Gen. Math. Notes, Vol. 18, No. 1, September, 2013, pp. 64-87
ISSN 2219-7184; Copyright © ICSRS Publication, 2013
www.i-csrs.org
Available free online at http://www.geman.in
Lorentzian Beltrami-Meusnier Formula
Soley Ersoy1 and Murat Tosun2
1, 2
Department of Mathematics
Faculty of Arts Sciences, Sakarya University
54187, Sakarya, Turkey
1
E-mail: sersoy@sakarya.edu.tr
2
E-mail: tosun@sau.edu.tr
(Received: 4-6-13 / Accepted: 9-7-13)
Abstract
In this paper, the relationship between the arbitrary sectional curvatures and
normal sectional curvatures of generalized timelike ruled surface with timelike
generating space in n-dimensional Minkowski space ℝ 1n are investigated. Three
different types of relation are obtained and called I., II., and III. type of
Lorentzian Beltrami-Meusnier formula.
Keywords: Sectional Curvature, Ruled surface, Beltrami-Meusnier Formula.
1
Introduction
Meusnier formula and Beltrami formula are well known theorems from classical
surface theory, (see for example, [3], [7], etc.) An analogue of Meusnier formula
was obtained in [6] by the way of applying this formula to tangential sections of
generalized ruled surface in n − dimensional Euclid space Εn , and was called as
Beltrami-Meusnier formula. In [5], authors calculated the first fundamental form
and the metric coefficients of generalized timelike ruled surfaces with timelike
generating space given in [1] and [2]. Moreover, according to Christoffel Symbols,
Lorentzian Beltrami-Meusnier Formula
65
Riemann-Christoffel curvatures of these surfaces were obtained. Furthermore, the
principal sectional curvatures of generalized timelike ruled surface with timelike
generating space and central ruled surface were found to be with respect to the
determinant of the first fundamental form of the surface for spacelike and timelike
tangential sections, separately. Lastly, four different types of Lorentzian BeltramiEuler formulas were constituted for generalized timelike ruled surface with
timelike generating space in [5]. In this paper, we will consult to these theorems,
in case of necessity. To avoid from repetition of the basic concepts in MinkowskiLorentz space, which are known from [8] and [9], will not be repeated.
2
Generalized Timelike Ruled Surfaces with Timelike
Generating Space in n-Dimensional Minkowski Space
A (k +1) − dimensional timelike ruled surface in n − dimensional Minkowski
space, ℝ 1n is given parametrically as
k
φ ( t , u1 ,...uk ) = α ( t ) + ∑ ui ei ( t )
(2.1)
i =1
rank (φt , e1 ( t ) ,..., ek ( t ) ) = k + 1
where the base curve α is a spacelike curve and generating space
E k (t ) = Sp{e1 (t ),..., ek (t )} is a timelike subspace.
{
•
•
}
A ( t ) = Sp e1 ( t ) ,..., ek ( t ) , e1 ( t ) ,..., e k ( t )
is called asymptotic bundle of M with respect to Ek (t ) . It is clear that A(t ) is a
timelike subspace. If dim A ( t ) = k + m , 0 ≤ m ≤ k , then one can find an
orthonormal
base
A(t )
containing
Ek (t )
such
Furthermore, for the orthonormal
for
{e1 (t ),...ek (t ), a k +1 (t ),...a k + m (t )} .
{e1 ( t ) ,… , ek ( t )} , the following equations hold [2]
i
k
eσ = ∑ ασµ eµ + κσ ak +σ
k
em + ρ = ∑ α ( m + ρ ) µ eµ
base
, 1≤σ ≤ m
µ =1
i
as
(2.2)
, 1≤ ρ ≤ k − m
µ =1
where
and
The subspace
ε µα vµ = −εν α µ v
(2.3)
κ1 > κ 2 > … > κ m > 0.
(2.4)
66
Soley Ersoy et al.
{
•
•
}
•
Sp e1 ( t ) ,..., ek ( t ) , e1 ( t ) ,..., e k ( t ) , α ( t )
is called tangential bundle of M with respect to Ek (t ) and denoted as T (t ) . It can
be easily seen that
k + m ≤ boyT (t ) ≤ k + m + 1 , 0 ≤ m ≤ k .
If dimT ( t ) = k + m , then {e1 (t ),...ek (t ), a k +1 (t ),...a k + m (t )} is the base for both
asymptotic and tangential bundles. If dim T ( t ) = k + m + 1 , then one can find an
orthonormal base for T (t ) as {e1 (t ),...ek (t ), a k +1 (t ),...a k + m (t ), a k + m+1 (t )} . For both
cases tangential bundle T (t ) is a timelike subspace, [2].
Let dim T (t ) = k + m + 1 . In this case (k + 1) − dimensional timelike ruled surface
has a (k − m ) − dimensional subspace called central space of M and denoted as
Z k − m (t ) ⊂ E k (t ) . The subspace Z k − m (t ) is either spacelike or timelike subspace. If
the base curve α of M is chosen to be the base curve and Z k − m (t ) to be
generating space, we get a (k − m + 1) − dimensional ruled surface contained by
M in ℝ 1n . This is denoted by Ω and called the central ruled surface. If Z k − m (t ) is
spacelike (timelike) then central ruled surface Ω becomes spacelike (timelike)
ruled surface, [1].
A base curve α of ( k + 1) -dimensional ruled surface M is a base curve of central
surface Ω ⊂ M as well, iff its tangent vector has the form
i
k
α ( t ) = ∑ ζ ν eν + ηm +1ak + m +1
ν =1
,
ηm +1 ≠ 0
(2.5)
where ηm +1 ≠ 0 , a k +m +1 is a unit vector well defined up to the sign with the
property that {e1 ,..., ek , a k +1 ,..., a k + m , a k + m +1 } is an orthonormal base of the
tangential bundle of M , [1]. The tangential space of M at the central points is
perpendicular to the asymptotic bundle A ( t ) . Considering the equation (2.1) at
the central point of central ruled surface Ω ⊂ M we see that
uσ = 0
,
1≤σ ≤ m.
(2.6)
For the spacelike base curve α of (k + 1) − dimensional timelike ruled surface M ,
if η m+1 ≠ 0 we call m − magnitudes
Lorentzian Beltrami-Meusnier Formula
Pσ =
ηm +1
κσ
67
,
1≤σ ≤ m
(2.7)
is called the σ th principal distribution parameter of M , [1].
The canonical base of the tangential of M is
k
m
 k 


