Directory UMM :Journals:Journal_of_mathematics:DM:

Relationship between Laplacian

Operator and D’Alembertian Operator

Relaci´

on entre los Operadores Laplaciano y D’Alembertiano

Graciela S. Birman

∗

₍

_{gbirman@exa.unicen.edu.ar}

₎

Graciela M. Desideri (

gmdeside@exa.unicen.edu.ar

)

NUCOMPA - Fac. Exact Sc. - UNCPBA Pinto 399 - B7000GHG, Tandil - Argentina.

Abstract

Laplacian and D’Alembertian operators on functions are very impor-tant tools for several branches of Mathematics and Physics. In addition to their relevance, both operators are very used in vector calculus.

In this paper, we show a relationship between the Laplacian and the D’Alembertian operators, not only on functions but also on vector fields defined on hypersurfaces in them-dimensional Lorentzian spaces.

We also define theBk1,...,kl

m -product andBm-congruence.

Key words and phrases:Laplacian, D’Alembertian, Lorentzian space, operator,Bk1,...,kl

m -product.

Resumen

Los operadores Laplaciano y D’Alembertiano aplicados a funciones son herramientas muy importantes en varias ramas de la Matem´atica y de la F´ısica. Sumada a su relevancia, ambos operadores se destacan por ser muy utilizados en el c´alculo vectorial.

En este art´ıculo mostramos la relaci´on entre los operadores La-placiano y D’Alembertiano tanto sobre funciones como sobre campos vectoriales definidos sobre hipersuperficies del espacio Lorentzianom -dimensional. Adem´as, definimos losBk1,...,kl

m - productos y laBm-

con-gruencia entre operadores.

Palabras y frases clave: Laplaciano, D’Alembertiano, espacios Lo-rentzianos, operador,Bk1,...,kl

m -producto.

Received 2002/09/03. Revised 2003/12/05. Accepted 2004/01/05. MSC (2000): Primary 53B30.

∗_{Partially supported by Consejo Nacional de Investigaciones Cient´ıficas y Tecnol´ogicas de}

1

Introduction

In the last three decades the interest in Lorentzian geometry has increased, [1]. We will concentrate on two differential operators of particular interest here: the Laplacian and the D’Alembertian.

Laplacian and D’Alembertian operators on functions are very important tools for several branches of Mathematics and Physics, specificly in investi-gating many geometrical and physical properties. In addition to relevance, both operators are very used in vector calculus.

Moreover, the Laplacian operator on functions is quite different from the Laplacian operator on vector fields and the D’Alembertian on functions is quite different from the D’Alembertian on vector fields.

There are many interesting vector fields in differential geometry, for ex-ample the mean curvature vector field. In [5], Bang-yen Chen developed the Laplacian on vector fields, and he studied its application on mean curvature vector field for submanifolds in Riemannian space. In [3], we studied the Laplacian operator of the mean curvature vector fields on surfaces in the 3-dimensional Lorentzian space, R31,and we showed the Laplacian operator of

the mean curvature vector fields on the non-lightlike surfacesS2

1,H02, S11×R, H1

0×R,R11×S1 andR21.

The purpose of this article is to show the relationship between the Lapla-cian and the D’Alembertian operators, not only on functions but also on vector fields for non null hypersurfaces in the n+ 1-dimensional Lorentzian space.

In order to do that we will first give the definitions of these operators on functions in both Euclidean and Lorentzian spaces.

In the third section, we will generalize the Laplacian and the D’Alembertian on vector fields of Riemannian geometry to Lorentzian geometry, specifically of the hypersurfaces in Riemannian space to non null hypersurfaces in then+ 1-dimensional Lorentzian space, Rn+11 . We will introduce the Bn+1k -product,

from which the relationship between Laplacian and D’Alembertian derives.

In the fourth section, we will study theBk1,...,kl

n+1 -product . We will show

that the Bk1,...,kl

n+1 -product becomes aBn+1-congruence.

In the fifth section we will show many examples of operators on vector fields and Bk

2

Preliminaries and definitions

LetRnbe then-dimensional Euclidean space with natural coordinatesu1,. . . ,un. In classical notation, the metric tensor is

g=gijdui_⊗duj _{withg=diag(+1, . . . ,+1)}

The Laplacian and D’Alembertian operators on functions definided onRn are well known operators, defined as follows.

