On hypersurfaces of the semi-symmetric classes in a real space form

実空間型内の準対称類超曲面について

On hypersurfaces of the semi-symmetric classes in a real space form

数学専攻 王 慧峰

Huifeng Wang

Let ˜ M

(c) be an (n+1)-dimensional real space form with constant curvature c (i.e. complete, simply connected Riemannian manifold with constant sectional curvature, say c). For each real number c and each integer n > 1 there is (up to isometry) exactly one n-dimensional real space form with constant curvature c.

The real space forms are:

(1) If c = 0, then ˜ M

(c) is an Euclidean space E

. (2) If c < 0, then ˜ M

(c) is a real hyperbolic space H

(c).

(3) If c > 0, then ˜ M

(c) is an Euclidean sphere S

(c) .

Let f : M

−→ M ˜

(c) be an isometric immersion of an n-dimensional Riemannian manifold M

in ˜ M

(c). For simplicity, we say that M

is a hypersurface immersed in ˜ M

(c), and for all local formulas and computations, we may consider f as an imbedding and thus identify x ∈ M with f (x) ∈ M ˜

(c). The tangent space T

(M ) is identified with a subspace of the tangent space T

( ˜ M ), and the normal space T

is the subspace of T

( ˜ M ) consisting of all X ∈ T

( ˜ M ) which are orthogonal to T

(M ) with respect to the Riemannian metric g.

For an arbitrary point x

∈ M

, we may choose a field ξ of unit normal vectors defined in a neighborhood U of x

. The second fundamental form h and the corresponding symmetric opreator A are defined and related to covariant differentiations ˜ ∇ and ∇ in ˜ M

(c) and M

, respectively, by the following formulas:

∇ ˜

Y = ∇

Y + h(X, Y ), (1)

∇ ˜

ξ = −AX, (2)

where X and Y are vector fields tangent to M

. The Gauss equation is:

R(X, Y ) = c(X ∧ Y ) + AX ∧ AY, X, Y ∈ T

(M ), (3)

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where X ∧ Y denotes the skew-symmetric endomorphism of T

(M ). And The Codazzi equation is expressed by

(∇

A)(Y ) = (∇

A)(X ). (4)

M

is called a locally symmetric space if the curvature tensor R of M

satisfies

∇R = 0.

For example, a piece of the product of two spaces with constant curvature is a locally symmetric space. Then it is naturally considered the question whether M

is ∇R = 0 under the weaker condition of

R(X, Y ) · R = 0

for all tangent vectors X and Y than ∇R = 0. With regard to this, Nomizu[2]

proved the following: Let M be an n-dimensional, connected, complete Rie- mannian manifold which is isometrically immersed in a Euclidean space R

so that the type number k is greater than 2 at least at one point. If M satisfies the condition of R(X, Y ) · R = 0, then it is of the form M = S

× R

, where S

is a hypersphere in a Euclidean subspace R

of R

and R

is a Eu- clidean subspace orthogonal to R

.

When c 6= 0 and M

is a hypersurface in ˜ M

(c), n > 2, with R(X, Y )·R = 0, either rankA = n for any x ∈ M or rankA ≤ 1 for any x ∈ M . Assume that the hypothesis as above and in addition that at each point exactly two principal curvatures are distinct and have multiplicities > 1. Then M is locally isometric to a product of two spaces with constant curvature.

Moreover, we shall assume that M is oriented (so that a unit normal vector field ξ is defined on the whole M ) and that the type number k(x) is greater than 3 everywhere on M . It is known that the function k(x) is locally constant and hence is a constant, say k, since M is connected. We may also speak of the differentiable function λ(x) which assings to each x ∈ M the non-zero eigenvalue of A at x.

Thus, at each x ∈ M , λ(x) is the non-zero eigenvalue of A with multiplicity k and 0 is the eigenvalue with multiplicity n − k. We define two distributions on M as follows:

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T

(x) = {X ∈ T

(M ); AX = 0}, T

(x) = {X ∈ T

(M ); AX = λ(x)X }.

We have T

(M ) = T

(x) + T

(x) (direct sum). For any Z ∈ T

(M ), Z

and Z

will denote the components of Z in T

(x) and T

(x), respectively. We prepare the following Lemmas without proof.

Lemma 1. T

and T

are differentiable.

Lemma 2. T

and T

are involutive.

Lemma 3. If X belongs to T

(x), then Xλ = 0.

Lemma 4. If X ∈ T

, Y ∈ T

, then A(∇

Y ) = −(Y λ)X . Lemma 5. (1)If Y ∈ T

, then ∇

(T

) ⊂ T

.

(2)If Y ∈ T

, then ∇

(T

) ⊂ T

.

(3)If Y ∈ T

, X ∈ T

and [X, Y ] = 0, then ∇

Y ∈ T

.

Lemma 6. If Y λ = 0 for every Y ∈ T

, then X ∈ T

implies ∇

(T

) ⊂ T

and ∇

(T

) ⊂ T

.

Lemma 7. Let Y and Z be vector fields belonging to T

such that ∇

Z =

∇

Y = 0. If there is a non-vanishing vector field X belonging to T

such that [X, Y ] = [X, Z] = 0, then (Y Z )( 1

λ ) = 0.

If the curvature tensor R of the riemannian manifold M satisfies

R(X, Y ) · R = a(X ∧ Y ) · R

for any tangent vector X and Y, then M is called semi-symmetric classes where a is constant and X ∧ Y is defined by (X ∧ Y )Z = g(Y, Z)X − g(X, Z)Y .

The purpose of this paper is to classify hypersurfaces of semi-symmetric classes in a real space form. We notice that this condition is weaker than R(X, Y )R = 0.

We prove the following theorem.

Theorem. Let M

be a complete hypersurface in M ˜

(c). We assume that a is a non-zero constant. If M

is not of constant curvature c and if R(X, Y )R = a(X ∧ Y )R on M

, then M

is totally umbilical and a space form.

The author would like to express his sincere gratitude to Professor Y.

Matsuyama for his valuable suggestions during the preparation of this paper.

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References

[1] B. Y. Chen, Geometry of Submanifold, New York, Dekker(1973)

[2] Katsumi. Nomizu, On hypersurfaces satisfying a certain condition on the curvature tensor, (Received August 22, 1967)

[3] S. Kobayashi and K. Nomizu, Foundations of differential geometry, Vol. II, Interscience Tracts No. 15, John Wiley and Sons, New York, (1963).

[4] K. Nomizu and B. Smyth, A formula of simons’ type and hypersurfaces with constant mean curvature, J. Differential Geometry, 3(1969)367-377 [5] Patrick J. Ryan, Homogeneity and some curvature conditions for hypersur-

faces, Tohoku Math. Journ, 21(1969), 363-388

[6] Patrick J. Ryan, Hypersurfaces with parallel ricci tensor, Osaka J. Math.

8(1971), 251-259

[7] YOSHIO MATSUYAMA, Complete Hypersurfaces With RS = 0 In E

, Proceedings of The American Mathematical Society, Volume 88, Number 1,May 1983

[8] Norifumi Yokoyama and Yoshio Matsuyama, On a Kaehler Hypersurface with the Cyclic Ricci Semi-symmetric Tensor, Acta Mathematica Sinica, English Series, Oct.,2009, Vol.25, No.10, pp.1591-1594

Department of Mathematics Chuo University 1-13-27 Kasuga Bunkyo-ku Tokyo 112-8551 Japan

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