ζ
+
α
u
e
+
∑  ν ∑ νµ µ  ν ∑ uσ κσ ak +σ + η m +1ak + m +1 , e1 , e2 ,… , ek  .
µ =1
σ =1
 ν =1 


(2.8)
We can evaluate the first fundamental form of M and the metric coefficients with
respect to this canonical base. In conventional notation we choose u0 = t and
calculate the metric coefficients of M as follows;
k


g 00 = − g + ∑ εν  ζ ν + ∑ ανµ uµ 
ν =1
µ =1


k
2
k


gν 0 = εν  ζ ν + ∑ ανµ uµ 
µ =1


gνµ = εν δνµ
m
g = det  g ij  = −∑ ( uσ κσ ) − η m2 +1
2
,
1 ≤ν ≤ k
,
0 ≤ν , µ ≤ k
,
0 ≤ i, j ≤ k .
(2.9)
σ =1
In addition to these, the coefficients of the inverse matrix  g ij  of the matrix
 g ij  , 0 ≤ i, j ≤ k , are as follows
g 00 = − g −1
k


gν 0 =  ζ ν + ∑ ανµ uµ  g −1
µ =1


, 1 ≤ν ≤ k
k
k




gνλ =  εν δνλ g −  ζ ν + ∑ ανµ uµ  ζ λ + ∑ α λµ u µ   g −1


µ =1
µ =1




, 1 ≤ ν , λ ≤ k.
(2.10)
Substituting the equations (2.9) and (2.10) into the Koszul equation (given in [4])
the Christoffel symbols are obtained for 1 ≤ ν , µ , λ ≤ k ,
68
Soley Ersoy et al.
Γ 000 =
k
k

 ∂g
1  ∂g
+
ζ
+

 ν ∑ ανµ uµ 
∑
2 g  ∂u0 ν =1 
µ =1
 ∂uν
λ
Γ 00
=
k
k
k
  ∂g

 ∂g
1  
+ ∑  ζ ν + ∑ ανµ uµ 
 −  ζ λ + ∑ α λµ u µ  
2 g  
µ =1
µ =1
  ∂u0 ν =1 
 ∂uν
0
Γνµ
= Γ 0µν

,




k i
k
 i
 k 

∂g  
1
+2 g   ζ λ + ∑ α λµ uµ  + ∑  ζ ν + ∑ ανµ uµ  α λν + ε λ
 ,

2 ∂uλ  
µ =1
µ =1
 ν =1 


= 0,
(2.11)
λ
Γνµ
= Γ λµν = 0,
Γ λ0 0 = Γ 00 λ =
1 ∂g
,
2 g ∂uλ
Γνλ 0 = Γ λ0ν =
k

 ∂g
1  
+ 2 g (α λν )  .
 −  ζ λ + ∑ α λµ uµ 
2 g  
µ =1
 ∂uν

Adopting that the base of tangential space in the neighborhood of coordinate
∂
system {u0 , u1 ,… , uk } is {∂ 0 , ∂1 ,… , ∂ k } (
= ∂ i , 0 ≤ i ≤ k ), the Riemann
∂ui
curvature tensor of the generalized timelike surface M is given by
k
k
 ∂ r