Definition 1. Letu1, . . . , unbe the natural cordinates inRn.The differential operators

∆ =

n

X

i=1 ∂2 ∂u2

i

(1)

¤=−∂

2

∂u2 1

+

n

X

i=2 ∂2 ∂u2

i

(2)

are called the Laplacian operator and the D’Alembertian operator in Rn, respectively. They are defined on smooth real-valued functions onRn.

Let (Rn

1,g) be ann-dimensional Lorentzian space of zero curvature where

the signature of gis (−,+, . . . ,+). We will indicate withh,ithe correspond-ing inner product.

In Lorentzian spaces there are three kinds of vectors: timelike, spacelike and lightlike, according to the inner product of the vector with itself is nega-tive, positive or zero, respectively.

We say that a hypersurface M in Rn1 is spacelike or timelike if at every

point p∈M its tangent space Tp(M) is spacelike or timelike, that is if the normal vector is timelike or spacelike, respectively, (cf. [2] for more details). We will call these hypersurfacesnon null hypersurfacesfrom now onwards.

ConsideringRn _=Rn

0,we denote the set of all smooth real-valued functions

onRn

ν withF(Rnν), whereν: 0,1.

It is natural then to define Laplacian and D’Alembertian operators on functions in the Lorentzian space Rn

1. Some operators on functions in the

Lorentzian space Rn1 are well known, (cf. [1] and [7]).

Definition 2. Letu1, . . . , unbe the natural coordinates inRn

1. The

differen-tial operators ∆ and ¤are given by:

∆ =

n

X

i=1 εi ∂

2

∂u2 i

and

¤=−ε1 ∂ 2

∂u2 1

+

n

X

i=2 εi ∂

2

∂u2 i

, (4)

respectively, whereεi=

½

−1 if i= 1, +1 if 2≤i≤n. . Both operators are defined on functionsf ∈ F(Rn

1).

According to Definition 1 and Definition 2, the Laplacian operator is de-fined by using the tensor metric of the respective structure. In some contexts, the Laplacian is defined with opposite sign and others name are used to call it (cf. [7]).

3

Relationships between the Laplacian and

D’Alembertian operators

We denote the Laplacian and the D’Alembertian operators on functions inRn1

with ∆n

1 and¤n1,and on functions inRn with ∆n0 and¤n0,respectively.

Proposition 3. According to Definitions 1 and 2, ∆n

1 (f) =¤n0 (f)and∆n0(f) =¤n1 (f).

Proof. By Definition 2,¤n

0 (f) =−∂

2

f ∂u2

1 +

Pn i=2

∂2

f ∂u2

i.

By Definition 1, ∆n

1 (f) =−∂

2

f ∂u2

1 +

Pn i=2

∂2

f ∂u2

i.

Thus ∆n

1 (f) =¤n0 (f).

Similarly,¤n1 (f) =−

³

−∂_∂u2f2 1

´

+Pn_i=2∂_∂u2f2

i =

Pn i=1

∂2

f ∂u2

i = ∆

n 0 (f).

The Laplacian operator on vector fields for submanifolds in Riemannian manifolds is known (cf. [5]). Now, we show the Laplacian and D’Alembertian operators on vector fields for hypersurfaces in an+ 1-dimensional Lorentzian space of zero curvature,Rn+11 .

LetM be an n-dimensional non null hypersurface inRn+11 with induced

connection∇. Let

Ξ (M) =©X :M →Rn+11 ; X is a vector field andX(p)∈Rn+11

ª

and

Ξ (M) =

½

X :M → [

p∈M

Tp(M) ;X is vector field and X(p)∈Tp(M)

¾

We sayE1, . . . , En is a basis of Ξ (M) andEn+1 is the unit normal vector field on M if at every point p∈M, {E1(p), . . . , En(p)} is a basis ofTp(M) andEn+1(p) is the unit normal vector atp, respectively. Thus,E1, . . . , En+1

is a basis of Ξ (M). If{E1(p), . . . , En(p)} is a orthonormal basis ofTp(M) and En+1(p) is the unit normal vector at p,∀p ∈ M, E1, . . . , En+1 is an orthonormal basis of Ξ (M)

We recall the well known fact that if X ∈Ξ (M) andEi ∈ Ξ (M),then ∇EiX is vector field of Ξ (M). Consequently, ifX ∈Ξ (M) andEij ∈Ξ (M)

then ∇Ei1· · · ∇EimX ∈ Ξ (M). Thus it is possible to define the Laplacian and the D’Alembertian operators on vector fields of Ξ (M).

