On hypersurfaces in a real space form with the Ricci tensor satisfying certain conditions

ある条件を満たすリッチテンソルをもつ実空間型内の超曲面に ついて

On hypersurfaces in a real space form with the Ricci tensor satisfying certain conditions

数学専攻 押尾 周平

Shuhei Oshio Let ˜ M

(˜ c) be an (n + 1)-dimensional space form of constant curva- ture ˜ c (i.e. complete, connected, simply connected Riemannian manifold of constant curvature, say, ˜ c). For each real number ˜ c, there is (up to isom- etry) exactly one space form in every dimension of sectional curvature ˜ c.

The space forms of sectional curvature ˜ c are denoted by S

(˜ c), R

and H

(˜ c) depending on whether ˜ c is positive, zero or negative, respectively.

S

(˜ c) is a Euclidean sphere of constant curvature ˜ c. R

is a Euclidean space. H

(˜ c) is a hyperbolic space of constant curvature ˜ c.

Let M

be a hypersurface in a space form ˜ M

(˜ c). Let ∇ and S be the covariant differentiation on M

and the Ricci tensor of M

, respectively.

P. J. Ryan classified these hypersurfaces with regard to the parallel Ricci tensor, i.e., ∇ S = 0. He proved that if the Ricci tensor S of M

is parallel and the mean curvature is constant, then M

has the parallel second fun- damental tensor, that is, M

is locally symmetric and M

is the product manifold of two space forms.

For each point x

∈ M

, we choose an unit normal vector field ξ defined in a neighborhood U(x

) of x

. Let ˜ ∇ (resp. ∇ ) be the covariant differen- tiation on ˜ M

(˜ c) (resp. M

). Then for any vector fields X, Y ∈ T

M

, we have

∇ ˜

Y = ∇

Y + g(AX, Y )ξ,

∇ ˜

ξ = − AX,

where g and A are the induced metric on M

and the (1, 1)-type symmetric tensor field called the second f undamental f orm, respectively.

Let R be the curvature tensor of M

. Then, for any vector fields X, Y and Z on U (x

), we have the following:

R(X, Y )Z = ˜ R(X, Y )Z + g(AY, Z )AX − g(AX, Z )AY, (1)

—Gauss equation ( ∇

A)Y = ( ∇

A)X,

—Codazzi equation where ˜ R is the curvature tensor of ˜ M

(˜ c). Since ˜ M

(˜ c) is of constant curvature ˜ c, ˜ R(X, Y )Z can be written as

R(X, Y ˜ )Z = ˜ c { g(Y, Z )X − g(X, Z)Y } . (2)

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In particular, if the second fundamental tensor A satisfies ( ∇

A)Y = 0

on a neighborhood of every point in M

, then we say that parallel second f undamental tensor.

Next, we denote the (0, 2)-type Ricci tensor of M

by S. For any point x of U (x

), S is defined by

S(X, Y ) =

∑n

i=1

g(R(X, e

)e

, Y ),

where { e

, . . . , e

} is an orthonormal basis of the tangent space T

M

. Us- ing the Gauss equation (1) and the equation (2), we obtain

S(X, Y ) = (n − 1)˜ cg(X, Y ) + (traceA)g(AX, Y ) − g(A

X, Y )

for any X tangent to M

on U (x

). Setting S(X, Y ) = g (SX, Y ), we can define the (1, 1)-type Ricci tensor of M

. We also denote the (1, 1)-type Ricci tensor by the same symbol S.

M

has the harmonic curvature if S satisfies ( ∇

S)Y = ( ∇

S)X for any X, Y tangent to M

. M

has the harmonic conformal curvature if S satisfies ( ∇

S)Y − ( ∇

S)X = 1

2(n − 1) { ( ∇

s)Y − ( ∇

s)X } for any X, Y tangent to M

. If S satisfies ( ∇

S)Y = 0 for any X, Y tangent to M

, then the Ricci tensor S is said to be parallel.

Let λ and µ are eigenvalues of A, then these are differentiable and have constant multiplicities. We define two distributions at each x ∈ M as follows:

T

(x) = { X ∈ T

M

| AX = λ(x)X } , T

(x) = { X ∈ T

M

| AX = µ(x)X } .

Lemma 1 [4]. T

and T

are differentiable.

