ_{i}(s) on M through p in the direction v_{i} (1 ≤ i ≤ 2n − 2) are circles in M_{n}(c), they satisfy

(3.1)  ˜∇2 ; γ̇_{i} γ̇_{i} = −k_{i}^{2}γ̇_{i}

for some positive constants k_{i}.

On the other hand, from (2.1) and (2.2) it follows that

(3.2)  ˜∇2 ; γ̇_{i} γ̇_{i} = g((∇γ̇_{i}A)γ̇_{i}, γ̇_{i})N − g(Aγ̇_{i}, γ̇_{i})Aγ̇_{i}.

Comparing the tangential components of (3.1) with (3.2), we have

g(Aγ̇_{i}, γ̇_{i})Aγ̇_{i} = k_{i}^{2}γ̇_{i}

so that we get

g(Av_{i}, v_{i})Av_{i} = k_{i}^{2}v_{i},

which implies

(3.3)  Av_{i} = ±k_{i}v_{i} (1 ≤ i ≤ 2n − 2).

Thus we obtain

(3.4)  g(Av_{i}, v_{j}) = 0 (1 ≤ i < j ≤ 2n − 2)

because vectors (v_{1}, v_{2}, …, v_{2n−2}) are orthonormal.

Let γ_{ij} = γ_{ij}(s) (1 ≤ i < j ≤ 2n − 2) be geodesics on M with γ_{ij}(0) = p and γ̇_{ij}(0) = (v_{i} + v_{j})/\sqrt{2}. Then by the same argument as the above we have

g(A(v_{i} + v_{j}), (v_{i} + v_{j}))A(v_{i} + v_{j}) = 2k_{ij}^{2}(v_{i} + v_{j})

for some positive constants k_{ij}. Hence we get

g(A(v_{i} + v_{j}), (v_{i} − v_{j})) = 0 (1 ≤ i < j ≤ 2n − 2).

Therefore, combining this with (3.4) we have

g(Av_{i}, v_{i}) = g(Av_{j}, v_{j}) (1 ≤ i, j ≤ 2n − 2).

This, together with (3.3), implies that AX = kX for all X ortogonal to ξ and for some constant k.

Moreover, ξ is also principal because g(Aξ, X) = g(ξ, AX) = g(ξ, kX) = 0 for all X ⊥ ξ.

Thus we see that M is η-umbilic at p and hence M is totally η-umbilic in M_{n}(c) since p is arbitrary. Therefore, owing to Theorem D, it follows that M is locally congruent to one of type A_{1} when c > 0, and of type A_{0} and A_{1} when c < 0. Theorem 3.3. Let M be a real hypersurface of M_{n}(c), c ≠ 0. Then M is locally congruent to one of type A_{1} and type A_{2} with r = π/4 when c > 0, and of type A_{0} and A_{1} when c < 0 if and only if there exist orthonormal vectors (v_{1}, v_{2}, …, v_{2n−2}) orthogonal to ξ at an arbitrary point p of M such that all geodesics γ_{i} = γ_{i}(s) on M with γ_{i}(0) = p and γ̇_{i}(0) = v_{i} (1 ≤ i ≤ 2n − 2) are circles in M_{n}(c) with the same curvature. Proof Let M be locally congruent to one of type A_{1} and type A_{2} with r = π/4 when c > 0, and of type A_{0} and A_{1} when c < 0. By Lemma 3.1, there exist orthonormal vectors (v_{1}, v_{2}, …, v_{2n−2}) orthogonal to ξ at an arbitrary point p of M such that all geodesics γ_{i} = γ_{i}(s) on M with γ_{i}(0) = p and γ̇_{i}(0) = v_{i} (1 ≤ i ≤ 2n − 2) are circles in M_{n}(c) with the same curvature λ.

Conversely, let γ_{i} = γ_{i}(s) (1 ≤ i ≤ 2n − 2) be geodesics on M with γ_{i}(0) = p and γ̇_{i}(0) = v_{i}. Then by assumption that all geodesics γ_{i} = γ_{i}(s) on M through p in the direction v_{i} (1 ≤ i ≤ 2n−2) are circles in M_{n}(c) with the same curvature k, the same argument as one in the proof of Theorem 3.2 gives

g(Av_{i}, v_{i})Av_{i} = k^{2}v_{i} (1 ≤ i ≤ 2n − 2),

where k is a positive constant. Then we get

(3.5)  Av_{i} = kv_{i} or Av_{i} = −kv_{i} (1 ≤ i ≤ 2n − 2).

