Spacelike Maximal Surfaces with Constant Scalar Normal Curvature in a Normal Contact Lorentzian Manifold

BULLETIN Bull. Malaysian Math. Soc. (Second Series) 21 (1998) 31-36 of the

MALAYSIAN MATHEMATICAL SOCIETY

Spacelike Maximal Surfaces with Constant Scalar Normal Curvature in a Normal Contact Lorentzian Manifold

IKAWA TOSHIHIKO

Nihon University, Dept. of Math., School of Medicine, Itabashi, Tokyo, Japan, 173 e-mail: [email protected]

Abstract. If the scalar normal curvature of a spacelike maximal surface in a 5-dimensional normal contact Lorentzian manifold with constant φ-sectional curvature is constant, then the surface is totally geodesic or nonpositively curved.

1. Introduction

On some odd dimensional manifolds, the normal contact Riemannian metric structure (or Sasakian structure) can be defined. The study of manifolds with this structure has a long history.

If we change the Riemannian metric of the Sasakian structure to a Lorentzian one, we can define the normal contact Lorentzian structure. This definition was given at the starting time of the study of the Sasakian structure. But practical study of it has not been given sufficiently yet (cf. [5], [7]). In [3], [4], we study the fundamental properties of manifolds with the normal contact Lorentzian structure.

In this paper, we shall study the scalar normal curvature for spacelike maximal surfaces in a 5-dimensional normal contact Lorentzian manifold of constant φ-sectional curvature and prove.

Theorem. Let M⁵ be a 5-dimensional normal contact Lorentzian manifold with constant φ-sectional curvature k and M² a spacelike maximal surface with vector field ξ normal to M². Assume that the scalar normal curvature K_N of M² in M⁵ is constant. Then M² is totally geodesic with Gauss curvature K= ^k−3₄ or a nonpositive curved surface.

2. Spacelike Submanifold

Let M be a (2n+1)-dimensional (n≥2) manifold. The normal contact Lorentzian structure ( , , , )φ ξ η g of M is given by a ( , )1 1-type skew-symmetric tensor field φ, a vector field ξ, a 1-form η and a Lorentzian metric g as

( ) ( )

φ η ξ φξ

ξ ξ η ξ

η φ ξ φ

φ η ξ

2 0

1

X X X

g X g X

Y g X Y X

Y Y X g X Y

X X

X

= − + =

= − = −

∇ = ∇ =

∇ = − −

( ) , ,

( , ) , ( ) ( , ),

( , ), ,

( ) ( , ) ,

(2.1)

where X is a vector field of M and ∇is the covariant derivative with respect to g ([3], [4]). When the curvature tensor field of K X Y Z( , ) of M has the following form

4K X Y Z( , ) = (k−3) ( ( , )g Y Z X − g X Z Y( , ) )

^{+ (k+1)}

(

^{η( ) ( )Y η Z X − η(X) ( )η Z Y+g X Z( , ) ( )ηY ξ} (2.2)

^{−g Y Z( , ) (η X)ξ + g(φY Z, )φX + g(φX Z, )φY − 2g(φX Y, )φZ}

)

, M is called a space of constant φ-sectional curvature k.

Let M be an n-dimensional submanifold of M. By ∇ we denote the covariant derivative of M determined by the induced metric on M. Let X(M) ( resp. X(M) )be the Lie algebra of vector fields on M resp. ( M) and X^⊥(M) the set of all vector fields normal to M.

The Gauss-Weingarten formulas are given by

∇XY = ∇XY + B X Y( , ), ∇XN = −AN(X) + D NX , (2.3)

X Y, ∈X(M), N∈X^⊥(M),

where D is the normal connection [6]. B is called the second fundamental form tensor and A the shape operator, and they satisfy

g A( ^N(X),Y) = g B X Y( ( , ), N). (2.4) If the induced metric on M is positive-definite, then M is called a spacelike

submanifold.

Let M be a spacelike submanifold of M with the vector field ξ normal to M, then from (2.1), (2.3) and (2.4), M satisfies following properties.

Proposition. Let M be an m-dimensional spacelike submanifold in a normal contact Lorentzian manifold M²ⁿ⁺¹ with structure ( , , , )φ ξ η g . Then

(i) The dimension m of M satisfies m≤n.

(ii) The shape operator of ξ direction is identically zero.

(iii) If X ∈X(M) then φX∈X^⊥(M).

