Journal of Applied Mathematics and Physics, 2013, 1, 55-59
http://dx.doi.org/10.4236/jamp.2013.13009 Published Online August 2013 (http://www.scirp.org/journal/jamp)
Some Results on the Differential Geometry of Spacelike
Curves in De-Sitter Space
Tunahan Turhan1*, Nihat Ayyildiz2
1Seydişehir Vocational School, Necmettin Erbakan University, Konya, Turkey
2Department of Mathematics, Süleyman Demirel University, Isparta, Turkey
Received June 14, 2013; revised July 15, 2013; accepted September 1, 2013
Copyright © 2013 Tunahan Turhan, Nihat Ayyildiz. This is an open access article distributed under the Creative Commons Attribu-
tion License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly
cited.
ABSTRACT
The differential geometry of curves on a hypersphere in the Euclidean space reflects instantaneous properties of sphere-
cal motion. In this work, we give some results for differential geometry of spacelike curves in 3-dimensional de-Sitter
space. Also, we study the Frenet reference frame, the Frenet equations, and the geodesic curvature and torsion functions
to analyze and characterize the shape of the curves in 3-dimensional de-Sitter space.
Keywords: De-Sitter Space; Frenet Equations; Frenet Reference Frame; Geodesic Curvature and Torsion; Local
Canonical Form
1. Introduction
Let


4
1234
,,,,1 4
i
xxxxxx i 

be a 4-di-
mensional vector space. The Lorentzian space


4
1
,, 4  is the 4-dimensional vector space
endowed with the pseudo scalar product
4

11223 344
,
x
yxyxyxyxy 
where 

1234
,,, ,
x
xxxxx

,,, ,yyyyy
4
1

4
1, 1234 [1].
The norm of a vector is defined by
,.
x
xxLet denote the 3-dimensional unitary
de-Sitter space, that is, is the hyperquadric,
[2,3],
3
1

34
11



34
11
,1xRxx .
Given 3-vector 

1234
,,, ,
x
xxxx

,4
1.

1234
,,, ,yyyyy
1234 in Then we can define the
wedge product

,,,zzzzz
x
yz as follows
123 4
123 4
123 4
123 4
eee e
x
xxx
xyz yyy y
zzz z


where is the canonical basis of [2]. 
1234
,,,eeee

4
1,
A spherical displacement can be specified by a unit
vector


,,
x
yz
uuuu
.
along the axis of the rotation and
a rotation angle

The Euler parameters of the rotation
defined in terms of and u

can be used to prescribe
a mapping of this rotation to a point in a higher dimen-
sional space [4-6]. The vector function




,
1234
,,,
X
uXX  XX
is given by
12
34
cosh ,sinh,
22
sinh,sinh .
22
x
yz
XXu
X
uX u





Let


,,
x
yz
uuuu denote a timelike rotation axis.
So, we get

2222
1234
22222
coshsinh 1.
22
xyz
XXXX
uuu



 
This means that the point lies on the hyperquadric
of radius 1 in Let us denote this hyperquadric
Our aim is to give an interpretation of the image of
the mapping
X
4
1.
3
1.


,Xu .

For this, we will examine the
differential geometry of curves on So, we will in-
troduce a Frenet frame for the curve and define the geo-
desic curvature and geodesic torsion functions which
characterize the shape of the curve. Also we will give
explicit formulas for the geodesic curvature and torsion
functions of the parameterized curve
3
1.

