Divulgaciones Matemáticas Vol. 12 No. 1(2004), pp. 35–52
Relationship between Laplacian
Operator and D’Alembertian Operator
Relación 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 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 Matemática y
de la Fı́sica. Sumada a su relevancia, ambos operadores se destacan por
ser muy utilizados en el cálculo vectorial.
En este artı́culo mostramos la relación entre los operadores Laplaciano y D’Alembertiano tanto sobre funciones como sobre campos
vectoriales definidos sobre hipersuperficies del espacio Lorentziano mk ,...,kl
dimensional. Además, 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 Cientı́ficas y Tecnológicas de
la República 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 Matemáticas Vol. 12 No. 1(2004), pp. 35–52
Relationship between Laplacian Operator and D’Alembertian Operator
2
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
Divulgaciones Matemáticas Vol. 12 No. 1(2004), pp. 35–52
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38
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
Divulgaciones Matemáticas Vol. 12 No. 1(2004), pp. 35–52
Relationship between Laplacian Operator and D’Alembertian Operator
39
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 Matemáticas Vol. 12 No. 1(2004), pp. 35–52
40
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 Matemáticas Vol. 12 No. 1(2004), pp. 35–52
(9)
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 Matemáticas Vol. 12 No. 1(2004), pp. 35–52
42
Graciela S. Birman, Graciela M. Desideri
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 Matemáticas Vol. 12 No. 1(2004), pp. 35–52
Relationship between Laplacian Operator and D’Alembertian Operator
43
¡ 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 Matemáticas Vol. 12 No. 1(2004), pp. 35–52
44
Graciela S. Birman, Graciela M. Desideri
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 Matemáticas Vol. 12 No. 1(2004), pp. 35–52
(16)
Relationship between Laplacian Operator and D’Alembertian Operator
45
¡ 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 Matemáticas Vol. 12 No. 1(2004), pp. 35–52
46
Graciela S. Birman, Graciela M. Desideri
P =
Pn+1
+··· +
Ã
Pn
D
j=1
i1 =1 qi1
i1 6=h
Ã
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
Ã
!
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
!
Ã
E
D m
Pn+1 Pn
Ej
+ · · · + j=1
i1 ,...,im =1 qi1 ,...,im ∇i1 ,...,im , Ej
i1 ,...,im 6=h,k
Ã
!
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 Matemáticas 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
Ã
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 Matemáticas 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
Ã
!
D m
E
Pn+1 Pn
+ · · · + j=1
Ej
i1 ,...,im =1 qi1 ,...,im ∇i1 ,...,im , Ej
Pn+1
+
Pn+1
Ã
Pn
Ã
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 Matemáticas 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 Matemáticas 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 Matemáticas 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 Matemáticas Vol. 12 No. 1(2004), pp. 35–52
52
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 Matemáticas Vol. 12 No. 1(2004), pp. 35–52