(iv) If m n= , then A^φX( )Y = A^φY(X), for X Y, ∈X(M).

Spacelike Maximal Surfaces

3. Local Formulas

We consider a spacelike surface M² in a 5-dimensional normal contact Lorentzian manifold M. Let {e e₁, ₂,e e₃, ₄, }ξ be an orthonormal frame field on M⁵so that

e₁, e₂ ∈X(M), e₁*:=φe₁=e₃, e₂*:=φe₂ =e₄. We shall make use of the following convention on the ranges of indices:

1 5 2

3 4 3 5

1

≤ ≤ ≤ ≤

≤ ^{∗ ∗} ≤ ≤ ≤

A B i j

i j

, , , , , ,

, , , , .

" "

" α β "

Let {w w w¹, ², ^1∗,w^2∗,w⁵} be the field of dual frames. Then the structure equations of M are given by

dw^A BwBA wB w w

BA A

B

= −

∑

^ε ∧ ^, + = ^0,

dwBA w w

C CA BC

BA

= −

∑

^ε ∧ + ^{Φ ,} Φ_B^A = ¹₂^{ε ε}_{C D}

∑

K_BCD^A w^C∧w^D,

K_BCD^A + K_BDC^A = 0.

Restricting these forms to M², we have w^α=0. Since 0=dw^α= −∑w_i^α ∧ wⁱ, by Cartan’s lemma we may write

wi h wij j h h

ij ji

α α α α

=

∑

^, = ^.

From these formulas we obtain

dwⁱ = −

∑

wⁱ_j ∧w^j, wⁱ_j + w_i^j = ^0, dwⁱ_j =

∑

w_kⁱ ∧w_j^k + ^Ωi_j^,

Ωⁱ_j 1 Rⁱ_jk w^k w

= 2

∑

A ∧ A,

^RijkA = ^KjkiA +

∑

^εα

(

^{h hik}α α^jA−^{h hi}α αA ^jk

)

. dw_β^α = −

∑

ε_γw_γ^α∧w_β^γ + ^Ωα_β,

Ω_β^α 1 _β^α

= 2

∑

^R k^wk∧^w

A A,

^RβαkA = ^KβαkA +

∑ (

^{h hikα iβ}A−^{h hiα}A ^ikβ

)

^.

An immersion is said to be maximal if ∑ihiiα=0 for all α.

We define hijkα nd a hijkα_A by

h_ijk^α w^k = dh_ij^α −

∑

h w_i^α _j −

∑

h w^α_{j i} +

∑

ε_βh w_ij^β _β^α

∑

A A

A A , (3.1) hijk w dhijk hjk wi hi kwkj

A A

A A

α = α −

∑

α −

∑

αA

∑

⁻

∑

^{h w}ijαA kA ⁺

∑

^hijk^w β β

βα

ε .

The Laplacian Δhijα is given by

Δhijα =

∑

hijkkα ^.

When M² is maximal in M⁵, that is ∑hkkα =0 for all α, Δhijα can be written as

( )

Δhijα Kkihjk Kk khij Kkjhki βα β

αβ β

βα β

=

∑

² − + ^{2 +}

^∑ (

^{Kmikm hmjα + Kkjkm hmiα + 2Kijkm hmkα}

)

( )

+

∑

^2hkm kiα ^{h h}β βmj− ^hmkα ^hkm ijβ ^hβ − ^{h h}mi mkα β ^hkjβ− ^{h h h}mjα kiβ βmk

. (3.2) The scalar normal curvature KN of M² is defined by

( )

KN =

∑

^{ε ε}α β βSαijSij Sij =

∑

h hik jk − h hjk ik βα

βα α β α β

, .

The covariant derivative of Sβαij

is defined by

( )

S_β^α_ijA =

∑

h_ik^α_Ah^β_jk + h h_ik^{α β}_jkA − h^α_jkAh_ik^β− h^α_jkh_ik^β_A ^. Then for the Laplacian of KN, we have the following formula

( )

1 2

ΔK_N =

∑

^{ε ε}α β Sβα_ijk 2 +

∑

S _ij h_ik h_ji − h_ik h_jk

βα α β β α

( ) _{A A A A}

+ ⁴

∑

^{ε ε}α β βα α β

S ij(Δhik)hjk.