X
t. For this aim,
*Corresponding author.
C
opyright © 2013 SciRes. JAMP

T. TURHAN, N. AYYILDIZ
56
we will use the exterior algebra of multivectors.
2. The Frenet Reference Frame
Let us consider a general parametrized spacelike curves
on denoted by
3
1


X
t

. We will focus on the geomet-
ric properties of
X
t. For this, we define arclenght
parameter
s
as

0
dd.
d
tX
s
t
t
t (2.1)
The integrand of Equation (2.1) is the magnitude of
the velocity of the point as it moves along the curve X

X
t. If d0
d
X
t then the function


s
t can be in-
verted to obtain


ts which allows the reparameteriza-
tion 





X
ts Xs. The magnitude of d
d
X
s
is
 
ddd 1.
ddd
Xts ts
sst

Now, we will use the unit speed form

X
s to define
the Frenet frame and the Frenet equations of the curve.
And so, we will give interpretation of these results in
terms of the general parameter . t
The Frenet frame of

X
s is the set of unit vectors,
E
, , and TN
B
defined in the following way. The
first vector,
E
is directed along the radius of the hy-
perquadric and is given by


.EXs
Note that

1Xs  and
E
is a spacelike vector.
The second vector, , is tangent to T

X
s and to
It is obtained by
3
1.
d
d
X
T
s

and is a spacelike vector. So, the curve

X
sT
in is
a spacelike curve. On the other hand, since is a unit
3
1

spacelike vector, its derivative d
d
T
s
will be normal to .
So,
T
d
d
T
s
will have a component along
E
given by
d,
d
TE
s which we compute by expanding the identity
d,
dTE s0
and we get
d,,
d
TETT
s 
The remaining component of d
d
T
s
orthogonal to both
E
and T is chosen as the direction of the unit timelike
vector , so we have
N
d
d.
d
d
TE
s
NTE
s



Here, we define the function d,
d
gTE
s
  which
measures the bend of


X
s out of the
E
T plane, to
be the geodesic curvature of


.
X
s
The remaining vector
B
of the Frenet Frame is ob-
tained by commuting the component of d
d
N
s
which is
not along either
E
or and choose the direction of T
B
along this component such that the frame taken in the
order , , TN
B
,
E
has positive orientation. The fact
that the component of d
d
N
s
in the direction
E
is zero
is obtained by expanding the identity

d,0,
d
d,,
d
NE
s
NENT
s

0.


On the other hand, by expanding d,
dNT s0

we
obtain
dd
,, ,
dd
,,
g
gg
NT
TN NN
ss
NN NE

 .
E


 
Therefore we find that the component of d
d
T
s
along
is T.
g

Finally we see that
B
is given by
d
d.
d
d
g
g
NT
s
BNT
s






The function d
d
g
NT
s
 
g

is defined as the geo-
desic torsion of


.
X
s The coefficient

is chosen as
either 1

or 1

to ensure that the determinant of the
matrix


,,TN ,EB is 1

, that is so that the Frenet
frame has positive orientation. The vector
E
has been
choosen as the last member of the frame for several rea-
sons, the primary one being that it is convenient to visu-
1.

T. TURHAN, N. AYYILDIZ 57
alize the 3-dimensional surface of the hyperquadric lo-
cally as the 3-dimensional Lorentzian space of its tangent
hyperplane. The vectors T, and N
B
lie in this
space and are analogous to the tangent, normal and bi-
normal vectors of a space curve in three dimensions. In
this way the geodesic curvature and torsion functions
g

and
g

are seen to be analogous to the curvature and
torsion of a space curve.
Then we have the following proposition.
Proposition 2.1. Let

X
s be a spacelike curve in
de-Sitter space Then the Frenet equations are
3
1.
d
d
d
d
d
d
d.
g
gN

d
T
s
NTB
s
B
s
ET
s






gg
NE

These equations may be viewed as a set of 16 linear
first-order differential equations in the components of ,
,
T
N
B
and
E
which, when the coefficients


g
s

and

g
s
 are specified functions of
s
, can be solved
to determine the curve


EXs in Thus the geo-
desic curvature and torsion functions,
3
1.


g
s

and

g
s

, of

X
s define it completely.
3. The Local Canonical Form
The local properties of a hyperquadrical curve


X
s in
the vicinity of a reference point 0
s
s can be obtained
by computing the series expansion of

X
s in the Fre-
net frame of the reference point


0.
X
s This form of

X
s
,e
is termed the local canonical form by Do Carmo
[7].
We choose the coordinate directions of the 4-dimen-
sional Lorentzian space containing denoted
by 1 2 3 4 where i has a in the i-th
coordinate position and zeros elsewhere, so that they
align with the Frenet frame
4
1
3
1,
1 ,e,e,e e