∂ r k s r
Rhlij = ∑ g rh 
Γ jl −
Γil − ∑ Γil Γ js + ∑ Γ sjl Γisr .
 ∂u

∂u j
r =0
s =0
s =0
 i

(2.12)
Considering the equations (2.9) and (2.11), Rij 00 , Rijνµ , Rν 0 µ 0 are found to be (in
terms of the determinant of the first fundamental form of M , the first and second
order partial differentials of g )
Rij 00 = 0
, 0 ≤ i, j ≤ k
Rijνµ = 0
, 0 ≤ i, j ≤ k , 1 ≤ ν , µ ≤ k
Rν 0 µ 0 =
1 ∂2 g
1 ∂g ∂g
−
2 ∂uν ∂uµ 4 g ∂uν ∂uµ
, 1 ≤ν , µ ≤ k,
(2.17)
Lorentzian Beltrami-Meusnier Formula
69
3
Lorentzian
Beltrami-Meusnier
Formula
for
Generalized Timelike Ruled Surfaces with Timelike
Generating Space in n-Dimensional Minkowski Space
Two-dimensional subspace Π of ( k + 1) − dimensional time-like ruled surface at
the point ξ ∈ TM (ξ ) is called tangent section of M at point ξ . If v and w form
a basis of the tangent section Π , then Q(v , w) = v , v w, w − v , w is a non-zero
quantity iff Π is non-degenerate. This quantity represents the square of the
Lorentzian area of the parallelogram determined by v and w . Using the square of
the Lorentzian area of the parallelogram determined by the basis vectors {v , w} ,
one has the following classification for the tangent sections of the time-like ruled
surfaces:
2
Q ( v , w ) = v , v w, w − v , w < 0 , (time-like plane),
2
Q ( v , w ) = v , v w, w − v , w
2
= 0 , (degenerate plane),
Q ( v , w ) = v , v w, w − v , w
2
> 0 , (space-like plane).
For the non-degenerate tangent section Π given by the basis {v , w} of M at the
point ξ
Kξ ( v , w ) =
Rvw v , w
Q ( v , w)
=
∑R
ijkm
βi γ j β k γ m
v , v w, w − v , w
(3.1)
2
is called sectional curvature of M at the point ξ , where v = ∑ β i
w = ∑γ j
∂
and
∂xi
∂
. Here the coordinates of the basis vectors v and w are
∂x j
( β0 , β1 ,…, βk )
and ( γ 0 , γ 1 ,… , γ k ) , respectively, [4].
Let the base curve α of timelike ruled surface M with timelike generating space
be also a base curve of central ruled surface Ω of M in ℝ 1n . In this case, a
normal tangent vector n of M orthogonal to Ek ( t ) is defined to be
m
n = ∑ uσ κσ ( t ) ak +σ ( t ) + ηm +1ak + m +1 ( t )
,
σ =1
(ηm+1 ≠ 0 )
(3.2)
at the point ∀ξ ( t , uν ) , where this normal tangent vector field is always spacelike
since it is orthogonal to generating space Ek ( t ) .
70
Soley Ersoy et al.
The central ruled surface Ω of the generalized timelike ruled surface M with
timelike generating space in ℝ 1n is either spacelike or timelike. Therefore, at the
central point ∀ζ ∈ Ω , any unit vector a is defined to be
a = λ0
n
+ λ1e1 + λ2 e2 + … + λs −1es −1 + λs es + λs +1es +1 + … + λk ek ,
n
a =1
where a and eσ , 1 ≤ σ ≤ m , are linearly independent. Here es is a timelike unit
vector which is either in subspace Fm ( t ) or in central subspace Z k −m ( t ) and we
write
a, a = λ02 + λ12 + λ22 + … + λs2−1 − λs2 + λs2+1 + … + λk2 = ±1.
This means that a is either unit spacelike or timelike vector. Thus, there exist the
following cases depending on whether central ruled surface Ω of M is spacelike
(i.e., Z k − m ( t ) , spacelike) or timelike (i.e., Z k − m ( t ) , timelike):
(a)
While the central space Z k − m ( t ) is spacelike,
(a1) Unit vector a is either spacelike or
(a2) Unit vector a is timelike.
(b)
While central Z k − m ( t ) is timelike,
(b1) Unit vector a is either spacelike or
(b2) Unit vector a is timelike.
Now let us consider these cases separately.
(a1) Let the central space be spacelike and unit vector a be spacelike vector:
Suppose that M is generalized timelike ruled surface with timelike generating
space and spacelike central ruled surface in ℝ 1n . The principle frame of M is
{e1 ,…, es ,…, em ,…, ek } and
generating space Ek ( t ) of
the normal tangent vector n is orthogonal to the
M , so at the central point ∀ζ ∈ Ω any spacelike
vector a can be written as follows
a = coshψ 0
k
n
+ ∑ coshψ ν eν + sinhψ s es and
n ν =1
ν ≠s
k
∑ cosh
ν
=0
ν ≠s
2
θν − sinh 2 θ s = 1
(3.3)
Lorentzian Beltrami-Meusnier Formula
71
where es , 1 ≤ s ≤ m , is a timelike vector in the subspace Fm ( t ) and the angles
ψ 0 ,ψ 1 ,… ,ψ s ,… ,ψ k are the hyperbolic angles between the spacelike unit vector a
and the vectors n, e1 ,… , es ,… , ek .
(a2) Let the central space be spacelike and unit vector a be timelike vector:
Suppose that M is generalized timelike ruled surface with timelike generating
space and spacelike central ruled surface. As in case (a1), at the point ∀ζ ∈ Ω
any timelike vector a is written as follows
a = sinhψ 0
k
n
+ ∑ sinhψ ν eν + coshψ s es and
n ν =1
ν ≠s
k
∑ sinh
ν
=0
ν ≠s
2
θν − cosh 2 θ s = −1
(3.4)
such that es , 1 ≤ s ≤ m , is a timelike vector in subspace Fm ( t ) and the hyperbolic
angles between the timelike unit vector a and vectors n, e1 ,… , es ,… , ek are
ψ 0 ,ψ 1 ,… ,ψ s ,… ,ψ k , respectively.
(b1) Let the central space be timelike and unit vector a be spacelike vector:
Suppose that M is generalized timelike ruled surface with timelike generating
space and timelike central ruled surface. At the ∀ζ ∈ Ω point any spacelike unit
vector a can be written as follows
a = coshψ 0
n
+
n
k
∑
ν
=1
ν ≠ m+ s
coshψ ν eν + sinhψ m + s em + s and
k
∑
ν
=0
ν ≠ m+ s
cosh 2 θν − sinh 2 θ m + s = 1 (3.5)
where em+ s , 1 ≤ s ≤ k − m , is timelike vector in the central space Z k −m ( t ) and the
angles ψ 0 ,ψ 1 ,… ,ψ m + s ,… ,ψ k are the hyperbolic angles between spacelike unit
vector a and base vectors n, e1 ,… , em + s ,… , ek , respectively.
(b2) Let the central space be timelike and unit vector a be timelike vector:
Suppose that M is generalized timelike ruled surface with timelike generating
space and timelike central ruled surface in ℝ 1n . As in the case (b1) we take n is
orthogonal to generating space Ek ( t ) so that any timelike vector n can be written
as follows
a = sinhψ 0
n
+
n
k
∑
ν
=1
ν ≠m+ s
sinhψ ν eν + coshψ m + s em + s and
k
∑
ν
=0
ν ≠m+ s
sinh 2 θν − cosh 2 θ m + s = −1 (3.6)
72
Soley Ersoy et al.
where em+ s , 1 ≤ s ≤ k − m , is timelike vector in the central space Z k −m ( t ) and the
hyperbolic angles ψ 0 ,ψ 1 ,… ,ψ m + s ,… ,ψ k are the angles between spacelike unit
vector a and the base vectors n, e1 ,… , em + s ,… , ek , respectively.
Therefore, taking the cases (a1) to (b2) into consideration, we give the following
theorems below, about the relationship between the curvatures of the section
( eσ , a ) , 1 ≤ σ ≤ m , and the σ th principal section ( eσ , n ) , 1 ≤ σ ≤ m .
Theorem 3.1: Let a be any spacelike unit vector of M generalized timelike ruled
surface with timelike generating space and with spacelike central ruled surface
and es , 1 ≤ s ≤ m , be timelike base vector within subspace Fm ( t ) , respectively.
i. Given the unit vector a which is linearly independent with the σ th spacelike
vector eσ , 1 ≤ σ ≤ m , ( σ ≠ s ), there exists the following relation between the
curvatures of timelike section ( eσ , a ) and the σ th spacelike principal section
( eσ , n ) at the point (ζ + ues ) ∈ M
(1 − cosh ψ ) K
2
σ
ζ + ueσ
( eσ , a ) = cosh 2 ψ 0 Kζ +ue ( eσ , n )
σ
where ψ 0 is the hyperbolic angle between a and n and ψ σ , 1 ≤ σ ≤ m , ( σ ≠ s )
are the hyperbolic angles between a and eσ , respectively.
ii. Given any unit vector a which is linearly independent with the s th timelike
vector es , 1 ≤ s ≤ m , at the point (ζ + ues ) ∈ M , there exists the following
relation between the curvatures of timelike section ( es , a ) and the s th timelike
principal section ( es , n )
(1 + sinh ψ ) K
2
s
ζ + ues
( es , a ) = cosh 2 ψ 0 Kζ +ue ( es , n )
s
where the hyperbolic angles between a and n and between a and es are ψ 0 and
ψ s respectively.
Proof: Let a be any spacelike unit vector of generalized timelike ruled surface
M with timelike generating space and with spacelike central ruled surface in
n − dimensional Minkowski space ℝ 1n .
i. If we consider the equation (3.1) then the curvature of timelike section is
Lorentzian Beltrami-Meusnier Formula
Kζ +ueσ ( eσ , a ) =
( eσ , a )
73
βσ βσ λ0 λ0 Rσ 0σ 0
eσ , eσ
a, a − eσ , a
2
, 1≤σ ≤ m , σ ≠ s .
(3.7)
at the point (ζ + ueσ ) ∈ M .
The coordinates of the ν th base vector eν , 1 ≤ ν ≤ k , and spacelike unit vector a
given by the equation (3.3) are ( β 0 , β1 ,… , βν ,… , β k ) and ( γ 0 , γ 1 ,… , γ s ,… , γ k ) ,
respectively. Thus, we may write
β 0 = eν , e0 = 0
and
γ 0 = a, e0 =
, βν = eν , eν = εν
coshψ 0
n
γ s = a, es = sinhψ s
, γ ν = a, eν = coshψ ν
, 1 ≤ν ≤ k
, 1 ≤ν ≤ k
, ν ≠s
, 1 ≤ s ≤ m.
When we substitute these last equations together with the equation (2.17) into the
equation (3.7) we find
2
cosh 2 ψ 0  1 ∂ 2 g 1  ∂g  