Thus we obtain the fact that ξ is principal because g(Aξ, v_{i}) = g(ξ, Av_{i}) = g(ξ, ±kv_{i}) = 0 for 1 ≤ i ≤ 2n − 2. Therefore M is a real hypersurface in M_{n}(c) with at most three distinct constant principal curvatures k, −k and α, where we have used Proposition A. Consequently, M is locally congruent to one of homogeneous real hypersurfaces of type A_{1}, A_{2} and B when c > 0, and of type A_{0}, A_{1}, A_{2} and B when c < 0. But the shape operators of homogeneous real hypersurfaces of type A_{2} of radius r(≠ π/4) and B when c > 0, and of type A_{2} and B when c < 0 do not satisfy (3.5). That is, M is locally congruent to one of type A_{1} and type A_{2} with r = π/4 when c > 0, and of type A_{0} and A_{1} when c < 0.

Replacing geodesics in Lemma 3.1 by circles, we have the following Lemma 3.4. Let M be a real hypersurface of type A in M_{n}(c), c ≠ 0. Take orthonormal vectors (v_{1}, v_{2}, …, v_{2n−2}) orthogonal to ξ at an arbitrary point p of M in such a way that (v_{1}, v_{2}, …, v_{2k}) (resp. (v_{2k+1}, v_{2k+2}, …, v_{2n−2})) are principal vectors with principal curvature λ (resp. μ). Then (v_{1}, v_{2}, …, v_{2n−2}) satisfy the following:

(1) All circles γ_{i} of an arbitrary curvature in M with γ_{i}(0) = p, γ̇_{i}(0) = v_{i} and the Frenet frame {γ̇_{i}, ξ} (1 ≤ i ≤ 2k) are circles of the curvature λ and the Frenet frame {γ̇_{i}, N} in M_{n}(c).

(2) All circles γ_{i} of an arbitrary curvature in M with γ_{i}(0) = p, γ̇_{i}(0) = v_{i} and the Frenet frame {γ̇_{i}, ξ} (2k + 1 ≤ i ≤ 2n − 2) are circles of the curvature μ and the Frenet frame {γ̇_{i}, N} in M_{n}(c).

Proof. Let γ_{i} = γ_{i}(s) (1 ≤ i ≤ 2n − 2) be circles on M with γ_{i}(0) = p and γ̇_{i}(0) = v_{i}. Then, from the assumption that all γ_{i} have the Frenet frame {γ̇_{i}, ξ}, it follows that each γ̇_{i} (1 ≤ i ≤ 2n − 2) is perpendicular to ξ.

Thus, owing to Theorem E, we get

∇γ̇_{i} ∥ Aγ̇_{i} − λγ̇_{i} ∥^{2} = 2g((∇γ̇_{i}A)γ̇_{i} + A(∇γ̇_{i} γ̇_{i}) − λ∇γ̇_{i} γ̇_{i}, Aγ̇_{i} − λγ̇_{i})
= 2g((∇γ̇_{i}A)γ̇_{i} + A(m_{i}ξ) − λm_{i}ξ, Aγ̇_{i} − λγ̇_{i})
= 2g((∇γ̇_{i}A)γ̇_{i} + (α − λ)m_{i}ξ, Aγ̇_{i} − λγ̇_{i}) = 0,

where each m_{i} is the curvature of γ̇_{i} (1 ≤ i ≤ 2k). Since Aγ̇_{i}(0) − λγ̇_{i}(0) = Av_{i} − λv_{i} = 0, we obtain Aγ̇_{i} − λγ̇_{i} = 0 (1 ≤ i ≤ 2k). Therefore, we see from (2.1) and (2.2) that

˜∇γ̇_{i} γ̇_{i} = g(Aγ̇_{i}, γ̇_{i})N = g(λγ̇_{i}, γ̇_{i})N = λN

and

˜∇γ̇_{i}N = −Aγ̇_{i} = −λγ̇_{i}.

This implies that γ_{i} (1 ≤ i ≤ 2k) are circles of the curvature λ and the Frenet frame {γ̇_{i}, N} in M_{n}(c). Here we note that k = n − 1 in type A_{1} when c > 0, and in type A_{0} and A_{1} when c < 0.