Definition 4. Let M be an n-dimensional non null hypersurface in Rn+11

with induced connection ∇. Let E1, . . . , En be an orthonormal basis of Ξ (M).

a) The Laplacian ∆ on vector fields of Ξ (M) is given by:

∆ =

n

X

i=1

εi∇Ei∇Ei, (5)

b) The D’Alembertian¤on vector fields of Ξ (M) is given by:

¤=−ε1∇E1∇E1+

n

X

i=2

εi∇Ei∇Ei, (6)

where εi=hEi, Eii, i= 1, . . . , n.

Now we introduce some notation which will be used later. Let∇1_i1 =∇Ei1, ∇2_i1,i2 =∇Ei1∇Ei2, . . . ,∇

m

i1,...,im =∇Ei1· · · ∇Eim,where 1≤i1, . . . , im≤n

and E1, . . . , En+1 is basis of Ξ (M). Let F(M) be the set of all smooth real-valued functions on M. Let

P(M) =©Q6= 0; Q=Pn_i1=1qi1∇

1

i1+· · ·+

Pn

i1,...,im=1qi1,...,im∇

m i1,...,im,

wherem=m(Q)<∞andqi1, . . . , qi1,...,im ∈ F(M)

ª

. We define a new application which produces a certain change of sign in some terms of the operators ofP(M). Since this application satisfies prop-erties of inner products, we shall call it “product”. We shall make use of this product when we relate the Laplacian and the D’Alembertian operators.

Definition 5. LetM be ann-dimensional non null hypersurface inRn+11 with

Fork: 0, . . . , n,theBk

n+1-product is an application onP(M) toP(M) which

is characterized by

¡

bk i1,...,im

¢

jt=

½

−εjt ifk∈ {i1, . . . , im}

εjt ifk /∈ {i1, . . . , im} , (7)

where εjt=hEj, Eti, j, t= 1, . . . , n+ 1, and{i1, . . . , im} ⊂ {1, . . . , n}.

We denote theBk

n+1-product withh,i k Bn+1.

The equalityQ=Pn+1_t=1 hQ, Etik_Bn+1EtmeansQX=Pn+1_t=1 hQX, Etik_Bn+1Et

for allX ∈Ξ (M). Hence, theBk

n+1-product is well defined.

Remark 6. TheBk

n+1-product isF(M)-bilinear.

Remark 7. From Definition 5, if∇m_i1,...,imX =Pn+1j=1X j

i1,...,imEj then we have

D

∇m_i1,...,imX, EtEk Bn+1

=Pn+1_j=1X_ij_1,...,imhEj, Etik_B

n+1 =Pn+1_j=1X_ij_1,...,im¡bk

i1,...,im

¢

jt=X t i1,...,im

¡

bk i1,...,im

¢

tt

=

½

−εttXt

i1,...,im ifk∈ {i1, . . . , im}

εttXt

i1,...,im ifk /∈ {i1, . . . , im}

.

The following theorem relates the Laplacian and the D’Alembertian oper-ators, which are defined in (5) and (6).

Theorem 8. LetM be ann-dimensional non null hypersurface inRn+11 with

induced connection ∇. Let E1, . . . , En+1 be an orthonormal basis of Ξ (M). Then, the Laplacian∆ and the D’Alembertian¤operators on vector fields of Ξ (M)are related by:

¤=

n+1

X

t=1

h∆, Eti1_Bn

+1Et (8)

and

∆ =

n+1

X

t=1

h¤, Eti1_Bn

+1Et. (9)

Proof. LetX ∈Ξ (M) and let ∇Ei∇EiX =

Pn+1 j=1X

j

iiEj. By (5) and (6),

Pn+1

t=1 h∆X, Eti 1

Bn+1Et=

Pn+1 t=1

­Pn

i=1εi∇Ei∇EiX, Et

®1 Bn+1Et =Pn+1_t=1 nPn_i=1εi­∇Ei∇EiX, Et

®1 Bn+1

o

Et

=Pn+1_t=1 nPn_i=1εiP_j=1n+1X_iij hEj, Eti1_B

n+1

o

Et.