−
2
 2 ∂uσ2 4 g  ∂uσ  
n

.
Kζ +ueσ ( eσ , a ) =
2
1 − cosh ψ σ
Since n = − g , we obtain
2
2
 1 ∂2 g
1  ∂g  


cosh ψ 0 −
+
 2 g ∂uσ2 4 g 2  ∂uσ  


( eσ , a ) =
1 − cosh 2 ψ σ
2
Kζ +ueσ
Considering equation (25) in ref. [5] we find the relation between curvature of the
σ th spacelike principal section ( eσ , n ) and curvature of timelike section ( eσ , a ) .
ii. Similarly, at the point (ζ + ues ) ∈ M , the timelike sectional curvature is given
by
β s β s λ0 λ0 Rs 0 s 0
, 1≤ s ≤ m, s ≠ σ .
(3.8)
Kζ +ues ( es , a ) =
2
es , es a, a − es , a
Substituting the equation (2.17) into the equation (3.8) and considering that
n = − g we obtain
2
74
Soley Ersoy et al.
2
 1 ∂2 g
1  ∂g  
(1 + sinh ψ s ) Kζ +ues ( es , a ) = cosh ψ 0  2 g ∂u 2 − 4 g 2  ∂u   .
s
 s 

2
2
Therefore, taking the equation (26) in ref. [5] into consideration we get the
relation between curvature of the s th timelike principal section ( es , n ) and
curvature of timelike section ( es , a ) .
Theorem 3.2: Let M be a generalized timelike ruled surface with timelike
generating space and with spacelike central ruled surface in n − dimensional
Minkowski space ℝ 1n . a and es , 1 ≤ s ≤ m , be any timelike unit vector of M and
timelike base vector within subspace Fm ( t ) , respectively. Given any timelike unit
vector a which is linearly independent with eσ , 1 ≤ σ ≤ m , σ ≠ s , at the point
(ζ + ueσ ) ∈ M , there exists the following relation between curvature of timelike
section ( eσ , a ) and curvature of the σ th principal spacelike section ( eσ , n )
(1 + cosh ψ ) K
2
σ
ζ + ueσ
( eσ , a ) = − sinh 2 ψ 0 Kζ +ue ( eσ , n ) .
σ
So that the angles ψ 0 and ψ σ are the hyperbolic angles between a and n and
between a and eσ , respectively.
Proof: At the point (ζ + ueσ ) ∈ M the curvature of timelike section ( eσ , a )
becomes
βσ βσ λ0 λ0 Rσ 0σ 0
, 1 ≤ σ ≤ m , σ ≠ s. ,
(3.9)
Kζ +ueσ ( eσ , a ) =
2
eσ , eσ a, a − eσ , a
The coordinates of the ν th base vector eν , 1 ≤ ν ≤ k , and timelike unit vector a
given by equation (3.4) are ( β 0 , β1 ,… , βν ,… , β k ) and ( γ 0 , γ 1 ,… , γ s ,… , γ k ) , and
we write the following equations
β 0 = eν , e0 = 0
and
γ 0 = a, e0 =
sinhψ 0
n
γ s = a, es = coshψ s
, βν = eν , eν = εν
, γ ν = a, eν = sinhψ ν
, 1 ≤ s ≤ m.
, 1 ≤ν ≤ k,
, 1 ≤ν ≤ k
, ν ≠ s,
Lorentzian Beltrami-Meusnier Formula
75
Substituting these equations together with the equation (2.17) into the equation
(3.9) and considering that n = − g , we reach
2
2
 1 ∂2 g
1  ∂g  
( −1 − cosh ψ σ ) Kζ +ueσ ( eσ , a ) = sinh ψ 0  − 2 g ∂u 2 + 4 g 2  ∂u   .
σ
 σ 

