On hypersurfaces in a real space form

On hypersurfaces in a real space form

Kentaro Noda

Department of Mathematics Chuo University

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Let ˜ M

(c) be an (n + 1)-dimensional real space form with constant cur- vature 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 a 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 a Euclidean sphere S

(c) .

Let M

be an n-dimensional Riemannian manifold isometrically immersed in M ˜

(c). Then we call a hypersurface such a M

. Let ∇ and R be the covari- ant differentiation in M

and the curvature tensor of M

, respectively.

In a recent work J. Simons has established a fomula for the Laplacian of the second fundamental form of a submanifold in a Riemannian manifold and has obtained an important application in the case of a minimal hypersurface in the sphere, for which the formula takes a rather simple form. The application is made by means of the Laplacian of the function f on the hypersurface, which is defined to be thhe square of the length of the second fundamental form.

In the paper, by a more direct route than Simons’ we first obtain the same type of formula in the case of a hypersurface M

immersed with constant mean curvature in a space ˜ M

(c) of constant sectional curvature, and then derive a new formula for the function f which involves the sectional curvature of M

. Moreover, we call that a Riemannian manifold is said to be Eienstein if the Ricci tensor is a constant multiple of the metric tensor, that is S = ρg.

We want to study Einstein hypersurface M

and a hypersurface M

with constant curvature. We prepoare the following theorem:

Theorem A. (Nomizu and Smith [4])

Let M

be a complete Riemannian manifold of dimension n with non-negative

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sectional curvature, and ϕ : M

→ R

an isometric immersion with constant mean curvature into a Euclidian space R

. If f = trace A

is constant on M

, then ϕ(M )is of the form S

× R

, 0 ≤ p ≤ n, where R

is an (n − p)- dimensional subspace of R

, and S

is a sphere in the Euclidian subspace perpendicular to R

. Except for the case p = 1, ϕ is an imbedding.

Theorem B. (Nomizu and Smith [4])

Let M

be an n-dimensional complete Riemannian manifold with non- negative sectional curvature, and ϕ : M

→ S

an isometric immersion with constant mean curvature. If f = trace A

is constant on M

, then either

(1) ϕ(M ) : a great or small sphere in S

, and ϕ is an imbedding or

(2) ϕ(M ) : a product sphere S

(r) × S

(s) , and ϕ is an imbedding for p ̸ = 1, n − 1.

Theorem C. (O’Neill and Stiel [5])

Let M

(c) be a hypersurface in M ˜

(c) with constant curvature. If c > 0 and M

is complete, then M

(c) is totally geodesic.

Theorem D. (Ryan [6])

Let M

, n > 2, be an Eienstein hypersurface in M ˜

(c). If ρ > (n − 1)c, then M

is umbilical with λ

= ρ

n − 1 − c and thus M

is of constant curvature ρ

n − 1 . If ρ = (n − 1)c, then t(x) ≤ 1 for all x and M

is constant curvature c. If ρ < (n − 1)c, then c > 0, ρ = (n − 2)c and M

is locally isometric to M

( n − 2

p − 1 c) × M

( n − 2

n − p − 1 c) where 1 < p < n − 1.

Theorem E. (Ryan [6])

Let M

be an Einstein hypersurface in S

(c), i.e., S = ρI. If ρ = (n − 1)c and M

is complete, then M

is totally geodesic.

The purpose of the paper is to prove the following theorems.

Theorem 1.

Let M

be an n-dimensional complete Riemannian manifold with non- negative sectional curvature, and ϕ : M

→ H

(c) an isometric immersion.

If trace A = constant and f = trace A

= constant on M

, then ∇ A = 0, (λ

− λ

)

(c + λ

λ

) = 0, M

is umbilic, M

= M

(c + λ

) and c + λ

≥ 0.

Theorem 2.

Let M

(c) be a hypersurface in M ˜

(c) with constant curvature. If c < 0 and complete, then M

(c) is totally geodesic.

Theorem 3.

Let M

be an Einstein hypersurface in H

(c), i.e., S = ρI . If ρ = (n − 1)c and M is complete, then M

is totally geodesic.

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Let ˜ M

(c) be an (n + 1)-dimensonal space form, i.e., a Riemannian man- ifold of constant curvature, say c. Let ϕ : M

−→ M ˜

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

in ˜ M

(c). For sim- plicity, we say that M

is a hypersurface immersed in ˜ M

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

with ϕ(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 vec- tors 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) where X ∧ Y denotes the skew-symmetric endomorphism of T

(M ). And The Codazzi equation is expressed by

( ∇

A)(Y ) = ( ∇

A)(X ). (4)

Moreover, we prepare the following lemmas to prove the Theorem. 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 0 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:

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.

Lemma 1. (Nomizu [4]) T

and T

are differentiable.

Lemma 2. (Nomizu [4]) T

and T

are involutive.

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Lemma 3. (Nomizu [4])

If X belongs to T

(x), then Xλ = 0.

Lemma 4. (Nomizu [4])

If X ∈ T

, Y ∈ T

, then A( ∇

Y ) = − (Y λ)X . Lemma 5. (Nomizu [4])

(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. (Nomizu [4])

If Y λ = 0 for every Y ∈ T

, then X ∈ T

implies ∇

(T

) ⊂ T

and ∇

(T

) ⊂ T

.

Lemma 7. (Nomizu [4])

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.

References

[1] Marcos Dajczer , Submanifolds and Isometric Immersions. Baced on the notes prepared by Mauricio Antonucci, Gilvan Oliveira, Paulo Lima-Filho and Rui Tojeiro. Mathematics Lecture Series, 13. Perish, Inc., Houston , Library of Congress

[2] Shoshichi Kobayashi and Katsumi Nomizu , Foundations of differential ge- ometry , Vol. , Wiley Classics Library Edition Published 199

[3]

[4] Katsumi Nomizu and Brian Smyth , A formula of Simons’ type and hyper- surfaces with constant mean curvature , J . Differential Geometry 3 (1969) , 367-377

[5] B. O’neill and E. Stiel , Isometric immersions of constant curvature mani- folds , Mich. math. Journ. 10 (1963) , 335-339

[6] Patrick J. Ryan , Homogeneity and some curvature conditions for hypersur- faces , Tohoku Math. Journ. 21 (1969) , 363-388

[7] Patrick J. Ryan , Hypersurfaces with parallel Ricci tensor , Osaka J . Math.

8 (1971) , 251-259

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