Divulgaciones Matem´ aticas Vol. 12 No. 1(2004), pp. 35–52

Relationship between Laplacian Operator and D’Alembertian Operator Relaci´on entre los Operadores Laplaciano y D’Alembertiano Graciela S. Birman ∗ ([email protected]) Graciela M. Desideri ([email protected]) NUCOMPA - Fac. Exact Sc. - UNCPBA Pinto 399 - B7000GHG, Tandil - Argentina. Abstract Laplacian and D’Alembertian operators on functions are very important 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 the m-dimensional Lorentzian spaces. k ,...,kl -product and Bm -congruence. We also define the Bm1 Key words and phrases: Laplacian, D’Alembertian, Lorentzian space, k1 ,...,kl -product. operator, Bm 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 Laplaciano y D’Alembertiano tanto sobre funciones como sobre campos vectoriales definidos sobre hipersuperficies del espacio Lorentziano mk ,...,kl dimensional. Adem´ as, definimos los Bm1 - productos y la Bm - congruencia entre operadores. Palabras y frases clave: Laplaciano, D’Alembertiano, espacios Lok1 ,...,kl rentzianos, operador, Bm -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 la Rep´ ublica Argentina.

36

1

Graciela S. Birman, Graciela M. Desideri

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 investigating 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 example 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 3dimensional Lorentzian space, R31 , and we showed the Laplacian operator of the mean curvature vector fields on the non-lightlike surfaces S12 , H02 , S11 × R, H01 × R, R11 × S 1 and R21 . The purpose of this article is to show the relationship between the Laplacian 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 the n + 1k dimensional Lorentzian space, Rn+1 . We will introduce the Bn+1 -product, 1 from which the relationship between Laplacian and D’Alembertian derives. k1 ,...,kl In the fourth section, we will study the Bn+1 -product . We will show k1 ,...,kl that the Bn+1 -product becomes a Bn+1 -congruence.

In the fifth section we will show many examples of operators on vector k fields and Bn+1 -products. Divulgaciones Matem´ aticas Vol. 12 No. 1(2004), pp. 35–52

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37

Preliminaries and definitions

Let Rn be the n-dimensional Euclidean space with natural coordinates u1 ,. . . ,un . In classical notation, the metric tensor is g = gij dui ⊗ duj

with g =diag(+1, . . . , +1)

The Laplacian and D’Alembertian operators on functions definided on Rn are well known operators, defined as follows. Definition 1. Let u1 , . . . , un be the natural cordinates in Rn . The differential operators n X ∂2 (1) ∆= ∂u2i i=1 n

¤=−

X ∂2 ∂2 + ∂u21 i=2 ∂u2i

(2)

are called the Laplacian operator and the D’Alembertian operator in Rn , respectively. They are defined on smooth real-valued functions on Rn . Let (Rn1 , g) be an n-dimensional Lorentzian space of zero curvature where the signature of g is (−, +, . . . , +). We will indicate with h , i the corresponding 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 negative, 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 hypersurfaces non null hypersurfaces from now onwards. Considering Rn = Rn0 , we denote the set of all smooth real-valued functions on Rnν with F (Rnν ), where ν : 0, 1. It is natural then to define Laplacian and D’Alembertian operators on functions in the Lorentzian space Rn1 . Some operators on functions in the Lorentzian space Rn1 are well known, (cf. [1] and [7]). Definition 2. Let u1 , . . . , un be the natural coordinates in Rn1 . The differential operators ∆ and ¤ are given by: ∆=

n X i=1

εi

∂2 ∂u2i

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Graciela S. Birman, Graciela M. Desideri

and

n X ∂2 ∂2 ¤ = −ε1 2 + εi 2 , ∂u1 i=2 ∂ui ½ −1 if i = 1, respectively, where εi = . +1 if 2 ≤ i ≤ n. Both operators are defined on functions f ∈ F (Rn1 ).

(4)

According to Definition 1 and Definition 2, the Laplacian operator is defined 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 in Rn1 with ∆n1 and ¤n1 , and on functions in Rn with ∆n0 and ¤n0 , respectively. Proposition 3. According to Definitions 1 and 2, ∆n1 (f ) = ¤n0 (f ) and ∆n0 (f ) = ¤n1 (f ). Pn ∂ 2 f ∂2f Proof. By Definition 2, ¤n0 (f ) = − ∂u 2 + i=2 ∂u2i . 1 P 2 n ∂ f ∂2f By Definition 1, ∆n1 (f ) = − ∂u 2 + i=2 ∂u2i . 1 Thus ∆n1 (f ) = ¤n0 (f )³. ´ P Pn n ∂2f ∂2f Similarly, ¤n1 (f ) = − − ∂u + i=2 ∂u 2 2 = i=1 1

i

∂2f ∂u2i

= ∆n0 (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 a n + 1-dimensional Lorentzian space of zero curvature, Rn+1 . 1 Let M be an n-dimensional non null hypersurface in Rn+1 with induced 1 connection ∇. Let © ª Ξ (M ) = X : M → Rn+1 ; X is a vector field and X (p) ∈ Rn+1 1 1 and Ξ (M ) =

½ X:M →

[

¾ Tp (M ) ; X is vector field and X (p) ∈ Tp (M ) .