We obtain, therefore, a relation between curvature of spacelike σ th principal
section ( eσ , n ) , 1 ≤ σ ≤ m , σ ≠ s , given by the equation (25) in ref. [5] and the
2
2
curvature of timelike section ( eσ , a ) and this completes the proof.
Theorem 3.3: Let M be a generalized timelike ruled surface with timelike
generating space and with spacelike central ruled surface in n − dimensional
Minkowski space ℝ 1n . a and em + s , 1 ≤ s ≤ k − m , be any spacelike unit vector of
M and timelike base vector within the central space Z k − m ( t ) , respectively. Given
the spacelike unit vector a which is linearly independent of spacelike base
vector eσ , 1 ≤ σ ≤ m , at the point (ζ + ueσ ) ∈ M , there exist the following relation
between the curvatures of timelike section ( eσ , a ) and the σ th spacelike principal
section ( eσ , n )
(1 − cosh ψ ) K
2
σ
ζ + ueσ
( eσ , a ) = cosh 2 ψ 0 Kζ +ue ( eσ , n )
σ
where the hyperbolic angles between a and n , between a and eσ are ψ 0 and
ψ σ , respectively.
Proof: At the point (ζ + ueσ ) ∈ M the curvature of timelike section ( eσ , a ) ,
1 ≤ σ ≤ m , is given by
Kζ +ueσ ( eσ , a ) =
βσ βσ λ0 λ0 Rσ 0σ 0
eσ , eσ
a, a − eσ , a
2
, 1 ≤ σ ≤ m.
(3.10)
The coordinates of the ν th base vector eν , 1 ≤ ν ≤ k , and spacelike unit vector are
( β0 , β1 ,… , βν ,…, βk )
and ( γ 0 , γ 1 ,… , γ m + s ,… , γ k ) , respectively so that
β 0 = eν , e0 = 0
and
γ 0 = a, e0 =
coshψ 0
n
, βν = eν , eν = εν
, γν = a, eν = coshψ ν
γ m + s = a, em + s = sinhψ m + s
, 1 ≤ s ≤ k − m.
, 1 ≤ν ≤ k,
, 1 ≤ ν ≤ k , ν ≠ m + s,
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Soley Ersoy et al.
When these equations together with the equation (2.17) are substituted into the
equation (3.10) one finds
2
cosh 2 ψ 0  1 ∂ 2 g 1  ∂g  


−
2
 2 ∂uσ2 4 g  ∂uσ  
n

.
Kζ +ueσ ( eσ , a ) =
2
1 − cosh ψ σ
Considering the last equation together with that n = − g yields us
2
2
 1 ∂2 g
1  ∂g  
(1 − cosh ψ σ ) Kζ +ueσ ( eσ , a ) = cosh ψ 0  − 2 g ∂u 2 + 4 g 2  ∂u   .
σ
 σ 

Therefore, we obtain the relation between curvature of spacelike σ th principal
section ( eσ , n ) , 1 ≤ σ ≤ m , given by the equation (25) in ref. [5] and the curvature
2
2
of timelike section ( eσ , a ) and this completes the proof.
Theorem 3.4: Let M be a generalized timelike ruled surface with timelike
generating space and with spacelike central ruled surface in n − dimensional
Minkowski space ℝ 1n , a be arbitrary timelike unit vector of M and em + s ,
1 ≤ s ≤ k − m , be a timelike base vector in central space Z k − m ( t ) , respectively.
Giving a spacelike vector eσ , 1 ≤ σ ≤ m , and linearly independent with timelike
unit vector a there exists a relation at the point (ζ + ueσ ) ∈ M ,
(1 + cosh ψ ) K
2
σ
ζ +ueσ
( eσ , a ) = − sinh 2 ψ 0 Kζ +ue ( eσ , n )
σ
between the curvatures of timelike section ( eσ , a ) and the σ th spacelike principal
section ( eσ , n ) . Here ψ 0 and ψ σ are the hyperbolic angles between a , n , and a ,
eσ , 1 ≤ σ ≤ m , respectively.
Proof: Taking 1 ≤ σ ≤ m , the sectional curvature at the point (ζ + ueσ ) ∈ M is
given by
βσ βσ λ0 λ0 Rσ 0σ 0
, 1 ≤ σ ≤ m.
(3.11)
Kζ +ueσ ( eσ , a ) =
2
eσ , eσ a, a − eσ , a
If the coordinates of the ν th base vector eν , 1 ≤ ν ≤ k , and the timelike unit vector
given by the equation (3.6) are ( β 0 , β1 ,… , βν ,…, β k ) and ( γ 0 , γ 1 ,… , γ m + s ,… , γ k ) ,
respectively, then we may write
β 0 = eν , e0 = 0
, βν = eν , eν = εν
, 1 ≤ν ≤ k
Lorentzian Beltrami-Meusnier Formula
and
γ 0 = a, e0 =
sinhψ 0
n
77
, γ ν = a, eν = sinhψ ν
, 1 ≤ ν ≤ k , ν ≠ m + s,
γ m+ s = a, em+ s = coshψ m+ s , 1 ≤ s ≤ k − m.
Substituting the last equations and the equation (2.17) into the equation (3.11) and
2
considering that n = − g we reach that
( −1 − cosh ψ ) K
2
σ
ζ + ueσ
2
 1 ∂2 g
1  ∂g  
+
( eσ , a ) = sinh ψ 0  −

 .
2 g ∂uσ2 4 g 2  ∂uσ  


2
Taking the equation (25) in ref. [5] into consideration, the relation between
curvature of the σ th principal spacelike section ( eσ , n ) , 1 ≤ σ ≤ m , and curvature
of timelike section ( eσ , a ) is found and this completes the proof.
Taking vector e given by equations (30), (31), (32) and (33) in ref. [5] to be the
unit vector in Ek ( t ) and unit vector a given by equations (3.3), (3.4), (3.5) and
(3.6) to be a unit vector at the central point ∀ζ ∈ Ω , the following theorems
related to the sectional curvatures of M could be given.
Theorem 3.5: Let M be a generalized timelike ruled surface with timelike
generating space and with spacelike central ruled surface in n − dimensional
Minkowski space ℝ 1n , taking vector a , which is linearly independent with
spacelike unit vector e in generating space Ek ( t ) of M , to be spacelike unit
vector at the central point ∀ζ ∈ Ω , the following relation holds between
curvature of nondegenerate section ( e, a ) of M and curvature of spacelike
section ( e, n )
Kζ ( e, a ) =
cosh 2 ψ 0
1 − e, a
2
Kζ ( e, n )
where ψ 0 is the angle between spacelike unit vector a and spacelike normal
tangent vector n .
Proof: Considering the equation (3.1) the sectional curvature at the central point
ζ ∈ Ω is given by
m
∑ βσ βσ λ λ Rσ σ
Kζ ( e, a ) =
σ =1
σ ≠s
0 0
0 0
+ β s β s λ0 λ0 Rs 0 s 0
e, e a, a − e, a
2
.
(3.12)
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Soley Ersoy et al.
If the coordinates of e given by equation (30) in ref. [5] and tangent vector a
given by equation (3.3) ( β 0 , β1 ,… , β s ,… , β k ) and ( γ 0 , γ 1 ,… , γ s ,… , γ k ) ,
respectively, then we may write
β 0 = e, e0 = 0,
β s = e, es = sinh θ s
, 1≤ s ≤ m
βν = e, eν = cosh θν
, 1 ≤ν ≤ k , ν ≠ s
and
γ 0 = a, e0 =
coshψ 0
,
n
γ s = a, es = sinhψ s
, 1≤ s ≤ m
γ ν = a, eν = coshψ ν
, 1 ≤ ν ≤ k , ν ≠ s.
Substituting the last equations together with the equation (2.17) into the equation
(3.12) we get
cosh 2 ψ 0  m
cosh 2 θσ
∑
2
 σ =1
n