Similarly in the case where M is of type A_{2}, we can show that γ_{i} (2k + 1 ≤ i ≤ 2n−2) are circles of the curvature μ and the Frenet frame {γ̇_{i}, N} in M_{n}(c).

4 Real hypersurfaces of type B

We denote by M_{n}(c) a complex space form with the metric of constant holomorphic sectional curvature 4c and M a real hypersurface in M_{n}(c), c ≠ 0. In this section, we are concerned with homogeneous real hypersurfaces of type B. Then, according to Takagi’s classification theorem [13] and Berndt’s one [1], the principal curvatures and their multiplicities of type B in M_{n}(c) are given as follows:

(i) In the case c > 0, type B has three distinct constant principal curvatures α = 2 cot 2r with multiplicity 1, λ = − tan(r − π/4) with multiplicity n − 1 and μ = cot(r − π/4) with multiplicity n − 1.

(ii) In the case c < 0, type B has three distinct constant principal curvatures α = 2 tanh(2r) with multiplicity 1, λ = tanh(r) with multiplicity n − 1 and μ = coth(r) with multiplicity n − 1.

Then we first have the following Lemma 4.1. Let M be a real hypersurface of type B in M_{n}(c), c ≠ 0. Take orthonormal vectors (v_{1}, v_{2}, …, v_{2n−2}) orthogonal to ξ at an arbitrary point p of M in such a way that (v_{1}, v_{2}, …, v_{n−1}) (resp. (v_{n}, v_{n+1}, …, v_{2n−2})) are principal vectors with principal curvature λ (resp. μ). Then (v_{1}, v_{2}, …, v_{2n−2}) satisfy the following:

(1) All circles γ_{i} of an arbitrary curvature in M with γ_{i}(0) = p, γ̇_{i}(0) = v_{i} and the Frenet frame {γ̇_{i}, ξ} (1 ≤ i ≤ n − 1) are circles of the curvature λ and the Frenet frame {γ̇_{i}, N} in M_{n}(c).

(2) All circles γ_{i} of an arbitrary curvature in M with γ_{i}(0) = p, γ̇_{i}(0) = v_{i} and the Frenet frame {γ̇_{i}, ξ} (n ≤ i ≤ 2n − 2) are circles of the curvature μ and the Frenet frame {γ̇_{i}, N} in M_{n}(c).

Proof. Let γ_{i} = γ_{i}(s) (1 ≤ i ≤ 2n − 2) be circles on M with γ_{i}(0) = p and γ̇_{i}(0) = v_{i}. Then, from the assumption that all γ_{i} have the Frenet frame {γ̇_{i}, ξ}, it follows that each γ̇_{i} (1 ≤ i ≤ 2n − 2) is perpendicular to ξ.

Thus, owing to Theorem F, we get

∇γ̇_{i} ∥ Aγ̇_{i} − λγ̇_{i} ∥^{2} = 2g((∇γ̇_{i}A)γ̇_{i} + A(∇γ̇_{i} γ̇_{i}) − λ∇γ̇_{i} γ̇_{i}, Aγ̇_{i} − λγ̇_{i}) =
= 2g((∇γ̇_{i}A)γ̇_{i} + A(k_{i}ξ) − λk_{i}ξ, Aγ̇_{i} − λγ̇_{i}) =
= 2g((∇γ̇_{i}A)γ̇_{i} + (α − λ)k_{i}ξ, Aγ̇_{i} − λγ̇_{i}) = 0,

where each k_{i} is the curvature of γ_{i} (1 ≤ i ≤ n − 1). Since Aγ̇_{i}(0) − λγ̇_{i}(0) = Av_{i} − λv_{i} = 0, we obtain Aγ̇_{i} − λγ̇_{i} = 0 (1 ≤ i ≤ n − 1). Therefore, we see from (2.1) and (2.2) that

˜∇γ̇_{i} γ̇_{i} = g(Aγ̇_{i}, γ̇_{i})N = g(λγ̇_{i}, γ̇_{i})N = λN

and

˜∇γ̇_{i}N = −Aγ̇_{i} = −λγ̇_{i}.

This implies that γ_{i} (1 ≤ i ≤ n − 1) are circles of the curvature λ and the Frenet frame {γ̇_{i}, N} in M_{n}(c).