p∈M

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We say E1 , . . . , En is a basis of Ξ (M ) and En+1 is the unit normal vector field on M if at every point p ∈ M , {E1 (p) , . . . , En (p)} is a basis of Tp (M ) and En+1 (p) is the unit normal vector at p, respectively. Thus, E1 , . . . , En+1 is a basis of Ξ (M ). If {E1 (p) , . . . , En (p)} is a orthonormal basis of Tp (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 ) and Ei ∈ Ξ (M ) , then ∇Ei X is vector field of Ξ (M ). Consequently, if X ∈ Ξ (M ) and Eij ∈ Ξ (M ) then ∇Ei1 · · · ∇Eim X ∈ 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+1 1 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

εi ∇Ei ∇Ei ,

(5)

i=1

b) The D’Alembertian ¤ on vector fields of Ξ (M ) is given by: ¤ = −ε1 ∇E1 ∇E1 +

n X

εi ∇Ei ∇Ei ,

(6)

i=2

where εi = hEi , Ei i , i = 1, . . . , n. 1

2

Now we introduce some notation which will be used later. Let ∇i1 = ∇Ei1 , m

∇i1 ,i2 = ∇Ei1 ∇Ei2 , . . . , ∇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 © Pn Pn 1 m P (M ) = Q 6= 0; Q = i1 =1 qi1 ∇i1 + · · · + i1 ,...,im =1 qi1 ,...,im ∇i1 ,...,im , ª where m = m(Q) < ∞ and qi1 , . . . , qi1 ,...,im ∈ F (M ) . We define a new application which produces a certain change of sign in some terms of the operators of P (M ). Since this application satisfies properties 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. Let M be an n-dimensional non null hypersurface in Rn+1 with 1 induced connection ∇. Let E1 , . . . , En+1 be an orthonormal basis of Ξ (M ). Divulgaciones Matem´ aticas Vol. 12 No. 1(2004), pp. 35–52

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Graciela S. Birman, Graciela M. Desideri

k For k : 0, . . . , n, the Bn+1 -product is an application on P (M ) to P (M ) which is characterized by ½ ¢ ¡ k −εjt if k ∈ {i1 , . . . , im } bi1 ,...,im jt = , (7) εjt if k ∈ / {i1 , . . . , im }

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

k We denote the Bn+1 -product with h, iBn+1 . Pn+1 Pn+1 k k The equality Q = t=1 hQ, Et iBn+1 Et means QX = t=1 hQX, Et iBn+1 Et k -product is well defined. for all X ∈ Ξ (M ). Hence, the Bn+1 k -product is F (M )-bilinear. Remark 6. The Bn+1 Pn+1 m Remark 7. From Definition 5, if ∇i1 ,...,im X = j=1 Xij1 ,...,im Ej then we have D m Ek Pn+1 k ∇i1 ,...,im X, Et = j=1 Xij1 ,...,im hEj , Et iBn+1 Bn+1 ¡ ¢ ¡ ¢ Pn+1 = j=1 Xij1 ,...,im bki1 ,...,im jt = Xit1 ,...,im bki1 ,...,im tt ½ −εtt Xit1 ,...,im if k ∈ {i1 , . . . , im } = . εtt Xit1 ,...,im if k ∈ / {i1 , . . . , im }

The following theorem relates the Laplacian and the D’Alembertian operators, which are defined in (5) and (6). Theorem 8. Let M be an n-dimensional non null hypersurface in Rn+1 with 1 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 1 ¤= h∆, Et iBn+1 Et (8) t=1

and ∆=

n+1 X

1

h¤, Et iBn+1 Et .

t=1

Pn+1 Proof. Let X ∈ Ξ (M ) and let ∇Ei ∇Ei X = j=1 Xiij Ej . By (5) and (6), ®1 Pn+1 Pn+1 ­Pn 1 t=1 h∆X, Et iBn+1 Et = t=1 i=1 εi ∇Ei ∇Ei X, Et Bn+1 Et o ®1 ­ Pn+1 nPn = t=1 i=1 εi ∇Ei ∇Ei X, Et Bn+1 Et o Pn+1 nPn Pn+1 j 1 = t=1 ε X hE , E i j t Bn+1 Et . ii i=1 i j=1 From the orthonormality condition of the basis of Ξ (M ), Divulgaciones Matem´ aticas Vol. 12 No. 1(2004), pp. 35–52

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Relationship between Laplacian Operator and D’Alembertian Operator Pn+1 nPn

41

o 1 Xiij hEj , Et iBn+1 Et o Pn+1 n Pn+1 t = t=1 −ε1 X11 hEt , Et i + i=2 εi Xiit hEt , Et i Et ® ® Pn+1 ­ Pn+1 j Pn+1 Pn+1 ­ Pn+1 j = − t=1 ε1 j=1 X11 Ej , Et Et + t=1 i=2 εi j=1 Xii Ej , Et Et ® ® Pn+1 Pn+1 ­ Pn+1 ­ = −ε1 t=1 ∇E1 ∇E1 X, Et Et + i=2 εi t=1 ∇Ei ∇Ei X, Et Et Pn+1 = −ε1 ∇E1 ∇E1 X + i=2 εi ∇Ei ∇Ei X = ¤X. Pn+1 1 Therefore, ¤ = t=1 h∆, Et iBn+1 Et . Analogously, Pn+1 1 t=1 h¤X, Et iBn+1 Et ®1 ®1 ª ­ ­ Pn Pn+1 © = t=1 − ε1 ∇Ei ∇Ei X, Et Bn+1 + i=2 εi ∇Ei ∇Ei X, Et Bn+1 Et ª Pn+1 © Pn+1 j Pn Pn+1 1 1 = t=1 −ε1 j=1 X11 hEj , Et iBn+1 + i=2 εi j=1 Xiij hEj , Et iBn+1 Et Eª DP Pn+1 © Pn n+1 j Et = t=1 j=1 Xii Ej , Et i=1 εi © Pn+1 ­ ® ª Pn = i=1 εi t=1 ∇Ei ∇Ei X, Et Et Pn = i=1 εi ∇Ei ∇Ei X = ∆X. t=1