Kζ ( e, a ) =
 1 ∂ 2 g 1  ∂g  2 
 1 ∂ 2 g 1  ∂g  2  
2

 + sinh θ s 

−
−
 2 ∂uσ2 4 g  ∂uσ  
 2 ∂us2 4 g  ∂us   




.
2
1 − e, a
Considering the last equation and n = − g we reach
2
2
2
 m
 1 ∂2 g
 1 ∂2 g
1  ∂g  
1  ∂g   
2
2





+
+ sinh θ s −
+
cosh ψ 0 ∑ cosh θσ −
 2 g ∂uσ2 4 g 2  ∂uσ  
 2 g ∂us2 4 g 2  ∂us   
 σ =1





Kζ ( e, a ) =
.
2
1 − e, a
2
Considering curvature of the σ th spacelike principal section ( eσ , n ) , 1 ≤ σ ≤ m ,
σ ≠ s , given by the equation (25) in ref. [5] and the curvature of the s th timelike
principal section ( es , n ) , 1 ≤ s ≤ m , given by the equation (26) in ref. [5] at the
central point ζ ∈ Ω we find
 m

cosh 2 ψ 0  ∑ cosh 2 θσ Kζ ( eσ , a ) − sinh 2 θ s Kζ ( es , a ) 
 σ =1
.
Kζ ( e, a ) =
2
1 − e, a
Lorentzian Beltrami-Meusnier Formula
79
If we take I. type Lorentzian Beltrami-Euler formula given by theorem in ref. [5]
we complete the proof.
Theorem 3.6: Let M be a generalized timelike ruled surface with timelike
generating space and spacelike central ruled surface in n − dimensional
Minkowski space ℝ 1n and a be a timelike unit vector which is linearly
independent with spacelike unit vector e in generating space Ek ( t ) of M . The
relationship between curvature of timelike section
spacelike section ( e, n ) of M at ∀ζ ∈ Ω is
Kζ ( e, a ) = −
sinh 2 ψ 0
1 + e, a
2
( e, a )
and curvature of
Kζ ( e, n )
where ψ 0 is an angle between timelike vector a and spacelike normal tangent
vector n .
Proof: At the point ζ ∈ Ω the sectional curvature is given by
m
∑ βσ βσ λ λ Rσ σ
Kζ ( e, a ) =
σ =1
σ ≠s
0 0
0 0
+ β s β s λ0 λ0 Rs 0 s 0
e, e a, a − e, a
2
.
(3.13)
If the coordinates of e given by equation (30) in ref. [5] and the unit vector a
given by equation (3.4) are ( β 0 , β1 ,… , β s ,… , β k ) and ( γ 0 , γ 1 ,… , γ s ,… , γ k ) ,
respectively, then we obtain
β 0 = e, e0 = 0,
β s = e, es = sinh θ s
, 1≤ s ≤ m
βν = e, eν = cosh θν
, 1 ≤ ν ≤ k , ν ≠ s,
and
γ 0 = a, e0 =
sinhψ 0
,
n
γ s = a, es = coshψ s
, 1≤ s ≤ m
γ ν = a, eν = sinhψ ν
, 1 ≤ ν ≤ k , ν ≠ s.
Substituting the last equations together with equation (2.17) into equation (3.13)
and considering that n = − g we reach
2
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Soley Ersoy et al.
2
2
 m
 1 ∂2 g
 1 ∂2 g
 ∂g  
 ∂g   
1
1
2
2
 + sinh θ s  −

+
+
sinh ψ 0  ∑ cosh θσ  −
 2 g ∂uσ2 4 g 2  ∂uσ  
 2 g ∂us2 4 g 2  ∂us   
 σ =1





Kζ ( e, a ) = −
.
2
1 + e, a
2
Considering the curvature of the σ th spacelike principal section
( eσ , n )
,
1 ≤ σ ≤ m , σ ≠ s , which is given by equation (25) in ref. [5], and curvature of the
s th timelike principal section ( es , n ) , 1 ≤ s ≤ m , is given by (26) in ref. [5], into
the last equation at the point ζ ∈ Ω , we get
 m

sinh 2 ψ 0  ∑ cosh 2 θσ Kζ ( eσ , a ) − sinh 2 θ s Kζ ( es , a ) 
 σ =1
.
Kζ ( e, a ) = −
2
1 + e, a
The last equation together with the I. type Lorentzian Beltrami-Euler formula
completes the proof.
Theorem 3.7: Let M be a generalized timelike ruled surface with timelike
generating space and spacelike central ruled surface in n − dimensional
Minkowski space ℝ 1n . If we suppose the vector a , which is linearly independent
with timelike unit vector e in the generating space Ek ( t ) to be a spacelike unit
vector at the point ∀ζ ∈ Ω there exists the following relation between curvature
of timelike section ( e, a ) and curvature of timelike section ( e, n ) of M
Kζ ( e, a ) =
cosh 2 ψ 0
1 + e, a
2
Kζ ( e, n )
where ψ 0 is an angle between spacelike vector a and spacelike normal tangent
vector n .
Proof: At the central point ζ ∈ Ω , the curvature of the timelike section ( e, a ) is
given by
m
∑ βσ βσ λ λ Rσ σ
Kζ ( e, a ) =
σ =1
σ ≠s
0 0
0 0
+ β s β s λ0 λ0 Rs 0 s 0
e, e a, a − e, a
2
.
(3.14)
If the coordinates of e given by equation (30) in ref. [5] and the tangent vector a
are ( β 0 , β1 ,… , β s ,… , β k ) and ( γ 0 , γ 1 ,… , γ s ,… , γ k ) , respectively, then we get
Lorentzian Beltrami-Meusnier Formula
81
β 0 = e, e0 = 0,
β s = e, es = cosh θ s
, 1 ≤ s ≤ m,
βν = e, eν = sinh θν
, 1 ≤ ν ≤ k , ν ≠ s,
and
γ 0 = a, e0 =
coshψ 0
,
n
γ s = a, es = sinhψ s
, 1 ≤ s ≤ m,
γ ν = a, eν = coshψ ν
, 1 ≤ ν ≤ k , ν ≠ s.
Substituting the last equations together with the equation (2.17) into the equation
(3.14) and considering that n = − g we find
2
2
2
 m
 1 ∂2 g
 1 ∂2 g
1  ∂g  
1  ∂g   
2



cosh 2 ψ 0  ∑ sinh 2 θσ  −
+
+
cosh
θ
−
+
s
 2 g ∂uσ2 4 g 2  ∂uσ  
 2 g ∂us2 4 g 2  ∂us   
 σ =1