P roof. Let { X

, . . . , X

} be a basis of T

(x

) and { X

. . . , X

} be a basis of T

(x

) for any point x

∈ M

. We extend X

’s to vector fields on M and define vector fields

Y

= (A − µI)X

f or 1 ≤ i ≤ p, Y

= (A − λI)X

f or p + 1 ≤ j ≤ n,

where I denotes the identify transformation. At x

, we have Y

= (λ − µ)X

for 1 ≤ i ≤ p and Y

= (µ − λ)X

for p + 1 ≤ j ≤ n. Thus Y

, . . . , Y

are linearly independent at x

and hence in a neighborhood U of x

. At each point of U , we have

(A − λI )Y

= (A − λI )(A − µI )X

= (A

− (λ + µ)A + λµ)X

= 0.

2

since t

− (λ+µ)+λµ is the minimal polynomial of A. Similarly, (A − µI)Y

= 0. Hence { Y

, . . . , Y

} is a basis of T

and { Y

, . . . , Y

} is a basis of T

. Therefore T

and T

are differentiable.

Lemma 2 [4]. T

and T

are involutive.

P roof. Suppose X, Y ∈ T

. Then we obtain

( ∇

A)Y = ∇

(AY ) − A( ∇

Y ), ( ∇

A)X = ∇

(AX ) − A( ∇

X).

Since the Codazzi equation, we get

A( ∇

Y ) − A( ∇

X) = ∇

(AY ) − ∇

(AX ).

Hence we obtain

A[X, Y ] = A( ∇

Y ) − A( ∇

X)

= ∇

(AY ) − ∇

(AX).

However, X, Y ∈ T

so

A[X, Y ] = (Xλ)Y − (Y λ)X + λ[X, Y ] (A − λI)[X, Y ] = (Xλ)Y − (Y λ)X.

Since (A − λI)[X, Y ] ∈ T

and (Xλ)Y − (Y λ)X ∈ T

, we get (A − λI )[X, Y ] = 0

A[X, Y ] = λ[X, Y ].

Therefore [X, Y ] ∈ T

, thus T

is involutive. Similarly, T

is involutive.

Lemma 3 [4]. If X ∈ T

(x ) and dimT

(x ) ≥ 2 , then X λ = 0 .

P roof. Since dimT

≥ 2, we may choose Y ∈ T

such that X and Y are linearly independent. Extending X and Y to vector fields belonging to T

, we have

(Xλ)Y − (Y λ)X = 0, at x. Thus Xλ = Y λ = 0 at x.

Theorem A [5]. If M

is a hypersurface with parallel Ricci tensor in a space form M ˜

(˜ c) and M

is not of constant curvature c, then ˜ M

is either the product manifold of two space forms or rank A = 2 at all points of M

.

Theorem B [5]. Let M

be a hypersurface with constant mean curvature

3

of the dimension n > 1 in a space form M ˜

(˜ c). If the Ricci tensor of M

is parallel, then M

is locally symmetric and M

is the product manifold of two space forms.

We have recall the definitions of the harmonic curvature and the harmonic conformal curvature, again:

M

has the harmonic curvature if Ricci tensor S satisfies the condition of ( ∇

S)Y = ( ∇

S)X for any X, Y tangent to M

. And M

has the harmonic conf ormal curvature if Ricci tensor S satisfies the condition of ( ∇

S)Y − ( ∇

S)X = 1

2(n − 1) { ( ∇

s)Y − ( ∇

s)X } for any X, Y tangent to M

, where s is the scalar curvature.

The purpose of this paper is to classify hypersurfaces with the harmonic curvature and with the harmonic conformal curvature in a space form. We note that these condition are weaker than ∇ S = 0. We prove the following theorems:

Theorem 1. Let M

be a hypersurface of dimension n ≥ 3 with constant mean curvature and harmonic curvature in a space form M ˜

(˜ c). Then M

is parallel.

Theorem 2. Let M

be a hypersurface of dimension n ≥ 3 with constant mean curvature and harmonic conformal curvature in a space form M ˜

(˜ c).

Then M

is parallel.

References

[1] B. Y. Chen, Geometry of Submanif olds, Dekker, New York, 1973.

[2] Katsumi Nomizu, On hypersurf aces satisf ying a certain condition on the curvature tensor, Tohoku Math. Journ. 20 (1968), 46-59.

[3] Kobayashi and Nomizu, F oundations of Dif f erential Geometry, Vol I, Wiley Interscience, New York, 1963.

[4] Patrick J. Ryan, Homogeneity and some curvature conditions f or hypersurf aces, Tohoku Math. Journ. 21 (1969), 363-388.

[5] Patrick J. Ryan, Hypersurf aces with parallel Ricci tensor, Osaka J.

Math. 8 (1971), 251-259.

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