0,Ts

Ns

0,


0
Bs
and


0
Es of the reference point


0.
X
s Computing
the derivatives of

X
s to the third order we have


2
2
3
2
3
,
d,
d
dd ,
1TN
s


d d
d d
g
ggg
Xs E
XT
s
XT NE
s
s
XB
s





 
dd
.
These expressions lead to the Taylor series expansion
of

X
s in the vicinity of the reference position 0
s
s

.
For convenience, we denote the reference position as
00s

and obtain
  
0
0
23
2
23
2
2
11 1
00 00
1! 2!3!
0
1
10 1
0
01 11
0
00 26
d
00 d
0
1
10
1
0
01
0
00 2
00
g
gg
g
g
g
g
sXX sXsXs
ss s
s
s
s

 





 
 



 


 

 
 


 

 

  



 
 

 
 

 
 
 
X
00
1
3
6
gg
g
s









(3.1)
where 0
g

and 0
g

are the values of the curvature and
torsion of


X
s and 1
g

is the value of d
d
g
s

, all
evaluated at the reference position Equation (3.1)
allows a description of the shape of
0.s

X
s to various of
approximation, for example, to the zeroth order


X
s
is simply the point


0X, to the first order it is ap-
proximated by the tangent vector . We see that 0 T
g

defines the shape of


X
s to the second order which
defines the amount that it bends away from the
E
T

plane. The parameter 0
g

defines the amount that


X
s bends out of the subspace. To the
second order,
ETN


X
s is approximated by its osculating
circle which has the radius

given by

12
2
11
.
1
g
 

The function

12
21
g
   is called the total cur-
vature of


.
X
s The plane of this circle osculating
plane, is defined by the tangent vector and the unit
vector G
T
dd 1
.
dd
g
Ts
GN
Ts

E


 (3.2)
The rotation of the osculating plane about is given
by
T
d
d
G
s
where
d
d
ggg
GTG
s


B

 
 (3.3)
and


2
1
gg
GN   
.E (3.4)
If 10
g


and 00
g


then from (3.3) we see that

T. TURHAN, N. AYYILDIZ
58
this plane remains instantaneously fixed, i.e.
d.
d
GT
s


Furthermore, since d0
d
g
s


implies d0
ds


and
therefore that the total curvature is constant. A general
curve

X
s is approximated to the third order by an
osculating sphere To determine the center c of this
pseudo-sphere we first note that first, second and third
derivatives of
2
1.

X
s lie in the subspace spanned by the
three orthogonal unit vectors T, and , i.e. we
have
G* G
2
2
d
d
d
dg
XT
s
XNE G
s




and

3
2*
3
dd d
dd d
XGTG
ss s

G

 
where

*1
gggggg
ggg
GB
G

G
GB
 







B
and

2
22 .
gggggg
GB  

 

Here is a spacelike vector if if
not, respectively.
*
G

2
22 0,
gg
 


Assume that is a timelike vector. So, the radius
vector, , of the osculating sphere must
have the form
*
G

sRX c

*
123
RXsckTkGkG
where 123 are constants. These constants are de-
termined by the requirement that
,,kkk
R
have constant
magnitude to the third order. Differentiating
,const.RR we obtain
2
2
32
32
d,0,
d
ddd
,,
dd
d
ddd
,3,
d dd
XXc
s
XXX
Xc ss
s
XX
Xc s
ss

 
 
0,
0.
X
So, from these equations we conclude
1
2
3
0,
1,
.
k
k
k














Hence, we have


*
2
1
1.
ggg
Xs cGG
GG





 

 
B
For those curves with 0
g

 this relation simplifies
using (3.1) and (3.3)

22
2
11 .
ggg
2
X
scN E
 
 

 
 







(3.5)
So we have the following proposition.
Proposition 3.1. Let


X
s be a spacelike curve with
the geodesic curvature
g

and the total curvature

in
de-Sitter space Then the radius vector
3
1.