i=1 εi

Pn+1 j=1

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 Laplacian and D’Alembertian operators on vector fields when M is a n-dimensional non null hypersurface in Rn+1 . In a classical way, we denote gij = hEi , Ej i , ¡ ¢1 −1 1 ≤ i, j ≤ n + 1, and g ij = (gij ) . Definition 9. Let M be an n-dimensional non null hypersurface in Rn+1 1 with induced connection ∇. Let E1 , . . . , En be a basis of Ξ (M ). a) The Laplacian ∆ on vector fields of Ξ (M ) is given by: ∆=

n X

g ij ∇Ei ∇Ej .

(10)

i,j=1

b) The D’Alembertian ¤ on vector fields of Ξ (M ) is given by: ¤ = −g 11 ∇E1 ∇E1 −

n X i=2

n ¡ ¢ X g i1 ∇Ei ∇E1 + ∇E1 ∇Ei + g ij ∇Ei ∇Ej . (11) i,j=2

k Naturally, the Bn+1 -product must also be extended to general basis.

Definition 10. Let M be an n-dimensional non null hypersurface in Rn+1 1 with induced connection ∇. Let E1 , . . . , En+1 be an orthonormal basis of Divulgaciones Matem´ aticas Vol. 12 No. 1(2004), pp. 35–52

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k Ξ (M ). For k : 0, . . . , n, the Bn+1 -product is an application on P (M ) to P (M ) which is characterized by: ½ ¡ k ¢ −gjt if k ∈ {i1 , . . . , im } . (12) bi1 ,...,im jt = gjt if k ∈ / {i1 , . . . , im } k

k We denote the Bn+1 -product with h, iBn+1 . k Remark 11. Since h, i is F (M )-bilinear, the Bn+1 -product is F (M )-bilinear too. Pn+1 m Remark 12. If ∇i1 ,...,im X = j=1 Xij1 ,...,im Ej then we have D m Ek Pn+1 k ∇i1 ,...,im X, Et = j=1 Xij1 ,...,im hEj , Et iBn+1 Bn+1 ¡ ¢ Pn+1 = j=1 Xij1 ,...,im bki1 ,...,im jt ( P n+1 − j=1 gjt Xij1 ,...,im if k ∈ {i1 , . . . , im } Pn+1 = g Xj if k ∈ / {i1 , . . . , im }  Dj=1 jt i1 ,...,imE  − ∇m X, Et if k ∈ {i1 , . . . , im } D mi1 ,...,im E = .  ∇i1 ,...,im X, Et if k ∈ / {i1 , . . . , im }

Theorem 13. Let M be an n-dimensional non null hypersurface in Rn+1 1 with induced connection ∇. Let E1 , . . . , En be a basis of Ξ (M ) and let En+1 be the unit normal vector field. Then, ¤=

n+1 X

h∆, Et iBn+1 Et

1

(13)

1

(14)

t=1

and ∆=

n+1 X

h¤, Et iBn+1 Et .

t=1

Proof. Clearly, the Laplacian ∆ and the D’Alembertian ¤ are two operators of P (M ). Pn+1 r Er , where Let X ∈ Ξ (M ) , then ∇Ei ∇Ej X = r=1 Xij ­ ® P n+1 r Xij = s=1 g sr ∇Ei ∇Ej X, Es By (10) and (11), E1 Pn+1 Pn+1 DPn 1 ij ∇ ∇ X, E Et h∆X, E i E = g E E t t t i j t=1 Bn+1 t=1 i,j=1 Bn+1 o ® ­ Pn+1 nPn 1 ij ∇Ei ∇Ej X, Et B = t=1 Et i,j=1 g n+1 n o Pn+1 Pn P n+1 1 ij r = t=1 g X hE , E i r t ij i,j=1 r=1 Bn+1 Et Divulgaciones Matem´ aticas Vol. 12 No. 1(2004), pp. 35–52

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¡ 1¢ o r bij rt Et Xij o Pn+1 n Pn Pn Pn 1j r r r = t,r=1 − j=1 g X1j grt − i=2 g i1 Xi1 grt + i,j=2 g ij Xij grt Et ­ ® Pn ®o ­ Pn+1 nPn i1 1j ∇ ∇ X, E + g ∇ ∇ X, E Et = − t=1 g E E t E E t 1 j i 1 i=2 j=1 o ® ­ Pn+1 nPn ij + t=1 ∇Ei ∇Ej X, Et Et i,j=2 g Pn Pn Pn+1 Pn 1j = − j=1 g ∇E1 ∇Ej X− i=2 g i1 ∇Ei ∇E1 X+ t=1 i,j=2 g ij ∇Ei ∇Ej X = ¤X. Pn+1 1 Therefore, ¤ = t=1 h∆, Et iBn+1 Et . Pn+1 1 In similar way, t=1 h¤X, Ej iBn+1 Et ¡ 1¢ o ¢ ¡ Pn Pn+1 nPn i1 r 1 1j r + g X = − t,r=1 b g X i1 bi1 rt Et 1j rt 1j i=2 j=1 o ¢ ¡ Pn+1 nPn 1 ij r + t,r=1 i,j=2 g Xij bij rt Et o n Pn Pn Pn+1 r r grt Et grt − i=2 g i1 Xi1 = − t,r=1 − j=1 g 1j X1j o Pn+1 nPn ij r + t,r=1 i,j=2 g Xij grt Et ­ ®o Pn+1 nPn Pn ij g = t=1 ∇ ∇ X, E Et = i,j=1 g ij ∇Ei ∇Ej X = ∆X. E E t i j i,j=1 Pn+1 1 Therefore, ∆ = t=1 h¤, Et iBn+1 Et . =