Pn+1 t=1

nPn i=1εi

Pn+1 j=1 X

j

iihEj, Eti 1 Bn+1

o

Et

=Pn+1_t=1 n−ε1Xt

11hEt, Eti+

Pn+1

i=2 εiXiit hEt, Eti

o

Et

=−Pn+1_t=1 ε1­ Pn+1_j=1 X11j Ej, Et

®

Et+Pn+1_t=1 Pn+1_i=2 εi­ Pn+1_j=1X_iijEj, Et®Et

=−ε1Pn+1_t=1 ­∇E1∇E1X, Et

®

Et+Pn+1_i=2 εiP_t=1n+1­∇Ei∇EiX, Et

®

Et

=−ε1∇E1∇E1X+

Pn+1

i=2 εi∇Ei∇EiX =¤X.

Therefore,¤=Pn+1t=1 h∆, Eti 1

Bn+1Et. Analogously,

Pn+1

t=1 h¤X, Eti 1 Bn+1Et =Pn+1t=1

©

−ε1­∇Ei∇EiX, Et

®1 Bn+1+

Pn i=2εi

­

∇Ei∇EiX, Et

®1 Bn+1

ª

Et

=Pn+1_t=1 ©−ε1Pn+1_j=1X11j hEj, Eti 1 Bn+1+

Pn i=2εi

Pn+1 j=1X

j

iihEj, Eti 1 Bn+1

ª

Et

=Pn+1_t=1 © Pn_i=1εiDPn+1_j=1X_iijEj, EtE ªEt

=Pn_i=1εi© Pn+1_t=1 ­∇Ei∇EiX, Et

®

Etª

=Pn_i=1εi∇Ei∇EiX = ∆X.

From now onwards, we will extend Definition 5 and Theorem 8 to general, not necessary orthonormal basis. In order to do that we first define the Lapla-cian and D’Alembertian operators on vector fields whenM is an-dimensional non null hypersurface in Rn+11 . In a classical way, we denotegij =hEi, Eji,

1≤i, j≤n+ 1,and¡gij¢_{= (gij)}−1

.

Definition 9. Let M be an n-dimensional non null hypersurface in Rn+11

with induced connection ∇. LetE1, . . . , En be a basis of Ξ (M). a) The Laplacian ∆ on vector fields of Ξ (M) is given by:

∆ =

n

X

i,j=1 gij_∇

Ei∇Ej. (10)

b) The D’Alembertian¤on vector fields of Ξ (M) is given by:

¤=−g11∇E1∇E1−

n

X

i=2

gi1¡∇Ei∇E1+∇E1∇Ei

¢

+

n

X

i,j=2

gij∇Ei∇Ej. (11)

Naturally, theBk

n+1-product must also be extended to general basis.

Definition 10. LetM be an n-dimensional non null hypersurface in Rn+11

Ξ (M). For k : 0, . . . , n, the Bk

n+1-product is an application on P(M) to

P(M) which is characterized by:

¡

bk i1,...,im

¢

jt=

½

−gjt ifk∈ {i1, . . . , im}

gjt ifk /∈ {i1, . . . , im} . (12)

We denote theBk

n+1-product withh,ikBn+1. Remark 11. Sinceh,iisF(M)-bilinear, theBk

n+1-product is F(M)-bilinear

too.