RXsc

 of the osculating sphere is as in
Equation (3.5).
2
1

4. Arbitrary Parameterization
We now derive the local canonical form of


X
t with
respect to the arbitrary parameter . To do this we use
the Frenet equations and the fact that
t




dd d
dd
..
ddddd
Xt XtXt
ss
v
tstst


Computing the derivatives of


X
t to the third order,
we obtain






2
22
2
3232
32
33
,
d,
d
dd,
d
d
dd1
dd
d
dd
33
dd d
g
g
g
ggg
XtE
Xt vT
t
Xt vTvNvE
t
t
Xt vvT
tt
vv
vvNvBv
ts






 







 .E
t
(4.1)
Thus the local canonical form for


X
t at a refer-
ence point 0t

becomes, to third order,

T. TURHAN, N. AYYILDIZ
59
 

0
00
01
0
23
2
23
2
232
2
2
3
2
3
1d 1d1d
00 00
1! d2!3! dd
d
3d
10 d
d1
01 d
d.
00 26
0
00 d
d
3dd
g
gg
gg
g
XX X
3
3
X
tXt tt
ttt
v
vt
vv
vv
t
t
t
tv
vv
vv
ts




 





 

 


 
 

 

 

 

 


6. Conclusion
t





This work develops the differential geometry of space-
like curves on de-Sitter space in four-dimensional Lor-
entzian space. The motivation for this work is the fact
that the Euler parameters of spherical displacements can
be used to map them to points on
3
1.
REFERENCES

[1] B. O’Neill, “Semi-Riemann Geometry: With Applictions
to Relativity,” Academic Press, New York, 1983, 469 p.
5. Formula for Geodesic Curvature and
Torsion [2] T. Fusho and S. Izumiya, “Lightlike Surfaces of Space-
like Curves in de Sitter 3-Space,” Journal of Geometry,
Vol. 88, 2008, pp. 19-29.
http://dx.doi.org/10.1007/s00022-007-1944-5
We now compute the vector
2
2
dd
dd
XX
X
tt

which in
view of Equation (4.1) becomes
[3] M. Kasedou, “Singularities of Lightcone Gauss Images of
Spacelike Hypersurfaces in de Sitter Space,” Journal of
Geometry, Vol. 94, 2009, pp. 107-121.
http://dx.doi.org/10.1007/s00022-009-0001-y
2
3
2
dd .
ddg
XX
X
vTNE
tt
   (5.1)
[4] J. M. McCarthy, “The Differential Geometry of Curves in
an Image Space of Spherical Kinematics,” Mechanism
and Machine Theory, Vol. 22, No. 3, 1987, pp. 205-211.
http://dx.doi.org/10.1016/0094-114X(87)90003-6
Computing the scalar product of (5.1) with itself, we
obtain the following equation for the geodesic curvature
g
 :
[5] J. M. McCarthy and B. Ravani, “Differential Kinematics
of Spherical and Spatial Motions Using Kinematic Map-
ping,” Journal of Applied Mechanics, Vol. 53, No. 1,
1986, pp. 15-22. http://dx.doi.org/10.1115/1.3171705
22
22
2
3
dd dd
,
dd
dd
.
dd
,
dd
g
XX XX
X
X
tt
tt
XX
tt

 



[6] B. Ravani and B. Roth, “Mappings of Spatial Kinemat-
ics,” Journal of Mechanisms, Transmissions and Automa-
tion in Design, Vol. 106, No. 3, 1984, pp. 341-347.
http://dx.doi.org/10.1115/1.3267417
And a formula for geodesic torsion
g

can be given
as
23
22
dd d
det,,,.
ddd
g
g3
X
XX
Ettt



 


[7] M. P. Do Carmo, “Differential Geometry of Curves and
Surfaces,” Prentice-Hall, Englewood Cliffs, 1976, 503 p.
This relation can be also seen in [2].