4

Pn+1 nPn t=1

i,j=1

g ij

Pn+1 r=1

Bn+1 -congruence

Let M be an n-dimensional non null hypersurface in Rn+1 with induced con1 nection ∇. From now anwards, we consider E1 , . . . , En+1 vector fields such that En+1 (p) is the unit normal vector at p and {E1 (p) , . . . , En (p)} is a basis of Tp (M ) , at all p ∈ M . Definition 14. Let k1 , . . . , kl be integer numbers such that 0 ≤ k1 < · · · < k1 ,...,kl -product is characterized by: kl ≤ n. The Bn+1 ³ ´ c ,...,kl bki11,...,i = (−1) gjt , (15) m jt

with c = |{kt ; kt ∈ {i1 , . . . , im } and 1 ≤ t ≤ l}|. In classical way, we consider |∅| = 0. It is obvious that 0 ≤ c ≤ min {l, m} ≤ n. the

k ,...,kl

k1 ,...,kl We denote the Bn+1 -product with h, iB1n+1

. Since h, i is F (M )-bilinear,

k1 ,...,kl Bn+1 -product

is too F (M )-bilinear. Pn+1 k ,...,k k1 ,...,kl If Q ∈ P (M ) , we denote t=1 hQ, Et iB1n+1 l Et with Bn+1 (Q) . Divulgaciones Matem´ aticas Vol. 12 No. 1(2004), pp. 35–52

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k1 ,...,kl From Definition 14, we get Bn+1 (Q) is a differential operator of P (M ) . ¢ ¡ 0 Remark 15. Let us note that bi1 ,...,im jt = gjt , at all j, t : 1, . . . , n + 1 and 0 {i1 , . . . , im } ⊂ {1, . . . , n} . Thus Bn+1 (Q) = Q.

Definition 16. Let P, Q be two differential operators of P (M ) . We say that k P is Bn+1 -congruent to Q if P = Bn+1 (Q). ¡ k ¢ ¡ h ¢ Lemma 17. Let bi1 ,...,im uv , bj1 ,...,js rt be as in (15), then ¡

bki1 ,...,im

¢

  −guv grt ¡ h ¢ −guv grt b = j1 ,...,js rt uv  guv grt

if k ∈ {i1 , . . . , im } ∧ h ∈ / {j1 , . . . , js } , if k ∈ / {i1 , . . . , im } ∧ h ∈ {j1 , . . . , js } , in other case.

Proof. It follows from the table: k = i1 h = j1 (−guv ) (−grt ) h = j2 (−guv ) (−grt ) ... ... h = js (−guv ) (−grt ) h∈ / {j1 , . . . , js } (−guv ) grt

... k = im k∈ / {i1 , . . . , im } ... (−guv ) (−grt ) guv (−grt ) ... (−guv ) (−grt ) guv (−grt ) ... ... ... ... (−guv ) (−grt ) guv (−grt ) ... (−guv ) grt guv grt

k1 ,...,kl -products with Bn+1 , where 0 ≤ k1 < We denote the set of all Bn+1 · · · < kl ≤ n.

Concecutive application of products in Bn+1 result in another product in Bn+1 . Its proof is more dull than the idea itself. So we have developed it in steps. k Proposition 18. Let P, Q, R ∈ P (M ) such that P = Bn+1 (R) and R = h Bn+1 (Q), then

 k,h  Bn+1 (Q) if ¡ ¢ k h Q if P = Bn+1 Bn+1 (Q) =  h,k Bn+1 (Q) if

k < h, k = h, k > h,

where 0 ≤ k, h ≤ n. Divulgaciones Matem´ aticas Vol. 12 No. 1(2004), pp. 35–52

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¡ h ¢ k Proof. We will explicitly show that Bn+1 Bn+1 (Q) . Let Pn Pn 1 m Q = i1 =1 qi1 ∇i1 + · · · + i1 ,...,im =1 qi1 ,...,im ∇i1 ,...,im . Since P Pn+1 n+1 k h k P = Bn+1 (R) = t=1 hR, Et iBn+1 Et and R = j=1 hQ, Ej iBn+1 Ej , then ³P ´ n+1 h k P = Bn+1 hQ, E i E j j j=1 Bn+1 ¾ ¶ µ ½ Eh D 1 P P n+1 n k Ej + · · · = Bn+1 j=1 i1 =1 qi1 ∇i1 , Ej Bn+1 µ ½ ¾ ¶ Eh D Pn+1 Pn m k +Bn+1 q ∇ , E Ej . i ,...,i j i1 ,...,im m j=1 i1 ,...,im =1 1 Bn+1