Remark 12. If∇m_i1,...,imX=Pn+1_j=1 X_ij1,...,imEj then we have

D

∇m_i1,...,imX, EtEk Bn+1

=Pn+1_j=1X_ij_1,...,imhEj, Etik_B

n+1 =Pn+1_j=1X_ij_1,...,im¡bk

i1,...,im

¢

jt

=

(

−Pn+1_j=1gjtX_ij_1,...,im ifk∈ {i1, . . . , im}

Pn+1 j=1gjtX

j

i1,...,im ifk /∈ {i1, . . . , im}

=

 



−D∇m_i1,...,imX, Et

E

ifk∈ {i1, . . . , im}

D

∇mi1,...,imX, Et

E

ifk /∈ {i1, . . . , im} .

Theorem 13. Let M be an n-dimensional non null hypersurface in Rn+11

with induced connection ∇. Let E1, . . . , En be a basis ofΞ (M)and letEn+1

be the unit normal vector field. Then,

¤=

n+1

X

t=1

h∆, Eti1_Bn+1Et (13)

and

∆ =

n+1

X

t=1

h¤, Eti1_Bn+1Et. (14)

Proof. Clearly, the Laplacian ∆ and the D’Alembertian¤are two operators ofP(M).

LetX∈Ξ (M),then∇Ei∇EjX =

Pn+1

r=1XijrEr,where Xr

ij =

Pn+1 s=1gsr

­

∇Ei∇EjX, Es

®

By (10) and (11),

Pn+1

t=1 h∆X, Eti 1

Bn+1Et=

Pn+1 t=1

DPn

i,j=1gij∇Ei∇EjX, Et

E1

Bn+1

Et

=Pn+1t=1

nPn i,j=1gij

­

∇Ei∇EjX, Et

®1 Bn+1

o

Et

=Pn+1_t=1 nPn_i,j=1gijPn+1

r=1Xijr hEr, Eti 1 Bn+1

o

=Pn+1_t=1 nPn_i,j=1gijPn+1

LetM be ann-dimensional non null hypersurface inRn+11 with induced

con-nection ∇. From now anwards, we consider E1, . . . , En+1vector fields such that En+1(p) is the unit normal vector at p and {E1(p), . . . , En(p)} is a basis ofTp(M),at allp∈M.

Definition 14. Letk1, . . . , kl be integer numbers such that 0≤k1 <· · · < kl≤n. TheBk1,...,kl

n+1 -product is characterized by:

³

n+1 -product is tooF(M)-bilinear.

IfQ∈ P(M),we denote Pn+1_t=1 hQ, Etik1,...,kl

Bn+1 Et withB

k1,...,kl

From Definition 14, we getBk1,...,kl

n+1 (Q) is a differential operator ofP(M).

Remark 15. Let us note that ¡b0 i1,...,im

¢

jt =gjt, at all j, t: 1, . . . , n+ 1 and

{i1, . . . , im} ⊂ {1, . . . , n}.ThusB0

n+1(Q) =Q.

Definition 16. LetP, Qbe two differential operators ofP(M).We say that

P isBn+1-congruent toQifP =Bk n+1(Q).

Lemma 17. Let ¡bk i1,...,im

¢

uv,

¡

bh j1,...,js

¢

rt be as in (15), then

¡

bk i1,...,im

¢

uv

¡

bh j1,...,js

¢

rt=

 



−guvgrt ifk∈ {i1, . . . , im} ∧h /∈ {j1, . . . , js},

−guvgrt ifk /∈ {i1, . . . , im} ∧h∈ {j1, . . . , js}, guvgrt in other case.

Proof. It follows from the table:

k=i1 ... k=im k /∈ {i1, . . . , im}

h=j1 (−guv) (−grt) ... (−guv) (−grt) guv(−grt)

h=j2 (−guv) (−grt) ... (−guv) (−grt) guv(−grt)

... ... ... ... ...

h=js (−guv) (−grt) ... (−guv) (−grt) guv(−grt)

h /∈ {j1, . . . , js} (−guv)grt ... (−guv)grt guvgrt

We denote the set of all Bk1,...,kl

n+1 -products with Bn+1, where 0 ≤ k1 <

· · ·< kl≤n.

Concecutive application of products inBn+1 result in another product in

Bn+1.Its proof is more dull than the idea itself. So we have developed it in steps.

