A characterization of Kaehler submanifolds in a complex space form

複素空間型内のケーラー部分多様体の特徴づけ

A characterization of Kaehler submanifolds in a complex space form

数学専攻 長山 靖

Yasushi NAGAYAMA Let ˜ M (c) be a complex (n +p)-dimensional complex space form of constant holomorphic sectional curvature c (i.e. complete, simply connected Kaehler manifold with constant holomorphic sectional curvature, say, c). For each real number c, there is (up to holomorphic isometry) exactly one complex space form in every dimension with holomorphic sectional curvature c. The complex space forms of holomorphic sectional curvature c are denoted by P ^n+p (c), C ^n+p and D ^n+p depending on whether c is positive, zero or negative, respectively. P ^n+p (c) is the complex projective space with Fubini-Study metric of constant holomorphic sectional curvature c. C ^n+p is the complex Euclidean space. D ^n+p is the open unit ball in C ^n+p endowed with Bergman metric of constant holomorphic sectional curvature c.

Let M be a connected manifold of complex dimension n(≥ 2) isometrically and holomorphically immersed in a complex space form ˜ M (c) of comlpex dimension n + p. Then we call M a Kaehler hypersurface of ˜ M (c). The complex structure ˜ J and the Kaehler metric ˜ g of ˜ M (c) induce a complex structure J and Kaehler metric g on M , respectively. Let ∇(resp. ˜ ∇) denotes the covariant differentiation in M (resp. ˜ M (c)). Extend ξ to a normal vector field defined in a neighborhood U of x ∈ M and define −A ξ X to be the tangential component of ˜ ∇ X ξ for X ∈ T x M . A ξ X depends only on ξ at x and X , and we call A ξ the second fundamental form. Let R be the curvature tensor of M and X, Y and Z be the tangent vectors on M . Then we have the following relationships:

∇ ˜ X Y = ∇ X Y + X

α

g(A α X, Y )ξ α + X

α

g(JA α X, Y ) ˜ Jξ α , (1)

g(A α X, Y ) = g(X, A α Y ), (2)

∇ ˜ X ξ α = −A α X + X

β

s αβ (X ) ˜ Jξ β , (3)

s αβ + s βα = 0, (4)

A α J = −JA α , (5)

R(X, Y )Z = c

4 {g(Y, Z )X − g(X, Z)Y + g(JY, Z)JX (6)

− g(JX, Z)JY + 2g(X, JY )JZ}

+ X

α

g(A α Y, Z)A α X − X

α

g(A α X, Z)A α Y

+ X

α

g(JA α Y, Z )JA α X − X

α

g(JA α X, Z)JA α Y,

1

—— Gauss equation

(∇ X A α )Y − X

β

s αβ (X )JA β Y = (∇ Y A α )X − X

β

s αβ (Y )JA β X, (7)

—— Codazzi equation ,where we write A α = A ξ

.

Theorem.([1]) Let M ⁿ be an n-dimentional manifold which is minimally immersed in a real space form M ˜ ^n+p (c) with constant curvature c and R the curvature tensor of M ⁿ . We assume that

(R(X, Y )A ξ )Z = 0 for all vectors X,Y and Z tangent to M ⁿ and any normal vector ξ.

(1) The case of c > 0: If M ⁿ is compact (or the scalar curvature of M ⁿ is constant) and S ≥ pnc, then M ⁿ is parallel and S=pnc.

(2) The case of c ≤ 0: If the scalar curvature of M ⁿ is constant, then M ⁿ is totally geodesic.

In this paper, we consider a characterization of Kaehler submanifolds in a complex space form which satisfy (R(X, Y )A ξ )Z = 0 for all vectors X , Y and Z tangent to M and A ξ in the direction of any normal ξ. Then we obtain the following:

Theorem. Let M ⁿ be an n-dimensional manifold which is isometrically and holomorphically immersed in a complex space form M ˜ ^n+p (c) with constant holomorphic sectional curvature c and R the curvature tensor of M ⁿ . Let x be any point of M ⁿ . Then there exists a neighborhood U of x which local field ξ of normal vectors and the second fundamental form A ξ in the direction of ξ are defined on U . We assume that

(R(X, Y )A ξ )Z = 0 (8)

for all vectors X,Y and Z tangent to M ⁿ and any normal vector ξ. Then we obtain the following:

(1) The case of c 6= 0 : M is Einstein.