Let us note D that Eh Pn 1 i1 =1 qi1 ∇i1 , Ej

Bn+1

D

Pn

2

i1 ,i2 =1 qi1 ,i2 ∇i1 ,i2 , Ej

−

= Eh

D

Pn

i1 =1 qi1 i1 6=h

Bn+1

=

E D 1 E 1 ∇i1 , Ej − qh ∇h , Ej ,

D 2 E q ∇ , E i ,i j i ,i 1 2 i1 ,i2 =1 1 2

Pn

i1 ,i2 6=h D 2 D 2 E P E D 2 E n i2 =1 qh,i2 ∇h,i2 , Ej − i1 =1 qi1 ,h ∇i1 ,h , Ej + qh,h ∇h,h , Ej ,

Pn

and in the same way, D m Eh Pn q ∇ , E i ,...,i j i ,...,i 1 m i1 ,...,im =1 1 m Bn+1 D m E Pn = i1 ,...,im =1 qi1 ,...,im ∇i1 ,...,im , Ej i1 ,...,im 6=h

D m E q ∇ , E h,i ,...,i j h,i2 ,...,im 2 m i2 ,...,im =1 D m E Pn − i1 ,i3 ,...,im =1 qi1 ,h,i3 ,...,im ∇i1 ,h,i3 ,...,im , Ej D m E Pn − · · · − i1 ,...,im−1 =1 qi1 ,...,im−1 ,h ∇i1 ,...,im−1 ,h , Ej D m E Pn + i3 ,...,im =1 qh,h,i3 ,...,im ∇h,h,i3 ,...,im , Ej D m E Pn + i2 ,i4 ,...,im =1 qh,i2 ,h,i4 ,...,im ∇h,i2 ,h,i4 ,...,im , Ej E D m Pn + · · · + i1 ,...,im−2 =1 qi1 ,...,im−2 ,h,h ∇i1 ,...,im−2 ,h,h , Ej E D m Pn − i4 ,...,im =1 qh,h,h,i4 ,...,im ∇h,h,h,i4 ,...,im , Ej D m E Pn − · · · − i1 ,...,im−3 =1 qi1 ,...,im−3 ,h,h,h ∇i1 ,...,im−3 ,h,h,h , Ej D m E m + · · · + (−1) qh,...,h ∇h,...,h , Ej . −

Pn

We distinguish two cases: a) If k 6= h, Divulgaciones Matem´ aticas Vol. 12 No. 1(2004), pp. 35–52

46

Graciela S. Birman, Graciela M. Desideri

P =

Pn+1

+··· +

à Pn

D

j=1 i1 =1 qi1 i1 6=h à Pn+1 Pn j=1

Ek Bn+1

! Ej −

Pn+1 j=1

Ek D m , E ∇ q j i ,...,i i1 ,...,im m i1 ,...,im =1 1

D 1 Ek qh ∇h , Ej Ej Bn+1 !

Bn+1

i1 ,...,im 6=h

Ej

¶ Ek D m ∇ q , E Ej j h,i2 ,...,im j=1 i2 ,...,im =1 h,i2 ,...,im Bn+1 ¶ µ Ek D m Pn+1 Pn ∇ , E Ej − · · · − j=1 q j i1 ,...,im−1 ,h i1 ,...,im−1 =1 i1 ,...,im−1 ,h Bn+1 Ek D m Pn+1 m + · · · + j=1 (−1) qh,...,h ∇h,...,h , Ej Ej Bn+1 à ! D 1 E Pn+1 Pn = j=1 Ej i1 =1 qi1 ∇i1 , Ej i1 6=h,k E E Pn+1 D 1 Pn+1 D 1 − j=1 qk ∇k , Ej Ej − j=1 qh ∇h , Ej Ej ! à E D m Pn+1 Pn Ej + · · · + j=1 i1 ,...,im =1 qi1 ,...,im ∇i1 ,...,im , Ej i1 ,...,im 6=h,k à ! D m E Pn+1 Pn − j=1 Ej i2 ,...,im =1 qk,i2 ,...,im ∇k,i2 ,...,im , Ej i2 ,...,im 6=h   D m E Pn+1 Pn − j=1  i1 ,...,im−1 =1 qi1 ,...,im−1 ,k ∇i1 ,...,im−1 ,k , Ej  Ej −

µ Pn+1 Pn

1 ∇i 1 , E j

i1 ,...,im−1 6=h

E D m m ∇ , E Ej (−1) q j k,...,k k,...,k j=1 Ã ! D E Pn+1 Pn m − j=1 Ej i2 ,...,im =1 qh,i2 ,...,im ∇h,i2 ,...,im , Ej i2 ,...,im 6=k D m E Pn+1 m + · · · + j=1 (−1) qh,k,...,k ∇h,k,...,k , Ej Ej   D m E Pn+1 Pn − · · · − j=1  i1 ,...,im−1 =1 qi1 ,...,im−1 ,h ∇i1 ,...,im−1 ,h , Ej  Ej

+··· +

Pn+1

i1 ,...,im−1 6=k

E D m m , E Ej ∇ (−1) q j k,...,k,h k,...,k,h j=1 D m E Pn+1 m + · · · + j=1 (−1) qh,...,h ∇h,...,h , Ej Ej Pn 1 1 1 = i1 =1 qi1 ∇i1 − qk ∇k − qh ∇h , Ej + · · ·