Proposition 18. Let P, Q, R ∈ P(M) such that P = Bk

n+1(R) and R = Bh

n+1(Q), then

P =Bkn+1

¡

Bhn+1(Q)

¢

=

 



Bk,hn+1(Q) if k < h, Q if k=h, Bh,k_n+1(Q) if k > h,

(16)

Proof. We will explicitly show thatBk

Let us note that

Pn

and in the same way,

Pn

−Pn_i

From Definition 14, if k < h then the above expression is the same as

=Pn+1_j=1

In similar way to Proposition 18,

+

Equality (17) follows from Definition 14.

Corollary 20. LetBk1,...,kl

Proof. It follows from Proposition 18 and Corollary 19.

Now, we define an operation onBn+1× Bn+1.

Proof. By Proposition 18 and corollaries 19 and 20,◦is a well-defined opera-tion. By Proposition 18 and Corollary 20,◦is a conmutative operation. From Remark 15 and Proposition 18, the identity of (Bn+1,◦) is theB0

n+1-product.

By Proposition 18 and Corollary 19,

Bk1

Bk1,...,kl

n+1 ◦B k1,...,kl

n+1 =Bn+1k1 ◦ · · · ◦Bn+1kl ◦Bn+1k1 ◦ · · · ◦Bn+1kl

=³Bk1

n+1◦Bkn+11

´

◦ · · · ◦³Bkl

n+1◦Bn+1kl

´

=B0

n+1◦ · · · ◦Bn+10 =B0n+1.

Therefore³Bk1,...,kl

n+1

´−1

=Bk1,...,kl

n+1 .

Remark 23. The order ofBk1,...,kl

n+1 is

½

1 if Bk1,...,kl

n+1 =Bn+10 ,

2 in other case.

Theorem 24. Let P and Q be two Bn+1-congruent operators. There ex-ists Bk1,...,kl

n+1 ∈ Bn+1such that P = Bn+1k1,...,kl(Q),and this is an equivalence

relationship.

Proof. LetP, Q, R∈ P(M). By Proposition 22, for allP ∈ P(M) there exists

B0

n+1∈ Bn+1 such thatP =Bn+10 (P).Therefore,P isBn+1-congruent toP,

for allP ∈ P(M).

If P is Bn+1-congruent to Q then there exists Bk

n+1 ∈ Bn+1 such that P =Bk

n+1(Q). From Proposition 22Q=Bn+1k

¡

Bk n+1(Q)

¢

=Bk

n+1(P),that

is, there exists Bk

n+1 ∈ Bn+1 such thatQ=Bn+1k (P).Therefore,QisBn+1

-congruent toP.

IfP =Bk

n+1(R) and R=Bn+1h (Q), it follows from Proposition 18 that P =Bn+1k,h (Q),withB

k,h

n+1∈ Bn+1.

5

Examples

Lastly, we show some examples of the Laplacian and D’Alembertian operators on vector fields andBk

n+1-products.

In [3] the reader can find more information about non null surfaces of constant curvature in R31, mean curvature vector fields, and Laplacian on

mean curvature vector fields on non null surfaces inR31.

Example 25. Letx1, x2, x3be a coordinate system inR31such that{∂1, ∂2, ∂3}

is an orthonormal basis forR31,where ∂i = ∂x∂i. The pseudosphereS

2 1 in R31

is the surface defined by S2 1 =

©

(x1, x2, x3)∈R31 : −x21+x22+x23= 1

ª

. S2

1can be parametrized asx1= sinhω,x2= cosθcoshω,x3= sinθcoshω,

where ω∈Rand 0≤θ <2π.

The tangent vectors are expressed as follows:

∂ω=_∂∂_ω = coshω ∂1+ sinhωcosθ ∂2+ sinhωsinθ ∂3,

∂θ= ∂

∂θ =−coshωsinθ ∂2+ coshωcosθ ∂3.

The unit normal vector to the surfaceS2

Henceh∂ω, ∂ωi=−1,h∂ω, ∂θi= 0,h∂θ, ∂θi= cosh2ω andhN, Ni= 1.

According to Definition 9, the Laplacian operator on vector fields, ∆, for

S2

1 is given by:

∆ =−∇∂ω∇∂ω+

1

cosh2ω∇∂θ∇∂θ.