Kζ ( e, a ) = −
.
2
1 + e, a
If we substitute the curvature of the σ th spacelike principal section ( eσ , n ) ,
1 ≤ σ ≤ m , σ ≠ s , given by equation (25) in ref. [5], and the curvature of the sth
timelike principal section ( es , n ) , 1 ≤ s ≤ m , given by equation (26) in ref. [5] into
the last equation at the point ζ ∈ Ω , we get
 m

cosh 2 ψ 0  −∑ sinh 2 θσ Kζ ( eσ , a ) + cosh 2 θ s Kζ ( es , a ) 
 σ =1
.
Kζ ( e, a ) =
2
1 + e, a
Considering the II. type Lorentzian Beltrami-Euler formula given by Theorem 4.7
in ref. [5] completes the proof.
Theorem 3.8: Let M be a generalized timelike ruled surface with timelike
generating space and timelike central ruled surface in n − dimensional Minkowski
space ℝ 1n . Suggesting the vector a , which is independent with spacelike unit
vector e in the generating space Ek ( t ) , to be a spacelike unit vector at the point
∀ζ ∈ Ω , there exists the following relation between curvature of nondegenerate
section ( e, a ) and curvature of spacelike section ( e, n ) of M
Kζ ( e, a ) =
cosh 2 ψ 0
1 − e, a
2
Kζ ( e, n )
82
Soley Ersoy et al.
so that ψ 0 denotes the angle between spacelike unit vector a and spacelike
normal tangential vector n .
Proof: The curvature of the spacelike section ( e, a ) at the point ζ ∈ Ω is
m
βσ βσ λ λ Rσ σ
∑
σ
Kζ ( e, a ) =
0 0
=1
0 0
e, e a, a − e, a
2
.
(3.15)
If the coordinates of e from the equation (32) in [5] and the tangent vector a
given by equation (3.5) are ( β 0 , β1 ,… , β m + s ,… , β k ) and ( γ 0 , γ 1 ,… , γ m + s ,… , γ k ) ,
respectively, we find
β 0 = e, e0 = 0,
β m + s = e, em + s = sinh θ m + s , 1 ≤ s ≤ k − m,
βν = e, eν = cosh θν
and
γ 0 = a, e0 =
, 1 ≤ν ≤ k
, ν ≠ m + s,
coshψ 0
,
n
γ m + s = a, em + s = sinhψ m + s , 1 ≤ s ≤ k − m,
γ ν = a, eν = coshψ ν
, 1 ≤ ν ≤ k , ν ≠ m + s.
Considering that n = − g and substituting the last equations together with
equation (2.17) into the equation (3.15) yields
2
cosh 2 ψ 0
n
Kζ ( e, a ) =
2
 1 ∂ 2 g 1  ∂g 2 

cosh θσ 
−
∑
 2 ∂uσ2 4 g  ∂uσ  
σ =1

.
2
1 − e, a
m
2
Considering that the curvature of spacelike σ th principal section
1 ≤ σ ≤ m , given by equation (25) in ref. [5] at the point ζ ∈ Ω , we obtain
m
Kζ ( e, a ) =
cosh 2 ψ 0 ∑ cosh 2 θσ Kζ ( eσ , a )
σ =1
1 − e, a
2
( eσ , n )
,
Lorentzian Beltrami-Meusnier Formula
83
From III. type Lorentzian Beltrami-Euler formula, we see that there exists the
relation between the curvature of spacelike section ( e, a ) and curvature of
spacelike section ( e, n ) and this completes the proof.
Theorem 3.9: Let M be a generalized timelike ruled surface with timelike
generating space and timelike central ruled surface in n − dimensional Minkowski
space ℝ 1n . Considering that the vector a , which is independent with spacelike
unit vector e in the generating space Ek ( t ) , to be a timelike unit vector at the
point ∀ζ ∈ Ω , there exists the following relation between the curvature of
timelike section ( e, a ) and curvature of spacelike section ( e, n ) of M , such that
Kζ ( e, a ) = −
sinh 2 ψ 0
1 + e, a
2
Kζ ( e, n )
where ψ 0 is the angle between timelike unit vector a and spacelike normal
tangential vector n .
Proof: The curvature of timelike section ( e, a ) at the point ζ ∈ Ω is given by
m
Kζ ( e, a ) =
βσ βσ λ λ Rσ σ
∑
σ
=1
0 0
0 0
e, e a, a − e, a
2
.
(3.16)
Taking the coordinates of e from equation (32) in ref. [5] and the tangent vector
a to be ( β 0 , β1 ,… , β m + s ,… , β k ) and ( γ 0 , γ 1 ,… , γ m + s ,… , γ k ) , respectively, we reach
β 0 = e, e0 = 0,
β m + s = e, em + s = sinh θ m + s , 1 ≤ s ≤ k − m,
βν = e, eν = cosh θν
and
γ 0 = a, e0 =
, 1 ≤ν ≤ k
, ν ≠ m + s,
sinhψ 0
,
n
γ m + s = a, em + s = coshψ m + s , 1 ≤ s ≤ k − m,
γ ν = a, eν = sinhψ ν
, 1 ≤ ν ≤ k , ν ≠ m + s.
Considering n = − g and substituting the last equations together with the
equation (2.17) into the equation (3.16), we find
2
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Soley Ersoy et al.
2
 1 ∂2 g
1  ∂g  