+··· +

Pn+1

i 6=h,k

+

P1n

m i1 ,...,im =1 qi1 ,...,im ∇i1 ,...,im i1 ,...,im 6=h,k

−

Pn

m i2 ,...,im =1 qk,i2 ,...,im ∇k,i2 ,...,im i2 ,...,im 6=h

Divulgaciones Matem´ aticas Vol. 12 No. 1(2004), pp. 35–52

Relationship between Laplacian Operator and D’Alembertian Operator

47

Pn

m m m i1 ,...,im−1 =1 qi1 ,...,im−1 ,k ∇i1 ,...,im−1 ,k + · · · + (−1) qk,...,k ∇k,...,k i1 ,...,im−1 6=h Pn m m m − i2 ,...,im =1 qh,i2 ,...,im ∇h,i2 ,...,im + · · · + (−1) qh,k,...,k ∇h,k,...,k − · · · i2 ,...,im 6=k Pn m m m − i1 ,...,im−1 =1 qi1 ,...,im−1 ,h ∇i1 ,...,im−1 ,h + · · · + (−1) qk,...,k,h ∇k,...,k,h i1 ,...,im−1 6=k m m + · · · + (−1) qh,...,h ∇h,...,h n n X X 1 m = q i 1 ∇i 1 + · · · + qi1 ,...,im ∇i1 ,...,im i1 =1 i1 ,...,im =1

−

|

−

n X

{z

1

qi1 ∇i1 + · · · +

i1 =1

n X

{z

1

qi1 ∇i1 + · · · +

m

qi1 ,...,im ∇i1 ,...,im

i1 ,...,im =1

|

n X

}

∃j: ij =h ∧∀t: it 6=k n X

i1 =1

+

m

qi1 ,...,im ∇i1 ,...,im

i1 ,...,im =1

| −

}

∀j: ij 6=h ∧ ij 6=k n X

{z

1

}

∃j: ij =k ∧∀t: it 6=h n X

qi1 ∇i1 + · · · +

i1 =1

m

qi1 ,...,im ∇i1 ,...,im .

i1 ,...,im =1

|

{z

}

∃j: ij =h ∧ ∃t: it =k

From Definition 14, if k < h then the above expression is the same as k,h k,h h,k Bn+1 (Q), thus we write P = Bn+1 (Q). If k > h then we get P = Bn+1 (Q). b) If k = h, Ã Eh D 1 Pn+1 Pn P = j=1 i1 =1 qi1 ∇i1 , Ej i1 6=h

+··· +

Pn+1 j=1

à Pn

!

Bn+1

i1 ,...,im =1 qi1 ,...,im i1 ,...,im 6=h

D

Ej −

Pn+1 j=1

m ∇i1 ,...,im , Ej

D 1 Eh qh ∇h , Ej

Eh Bn+1

Bn+1

! Ej

¶ Eh D m , E ∇ Ej q j h,i ,...,i h,i2 ,...,im 2 m j=1 i2 ,...,im =1 Bn+1 µ Eh D m Pn+1 Pn , E − · · · − j=1 ∇ q j i ,...,i ,h i ,...,i ,h 1 m−1 i1 ,...,im−1 =1 1 m−1

−

µ Pn+1 Pn

+··· +

Pn+1 j=1

D m Eh m (−1) qh,...,h ∇h,...,h , Ej

Bn+1

Ej

Bn+1

Ej

Divulgaciones Matem´ aticas Vol. 12 No. 1(2004), pp. 35–52

¶ Ej

48

Graciela S. Birman, Graciela M. Desideri ! E E D 1 Pn+1 D 1 = j=1 ∇ , E E + q ∇ , E Ej q j j h j i i1 h i1 =1 1 j=1 i1 6=h à ! D m E Pn+1 Pn + · · · + j=1 Ej i1 ,...,im =1 qi1 ,...,im ∇i1 ,...,im , Ej Pn+1

+

Pn+1

Ã

Pn

à Pn

i1 ,...,im 6=h

! E D m Ej i2 ,...,im =1 qh,i2 ,...,im ∇h,i2 ,...,im , Ej

j=1

i2 ,...,im 6=h

+··· +

Pn+1 j=1

 P  n

D

i1 ,...,im−1 =1 qi1 ,...,im−1 ,h i1 ,...,im−1 6=h

m ∇i1 ,...,im−1 ,h , Ej

 E  Ej

D m E Pn+1 m+1 qh,...,h ∇h,...,h , Ej Ej + · · · + j=1 (−1) E´ Pn+1 ³DPn Pn 1 m ∇ ∇ = j=1 q + · · · + q , E Ej i i ,...,i j i1 ,...,im m i1 =1 1 i1 i1 ,...,im =1 1 = Q. 0 (Q) = Q. By Remark 15, P = Bn+1 Corollary 19. Let Q ∈ P (M ) and 0 ≤ k1 , k2 , k3 ≤ n, then  k3  Bn+1 (Q) if k1 = k2   k2   (Q) if k1 = k3 B ³ ´  n+1 k2 ,k3 k1 ,k2 ,k3 k1 Bn+1 Bn+1 (Q) = Bn+1 (Q) if k1 < k2 < k3 .  k2 ,k1 ,k3   B (Q) if k2 < k1 < k3  n+1   k2 ,k3 ,k1 (Q) if k2 < k3 < k1 Bn+1

(17)