(2) The case of c = 0 : A α = 0 for ^∀ α.

(3) The case of c < 0 : M is totally geodesic in a hyperbolic space.

(4) The case of c > 0 : p = 1 and M is P ⁿ (c), Q ⁿ (c) in P ⁿ⁺¹ .

The autor would like to express his sincere gratitude to Professor Y.Matsuyama for his valuable and proper suggestions during the preparation of this paper.

Now, we prepare the following results without proof.

Theorem A. (See[5],[6]).

If M ⁿ is a Kaehler hypersurface in a complex space form M ˜ ⁿ⁺¹ (c), then the following conditions are equivalent

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on M ⁿ :

(1) M ⁿ is locally symmetric.

(2) M ⁿ is Einstein.

Theorem B. (See[7]).

Let M ⁿ be a Kaehler hypersurface of complex dimention n ≥ 1 in a complex space form M ˜ ⁿ⁺¹ (c) of constant holomorphic sectional curvature c. M ⁿ is Einstein, then M ⁿ is locally symmetric and either M ⁿ is of constant holomorphic sectional curvature ˜ c and totally geodesic in M ˜ ⁿ⁺¹ (c) or M ⁿ is locally holomorphically isometric to the complex quadric Q ⁿ in the complex projective space P ⁿ⁺¹ (c), the latter case arising only when c > 0.

We next must introduce the concept of the first normal space of M ⁿ at x ∈ M ⁿ .

Definition.

For x ∈ M ⁿ , the first normal space, N 1 (x), is the orthogonal complement in T _x ^⊥ (M ⁿ ) of the set

N 0 (x) = {ξ ∈ T _x ^⊥ (M ⁿ )|A ξ = 0}.

We define a new inner product, <, >, on N 1 (x) by

< ξ, η >= traceA ξ A η for ξ, η ∈ N 1 (x).

One easily checks that <, > is a positive definite inner product on N 1 (x), and that for ξ, η ∈ N 1 (x),

< Jξ, Jη >=< ξ, η > and < ξ, Jξ >= 0

For ξ ∈ N 1 (x), we assume that A ² _ξ = λ ² I for λ > 0. Then it is easy to see that T x (M ) can be decomposed as

T x (M ⁿ ) = T _ξ ⁺ ⊕ T _ξ ⁻ where

T _ξ ⁺ = {X ∈ T x (M ⁿ )|A ξ X = λX} and T _ξ ⁻ = {X ∈ T x (M ⁿ )|A ξ X = −λX}

It is a simple matter to show that if X ∈ T _ξ ⁺ , then JX ∈ T _ξ ⁻ ; and if X ∈ T _ξ ⁻ , then JX ∈ T _ξ ⁺ . We employ the inner product <, > in the following theorem to prove that N 1 (x) has complex dimension no longer than 1 for all x ∈ M ⁿ .

Theorem C.(See[4])

Let x ∈ M ⁿ and let k be the complex dimension of N 1 (x). Then k ≤ 1.

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References

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[6] P.J.Ryan, A class of complex hypersurfaces, Colloquium Mathematicum 26 (1972), 177-182 [7] B.Smyth, Differential geometry of complex hypersurfaces, Ann. of Math., 85 (1967), 246-266

[8] Y.Matsuyama, A characterization of minimal Lagrangian submanifolds in a complex projective space, Int.

J. Pure Appl. Math. 44 (2008), no.3, 363-372.

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

[10] B.Smyth, A class of complex hypersurfaces, Ann. Of Math., 85 (1967), 246-266.

[11] K.Ogiue, Differential geometry of Kaehler submanifolds, Advances in Math. 13 (1974), 73–114.

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