+
sinh ψ 0 ∑ cosh θσ  −
 2 g ∂uσ2 4 g 2  ∂uσ  
σ =1

.
Kζ ( e, a ) = −
2
1 + e, a
m
2
2
If we consider the curvature of the σ th spacelike principal section ( eσ , n ) ,
1 ≤ σ ≤ m , given by equation (25) in ref. [5], we find
m
Kζ ( e, a ) = −
sinh 2 ψ 0 ∑ cosh 2 θσ Kζ ( eσ , a )
σ =1
1 + e, a
.
2
Considering the curvature of spacelike section ( e, n ) , which is given by III. type
Lorentzian Beltrami-Euler formula in ref. [5] completes the proof.
Theorem 3.10 Let M be a generalized timelike ruled surface with timelike
generating space and timelike central ruled surface in n − dimensional Minkowski
space ℝ 1n . Considering that the vector a , which is independent with timelike unit
vector e in the generating space Ek ( t ) of M , to be a timelike unit vector at the
central point ∀ζ ∈ Ω , there exists the following relation between curvature of
timelike section ( e, a ) and curvature of timelike section ( e, n )
Kζ ( e, a ) =
cosh 2 ψ 0
1 + e, a
2
Kζ ( e, n )
Where ψ 0 denotes the angle between spacelike unit vector a and spacelike
normal tangential vector n .
Proof: The curvature of timelike section ( e, a ) at the central point ζ ∈ Ω is given
by
m
Kζ ( e, a ) =
βσ βσ λ λ Rσ σ
∑
σ
=1
0 0
0 0
e, e a, a − e, a
2
.
(3.17)
If the coordinates of tangent vector a given by equation (3.5) and e given by (33)
in ref. [5] are ( β 0 , β1 ,… , β m + s ,… , β k ) and ( γ 0 , γ 1 ,… , γ m + s ,… , γ k ) , respectively,
then we write
Lorentzian Beltrami-Meusnier Formula
85
β 0 = e, e0 = 0,
β m + s = e, em + s = cosh θ m + s , 1 ≤ s ≤ k − m,
βν = e, eν = sinh θν
and
γ 0 = a, e0 =
, 1 ≤ ν ≤ k , ν ≠ m + s,
coshψ 0
,
n
γ m + s = a, em + s = sinhψ m + s , 1 ≤ s ≤ k − m,
γ ν = a, eν = coshψ ν
, 1 ≤ ν ≤ k , ν ≠ m + s.
Considering that n = − g and substituting the equation (2.17) together with the
last equations into the (3.17) we reach
2
2
 1 ∂2 g
1  ∂g  

cosh ψ 0 ∑ sinh θσ  −
+
 2 g ∂uσ2 4 g 2  ∂uσ  
σ =1

.
Kζ ( e, a ) = −
2
1 + e, a
m
2
2
If we use the curvature of the σ th spacelike principal section ( eσ , n ) , 1 ≤ σ ≤ m ,
given by the equation (25) in ref. [5] at the central point ζ ∈ Ω in the last
equation we see that
m
cosh ψ 0 ∑ sinh 2 θσ Kζ ( eσ , a )
2
Kζ ( e, a ) = −
σ =1
1 + e, a
2
.
Considering the IV. type Lorentzian Beltrami-Euler formula in Theorem 4.9, [5]
completes the proof.
From the last six theorems it can be easily seen that when the central ruled surface
of M is either spacelike or timelike, the relations obtained are seemed to be same.
Therefore we can give the following corollaries, respectively.
From Theorems 3.6 and 3.8 we give the following corollary.
Corollary 3.11: Let M be a generalized timelike ruled surface with timelike
generating space and (spacelike or timelike) central ruled surface in
n − dimensional Minkowski space ℝ 1n . The vector a is the spacelike unit vector,
which is linearly independent with spacelike unit vector e in Ek ( t ) of M at the
central point ∀ζ ∈ Ω . The curvature of spacelike section ( e, a ) of M depends on
the vectors e and a , Lorentzian Beltrami-Euler formula for the spacelike section
86
( e, n )
Soley Ersoy et al.
and the angle ψ 0 = ∢ ( a, n ) . In this case, there exists the following relation
between the curvature of spacelike section ( e, a ) and curvature of spacelike
section ( e, n ) of M
Kζ ( e, a ) =
cosh 2 ψ 0
1 − e, a
Kζ ( e, n ) .
2
This equation is called I. type Lorentzian Beltrami-Meusnier formula for
sectional curvature of generalized timelike ruled surface with timelike generating
space at the central point.
From Theorems 3.7 and 3.10 we give the following corollary.
Corollary 3.12: Let M be a generalized timelike ruled surface with timelike
generating space and (spacelike or timelike) central ruled surface in
n − dimensional Minkowski space ℝ 1n . The vector a be a timelike unit vector,
which is linearly independent with spacelike unit vector e in Ek ( t ) of M at the
central point ∀ζ ∈ Ω . In this case, the following relation holds between the
curvature of timelike section ( e, a ) and curvature of spacelike section ( e, n )
Kζ ( e, a ) = −
sinh 2 ψ 0
1 + e, a
2
Kζ ( e, n )
and this equation is called II. type Lorentzian Beltrami-Meusnier formula for
sectional curvature of generalized timelike ruled surface with timelike generating
space at the central point. Here, any curvature of timelike section ( e, a ) of M
depends on the vectors e and a , Lorentzian Beltrami-Euler formula for the
spacelike section ( e, n ) and the angle ψ 0 = ∢ ( a, n ) .
We can give the following corollary from Theorem 3.7 and 3.10.
Corollary 3.13: Let M be a generalized timelike ruled surface with timelike
generating space and (spacelike or timelike) central ruled surface in
n − dimensional Minkowski space ℝ 1n . We also suppose that the vector a is a
spacelike unit vector, which is linearly independent with timelike unit vector in
Ek ( t ) of M . In this special case any curvature of timelike section ( e, a ) of M
depends on the vectors e and a , Lorentzian Beltrami-Euler formula for the
timelike section ( e, n ) and the angle ψ 0 = ∢ ( a, n ) . The relation
Lorentzian Beltrami-Meusnier Formula
Kζ ( e, a ) =
87
cosh 2 ψ 0
1 + e, a
2
Kζ ( e, n )
holds between the curvature of timelike section ( e, a ) and curvature of timelike
section ( e, n ) and this equation is named III. type Lorentzian Beltrami-Meusnier
formula for sectional curvature of generalized timelike ruled surface with timelike
generating space at the central point.
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