Pn Pn 1 m Proof. Let Q = i1 =1 qi1 ∇i1 + · · · + i1 ,...,im =1 qi1 ,...,im ∇i1 ,...,im . In similar way to Proposition 18, n n ³ ´ X X 1 m k2 ,k3 k1 Bn+1 Bn+1 (Q) = qi1 ∇i1 + · · · + qi1 ,...,im ∇i1 ,...,im i1 =1

|

i1 ,...,im =1

{z

∀j: ij 6=k1 ∧ ij 6=k2 ∧ ij 6=k3

−

n X

1 qi1 ∇i1

i1 =1

|

+ ··· +

n X

m

qi1 ,...,im ∇i1 ,...,im

i1 ,...,im =1

{z

}

∃j: ij =k1 ∧∀t: it 6=k2 or ∃j: ij =k1 ∧∀t: it 6=k3 n n X X 1 m qi1 ,...,im ∇i1 ,...,im − qi1 ∇i1 + · · · + i1 ,...,im =1 i1 =1

|

{z

}

∃j: ij =k2 ∧∀t: it 6=k1 or ∃j: ij =k3 ∧∀t: it 6=k1

Divulgaciones Matem´ aticas Vol. 12 No. 1(2004), pp. 35–52

}

Relationship between Laplacian Operator and D’Alembertian Operator

+

n X i1 =1

|

1

qi1 ∇i1 + · · · +

n X

49

m

qi1 ,...,im ∇i1 ,...,im .

i1 ,...,im =1

{z

}

∃j: ij =k1 ∧ ∃t: it =k2 ∧ ∃r: ir =k3

Equality (17) follows from Definition 14. k1 ,...,kl Corollary 20. Let Bn+1 ∈ Bn+1 . Then ³ ³ ³ ´´ ´ k1 ,...,kl kl k1 k2 Bn+1 = Bn+1 Bn+1 · · · Bn+1 ··· .

(18)

Proof. It follows from Proposition 18 and Corollary 19. Now, we define an operation on Bn+1 × Bn+1 . Definition 21. Define ´◦ : Bn+1 × Bn+1 → ³ ³ Bn+1 by ´ k1 ,...,kl h1 ,...,ht k1 ,...,kl h1 ,...,ht Bn+1 ◦ Bn+1 (P ) = Bn+1 Bn+1 (P ) , for all P ∈ P (M ) . Proposition 22. (Bn+1 , ◦) is an abelian group. Proof. By Proposition 18 and corollaries 19 and 20, ◦ is a well-defined operation. By Proposition 18 and Corollary 20, ◦ is a conmutative operation. From 0 -product. Remark 15 and Proposition 18, the identity of (Bn+1 , ◦) is the Bn+1 By Proposition 18 and Corollary 19,  k1 0 ´  Bn+1 ◦ Bn+1 if k2 = k3 ³ k k3 k2 k1 2 ,k3 Bn+1 ◦ Bn+1 ◦ Bn+1 = if k2 < k3 B k1 ◦ Bn+1  n+1 k3 ,k2 k1 Bn+1 ◦ Bn+1 if k2 > k3  k1 B if k = k 2 3  n+1   k2  if k = k B  1 3 n+1   k3  B if k = k  1 2 n+1   k1 ,k2 ,k3   B if k < k < k3 1 2  n+1 k2 ,k1 ,k3 k1 ,k2 k3 Bn+1 if k2 < k1 < k3 = Bn+1 = ◦ Bn+1  k2 ,k3 ,k1   Bn+1 if k2 < k3 < k1    k 1 ,k3 ,k2  B if k1 < k3 < k2  n+1   k3 ,k1 ,k2   B if k 3 < k1 < k2   n+1 k3 ,k2 ,k1 Bn+1 if k3 < k2 < k1 ³ ´ k1 k2 k3 = Bn+1 ◦ Bn+1 ◦ Bn+1 . According to Corollary 20, ◦ is an associative operation. From the above properties, Divulgaciones Matem´ aticas Vol. 12 No. 1(2004), pp. 35–52

50

Graciela S. Birman, Graciela M. Desideri

k1 ,...,kl k1 ,...,kl kl kl k1 k1 Bn+1 ◦ Bn+1 ³ ´ = Bn+1 ³ ◦ · · · ◦ Bn+1´◦ Bn+1 ◦ · · · ◦ Bn+1

kl kl k1 k1 0 0 0 = Bn+1 ◦ Bn+1 ◦ · · · ◦ Bn+1 ◦ Bn+1 = Bn+1 ◦ · · · ◦ Bn+1 = Bn+1 . ³ ´−1 k1 ,...,kl k1 ,...,kl Therefore Bn+1 = Bn+1 .

½ Remark 23. The order of

k1 ,...,kl Bn+1

is

1 2

k1 ,...,kl 0 if Bn+1 = Bn+1 , in other case.

Theorem 24. Let P and Q be two Bn+1 -congruent operators. There exk1 ,...,kl k1 ,...,kl ists Bn+1 ∈ Bn+1 such that P = Bn+1 (Q) ,and this is an equivalence relationship. Proof. Let P, Q, R ∈ P (M ). By Proposition 22, for all P ∈ P (M ) there exists 0 0 (P ) . Therefore, P is Bn+1 -congruent to P, ∈ Bn+1 such that P = Bn+1 Bn+1 for all P ∈ P (M ) . k such that If P is Bn+1 -congruent to Q then there exists ¡ k Bn+1 ¢∈ Bn+1 k k k (P ) ,that P = Bn+1 (Q). From Proposition 22 Q = Bn+1 Bn+1 (Q) = Bn+1 k k (P ) . Therefore, Q is Bn+1 ∈ Bn+1 such that Q = Bn+1 is, there exists Bn+1 congruent to P. h k (Q) , it follows from Proposition 18 that (R) and R = Bn+1 If P = Bn+1 k,h k,h P = Bn+1 (Q) , with Bn+1 ∈ Bn+1 .