4. Proof of Theorem

Let M² be a spacelike maximal surface in a normal contact Lorentzian manifold M⁵ of constant φ-sectional curvature k. Then, from Proposition, we obtain

(h , ( , (

a

a h

a

a h

ij1 0 ij2 ij5

0

0 0

0 0 0 0

∗ = ∗

−

⎡

⎣⎢ ⎤

⎦⎥ = −

−

⎡

⎣⎢ ⎤

⎦⎥ = ⎡

⎣⎢ ⎤

⎦⎥

) ) ) . (4.1)

Spacelike Maximal Surfaces

From (3.2), it follows that

Δh k Δ Δ

a a h h

111 3

121

211

3 5

4 6 0

∗ ∗ ∗

= − − , = = ,

Δh k Δ Δ

a a h h

221 3

112

222

3 5

4 6 0

∗ ∗ ∗

= − − + , = = ,

Δh Δh k

a a

122

212 3 5 3

4 6

∗ ∗

= = − − + ,

Δh115 Δh Δh Δh

125

215

225

= = = = 0,

by virtue of (2.2).

From (3.1), we have

h₁₁₁^1∗ = −h₂₂₁^1∗ = −h₁₂₁^2∗ = −h₂₂₁^2∗

= −h122^∗ = −h^∗ = −h ^∗ = h ^∗ = a

1

212 1

112 2

222 2

,1, h₁₁₂^1∗ = −h₂₂₂^1∗ = −h₁₂₂^2∗ = −h₂₁₂^2∗

= h121^∗ = h^∗ = h ^∗ = −h ^∗ = a

1

211 1

111 2

221 2

,2, h1125 h h h

1210 2210

2220

= = = = 0, h₁₂₂⁰ = h₂₂₁⁰ = −a.

Since

S_{1 11}²∗ S_{1 22}² S_{1 12}² S_{1 21}² a²

∗

∗

∗

∗

∗

∗

= = 0, = − ∗ = 2 ,

by virtue of (4.1), we obtain

(Sβαijk) ² Sm ijk 2 0 Sijk

ε β

∑

⁼

∑

^{( A∗∗ )2 +}

∑

^{( 0 )2}

^{= 4 ((2}

(

^a2 1 ^{2 ((2a}2 ²

)

^a

2 32 4

), ) + ), ) − ,

( )

S_β^α_ij h_ik^α_Ah_jk^β_A − h_ik^β_Ah^α_jkA =

∑

^{2 ((2}

(

^a2 1 ^{2 ((2a}2 ²

)

), ) + ), )2 ,

( )

S_β^α_ij(Δh^α_jk)h_jk^β a k a

∑

^{= 8 4 3} ₄^{− −5 6 2 ,}

so that

( ) ( )

1

2 8 2 2 32 3 9

4 6

2 1 2 2

2 2 4 2

ΔK a a a k

N = ( ( ), ) + ( ( ), ) + − − a .

If we assume K_N≡constant, then since a is continuous, this equation reduces to a≡0 or a² ≥ ^k₈⁻³ everywhere. On the other hand, the Gauss curvature K of M² is given by K= ^k−3₄ −2a². Hence if a²≥ ^k₈⁻³ then K≤0 . This completes the proof.

Acknowledgement: The author is grateful to the referee for his valuable suggestions.

References

1. D.E. Blair, Contact manifolds in Riemannian geometry, Lecture Notes in Math. No. 509, Springer-Verlag, New York, 1967.

2. C-S. Houh, Some totally real minimal surfaces in CP², Proc. Amer. Math. Soc. 40 (1973), 240-244.

3. T. Ikawa and M. Erdogan, Sasakian manifolds with Lorentzian metric, Kyungpook Math. J. 35 (1996), 517-526.

4. T. Ikawa and J. Jun, On sectional curvatures of normal contact Lorentzian manifold, to appear.

5. I. Mihai and R. Rosca, On Lorentzian P-Sasakian manifolds, Classical Analysis, World Scientific, Singapore, 1992, 155-169.

6. B. O’Neill, Semi-Riemannian geometry with applications to relativity, Academic Press, New York, 1983.

7. R. Rosca, On pseudo-Sasakian manifolds, Rend. Math. Roma 4 (1984), 393-407.

Keywords: normal contact Lorentzian manifold, spacelike surface, scalar normal curvature.

1991 Mathematics Subject Classification: Primary 53C50, Secondary 53C42