The mean curvature vector fieldH forS12is given byH =−N (cf. [3]).

By applying toHthe Laplacian operator on vector fields forS2

1, we obtain

∆H = 2N−tanhω ∂ω=−2H−tanhω∂ω,(cf. [3]).

Example 26. Letx1, x2, x3be a coordinate system inR31such that{∂1, ∂2, ∂3}

is an orthonormal basis for R31, where∂i = ∂x∂i. The cylinderR

1

1×S1 in R31

is the surface defined by R11×S1=

©

(x1, x2, x3)∈L3 _{: x}2

2+x23= 1

ª

.

R11×S1can be parametrized asx1=t,x2= cosθ,x3= sinθ, wheret∈R

and 0< θ <2π.

The tangent vectors are expressed as∂t=∂1,∂θ=−sinθ ∂2+ cosθ ∂3. The unit normal vector to the surfaceR11×S1at (t, θ) isN = (0,cosθ,sinθ).

Hence,h∂t, ∂ti=−1,h∂θ, ∂ti= 0,h∂θ, ∂θi= 1 andhN, Ni= 1.

According to Definition 9, the D’Alembertian operator on vector fields for R11×S1 is given by¤=∇∂t∇∂t+∇∂θ∇∂θ.

The mean curvature vector fieldH is given byH =−1

2N (cf. [3]).

Since∇∂ω∇∂ωH = 0 and ∇∂θ∇∂θH =−

1

2∇∂θ(∂θ) =

1

2N,applying toH

the D’Alembertian operator on vector fields for R11×S1 we obtain ¤H = 1

2N =−H.

Example 27. Let ∆ be the Laplacian operator on vector fields for a surface

M inR31.We denote theBn+1-equivalence class of ∆ with [∆].If ∆ is defined

by

∆ =g11_∇

∂1∇∂1+g

12_∇

∂1∇∂2+g

21_∇

∂2∇∂1+g

22_∇ ∂2∇∂2, then,

B0

n+1(∆) = ∆, B1

n+1(∆) =−

¡

g11_∇

∂1∇∂1+g

12_∇

∂1∇∂2+g

21_∇ ∂2∇∂1

¢

+g22_∇ ∂2∇∂2 =−∆ + 2g22_∇

∂2∇∂2,

B2

n+1(∆) =g11∇∂1∇∂1−

¡

g12_∇

∂1∇∂2+g

21_∇

∂2∇∂1+g

22_∇ ∂2∇∂2

¢

=−∆ + 2g11_∇ ∂1∇∂1, and

B12

n+1(∆) =−g11∇∂1∇∂1+g

12_∇

∂1∇∂2+g

21_∇

∂2∇∂1−g

22_∇ ∂2∇∂2 = ∆−2¡g11_∇

∂1∇∂1+g

22_∇ ∂2∇∂2

¢

=−B 1

n+1(∆)+B 2

n+1(∆)

2 .

areBn+1-congruent operators. Therefore, [∆] =n∆,¤, B2

n+1(∆),−

¤+B2

n+1(∆)

2

o

Acknowledgement The authors want to thank one of the referees for his valuable comments.

References

[1] Been, J. K., Ehrlich, P. E., Easley, K. L. Global Lorentzian Geometry, Second Edition, Marcel Dekker Inc., New York, 1996.

[2] Birman, G. S. Integral Formulas in Semi-Riemannian Manifolds, To ap-pear.

[3] Birman, G. S., Desideri, G. M., Laplacian on Mean Curvature Vector Fields for some Non-Lightlike Surfaces in the 3-Dimensional Lorentzian Space, To appear in Actas del VII Congreso Monteiro.

[4] Birman, G., Nomizu, K., Trigonometry in Lorentzian Geometry, Amer. Math. Monthly91(6) (1984), 543–549.

[5] Chen, Bang-yen. Geometry of Submanifolds, Marcel Dekker, Inc., New York, 1973.

[6] Kupeli, D. N.,The Method of Separation of Variables for Laplace-Beltrami Equation in Semi-Riemannian Geometry, New Developments in Differ-ential Geometry (December, 1994), 279–290, Math. Appl., 350, Kluwer Acad. Publ., Dordrecht, 1996.