5

Examples

Lastly, we show some examples of the Laplacian and D’Alembertian operators k -products. on vector fields and Bn+1 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 in R31 . Example 25. Let x1 , x2 , x3 be a coordinate system in R31 such that {∂1 , ∂2 , ∂3 } ∂ is an orthonormal basis for R31 , where ∂i = ∂x . The pseudosphere S12 in R31 i © ª 2 3 is the surface defined by S1 = (x1 , x2 , x3 ) ∈ R1 : − x21 + x22 + x23 = 1 . S12 can be parametrized as x1 = sinh ω, x2 = cos θ cosh ω, x3 = sin θ cosh ω, where ω ∈ R and 0 ≤ θ < 2π. The tangent vectors are expressed as follows: ∂ω = ∂∂ω = cosh ω ∂1 + sinh ω cos θ ∂2 + sinh ω sin θ ∂3 , ∂θ = ∂∂θ = − cosh ω sin θ ∂2 + cosh ω cos θ ∂3 . The unit normal vector to the surface S12 at (ω, θ) is N = (sinh ω, cosh ω cos θ, cosh ω sin θ) . Divulgaciones Matem´ aticas Vol. 12 No. 1(2004), pp. 35–52

Relationship between Laplacian Operator and D’Alembertian Operator

51

Hence h∂ω , ∂ω i = −1, h∂ω , ∂θ i = 0, h∂θ , ∂θ i = cosh2 ω and hN, N i = 1. According to Definition 9, the Laplacian operator on vector fields, ∆, for S12 is given by: 1 ∆ = −∇∂ω ∇∂ω + ∇∂θ ∇∂θ . cosh2 ω The mean curvature vector field H for S12 is given by H = −N (cf. [3]). By applying to H the Laplacian operator on vector fields for S12 , we obtain ∆H = 2 N − tanh ω ∂ω = −2H − tanh ω∂ω , (cf. [3]). Example 26. Let x1 , x2 , x3 be a coordinate system in R31 such that {∂1 , ∂2 , ∂3 } ∂ is an orthonormal basis for R31 , where ∂i = ∂x . The cylinder R11 × S 1 in R31 i ª © is the surface defined by R11 × S 1 = (x1 , x2 , x3 ) ∈ L3 : x22 + x23 = 1 . R11 × S 1 can be parametrized as x1 = t, x2 = cos θ, x3 = sin θ, where t ∈ R and 0 < θ < 2π. The tangent vectors are expressed as ∂t = ∂1 , ∂θ = − sin θ ∂2 + cos θ ∂3 . The unit normal vector to the surface R11 ×S 1 at (t, θ) is N = (0, cos θ, sin θ) . Hence, h∂t , ∂t i = −1, h∂θ , ∂t i = 0, h∂θ , ∂θ i = 1 and hN, N i = 1. According to Definition 9, the D’Alembertian operator on vector fields for R11 × S 1 is given by ¤ = ∇∂t ∇∂t + ∇∂θ ∇∂θ . The mean curvature vector field H is given by H = − 12 N (cf. [3]). Since ∇∂ω ∇∂ω H = 0 and ∇∂θ ∇∂θ H = − 21 ∇∂θ (∂θ ) = 12 N, applying to H the D’Alembertian operator on vector fields for R11 × S 1 we obtain ¤H = 1 2 N = −H. Example 27. Let ∆ be the Laplacian operator on vector fields for a surface M in R31 . We denote the Bn+1 -equivalence class of ∆ with [∆] . If ∆ is defined by ∆ = g 11 ∇∂1 ∇∂1 + g 12 ∇∂1 ∇∂2 + g 21 ∇∂2 ∇∂1 + g 22 ∇∂2 ∇∂2 , then, 0 Bn+1 (∆) = ∆, ¡ ¢ 1 (∆) = − g 11 ∇∂1 ∇∂1 + g 12 ∇∂1 ∇∂2 + g 21 ∇∂2 ∇∂1 + g 22 ∇∂2 ∇∂2 Bn+1 = −∆ + 2g 22 ∇∂2 ∇∂2 , ¡ ¢ 2 Bn+1 (∆) = g 11 ∇∂1 ∇∂1 − g 12 ∇∂1 ∇∂2 + g 21 ∇∂2 ∇∂1 + g 22 ∇∂2 ∇∂2 = −∆ + 2g 11 ∇∂1 ∇∂1 , and 12 Bn+1 (∆) = −g 11 ∇∂1 ∇∂1 + g 12 ∇∂1 ∇∂2 + g 21 ∇∂2 ∇∂1 − g 22 ∇∂2 ∇∂2 ¡ ¢ B 1 (∆)+B 2 (∆) = ∆ − 2 g 11 ∇∂1 ∇∂1 + g 22 ∇∂2 ∇∂2 = − n+1 2 n+1 . are Bn+1 -congruent operators. Therefore, o n 2 [∆] = ∆, ¤, Bn+1 (∆) , −

2 ¤+Bn+1 (∆) 2

.

Divulgaciones Matem´ aticas Vol. 12 No. 1(2004), pp. 35–52

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Graciela S. Birman, Graciela M. Desideri

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 appear. [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. Monthly 91 (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 Differential Geometry (December, 1994), 279–290, Math. Appl., 350, Kluwer Acad. Publ., Dordrecht, 1996. [7] O’Neill, B. Semi-Riemannian Geometry with Applications to Relativity, Academic Press, New York, 1983.

Divulgaciones Matem´ aticas Vol. 12 No. 1(2004), pp. 35–52