Similarly we can show that γ_{i} (n ≤ i ≤ 2n − 2) are circles of the curvature μ and the Frenet frame {γ̇_{i}, N} in M_{n}(c). Theorem 4.2. Let M be a real hypersurface of M_{n}(c), c ≠ 0. Then M is locally congruent to one of type A and B if and only if there exist orthonormal vectors (v_{1}, v_{2}, …, v_{2n−2}) orthogonal to ξ at an arbitrary point p of M such that all circles γ_{i} = γ_{i}(s) in M with γ_{i}(0) = p, γ̇_{i}(0) = v_{i} and the Frenet frame {γ̇_{i}, ξ} (1 ≤ i ≤ 2n − 2) are circles in M_{n}(c) of the same curvature c_{i} (1 ≤ i ≤ 2k) or the same one c_{j} (2k + 1 ≤ j ≤ 2n − 2). Proof. Let M be locally congruent to one of type A and B in M_{n}(c).

First of all, let M be of type A in M_{n}(c) and let γ_{i} = γ_{i}(s) (1 ≤ i ≤ 2n−2) be circles in M with γ_{i}(0) = p, γ̇_{i}(0) = v_{i} and the Frenet frame {γ̇_{i}, ξ}. Then, owing to Lemma 3.4, there exist orthonormal vectors (v_{1}, v_{2}, …, v_{2n−2}) orthogonal to ξ at an arbitrary point p of M such that these circles γ_{i} = γ_{i}(s) (1 ≤ i ≤ 2n − 2) are circles in M_{n}(c) of the same curvature c_{i} = λ (1 ≤ i ≤ 2k) or the same one c_{j} = μ (2k + 1 ≤ j ≤ 2n − 2).

Next, let M be of type B in M_{n}(c). Then by Lemma 4.1, there exist orthonormal vectors (v_{1}, v_{2}, …, v_{2n−2}) orthogonal to ξ at an arbitrary point p of M such that all circles γ_{i} = γ_{i}(s) in M with γ_{i}(0) = p, γ̇_{i}(0) = v_{i} and the Frenet frame {γ̇_{i}, ξ} (1 ≤ i ≤ 2n − 2) are circles in M_{n}(c) of the same curvature c_{i} = λ (1 ≤ i ≤ n − 1) or the same one c_{j} = μ (n ≤ j ≤ 2n − 2).

Conversely, assume that there exist orthonormal vectors (v_{1}, v_{2}, …, v_{2n−2}) orthogonal to ξ at an arbitrary point p of M such that all circles γ_{i} = γ_{i}(s) in M with γ_{i}(0) = p, γ̇_{i}(0) = v_{i} and the Frenet frame {γ̇_{i}, ξ} (1 ≤ i ≤ 2n − 2) are circles in M_{n}(c) of the same curvature c_{i} (1 ≤ i ≤ 2k) or the same one c_{j} (2k + 1 ≤ j ≤ 2n − 2). Then the same argument as one in the proof of Theorem 3.2 gives

g(Av_{i}, v_{i})Av_{i} = c_{i}^{2}v_{i} (1 ≤ i ≤ 2k)

and

g(Av_{j}, v_{j})Av_{j} = c_{j}^{2}v_{j} (2k + 1 ≤ j ≤ 2n − 2),

where c_{i} and c_{j} are positive constants. Then we get

(4.1)  Av_{i} = ±c_{i}v_{i} and Av_{j} = ±c_{j}v_{j} (1 ≤ i ≤ 2k, 2k + 1 ≤ j ≤ 2n − 2).

Thus, by means of (4.1), we obtain the fact that ξ is principal because g(Aξ, v_{i}) = g(ξ, Av_{i}) = 0 for (1 ≤ i ≤ 2n − 2). Therefore M is a real hypersurface in M_{n}(c) with at most five distinct constant principal curvatures c_{i}, −c_{i}, c_{j}, −c_{j} and α, where we have used Proposition A. Consequently, M is locally congruent to one of homogeneous real hypersurfaces of type A_{1}, A_{2}, B, C, D and E when c > 0, and of type A_{0}, A_{1}, A_{2} and B when c < 0. But the shape operators of homogeneous real hypersurfaces of type C, D and E when c > 0 do not satisfy (4.1). That is, M is locally congruent to one of type A and B in M_{n}(c).

Acknowledgements. This paper was supported by YAKSOO research Foundation and TGRC-KOSEF.

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Department of Mathematics Inje University Kimhae 621-749, Korea E-mail:mathkim@ijnc.inje.ac.kr

Department of Applied Mathematics Pukyong National University Pusan 608-737, Korea