Mathematical Communications
6
(2001), 3945
39
Hyperspherical curves in
n
dimensional
k
isotropic
space
I
k
n
ˇ
Zeljka Milin
ˇ
Sipuˇ
s
∗
Abstract
.
In this paper we give the characterization of hyperspher
ical curves in
n
dimensional
k
isotropic space
I
k
n
.
Key words:
n
dimensional
k
isotropic space, osculating hypersphere,
hyperspherical curve
AMS subject classifications:
53A35
Received July 15, 2000
Accepted March 10, 2001
1.
Introduction
The
n
dimensional
k
isotropic space
I
k
n
is introduced in [5] where it is defined as a
pair (
A, V
) where
A
is a real
n
dimensional aﬃne space and
V
its corresponding
vector space decomposed in a direct sum of subspaces
V
=
U
1
⊕
U
2
,
dim
U
1
=
n
−
k
, dim
U
2
=
k
.
The space
U
1
is endowed with a scalar product
·
:
U
1
×
U
1
→
R
which is extended
on the whole
V
by
x
·
y
=
π
1
(
x
)
·
π
1
(
y
)
,
where
π
1
:
V
→
U
1
denotes the canonical projection. In such a way a semidefinite
scalar product on
V
is defined.
In this paper we describe the osculating hyperspheres of an admissible curve in
the space
I
k
n
. The theory of curves in
I
k
n
is developed in [2]. Furthermore, we study
the conditions under which an admissible curve is hyperspherical.
2.
Osculating hypersphere in
I
k
n
As it is shown in [5], a hypersphere in
I
k
n
is defined in aﬃne coordinates by the
equation
n
−
k
i
=1
x
2
i
+ 2
n
i
=1
α
i
x
i
+
α
0
= 0
,
(1)
∗
Department of Mathematics, University of Zagreb, Bijeniˇ
cka c. 30, HR 10 000 Zagreb, Croa
tia, email:
[email protected]
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40
ˇ
Z. Milin
ˇ
Sipuˇ
s
where
α
1
, . . . , α
n
∈
R
.
We distinguish the following types of hyperspheres in
I
k
n
([5], Theorem 7.1). If
(
α
n
−
k
+1
, . . . , α
n
) = (0
, . . . ,
0), then by an isotropic motion we obtain the normal
form of a parabolic hypersphere of type
l
,
l
∈ {
0
, . . . , k
−
1
}
,
n
−
k
i
=1
x
2
i
+
α
n
−
l
x
n
−
l
= 0
,
Its radius is defined by
−
α
n

l
2
,
α
n
−
l
= 0.
If
α
n
−
k
+1
=
· · ·
=
α
n
= 0, then by an isotropic motion we get a cylindrical
hypersphere
n
−
k
i
=1
x
2
i
=
r
2
.
Its radius is defined by
r
.
Definition 1.
Let
c
:
I
→
I
k
n
be a regular
C
r
curve and
f
(
x
) = 0
a regular
C
r
hypersurface,
r
≥
1
.
A point
P
0
(
t
0
)
is a point of contact of
r
th
order of the
curve
c
and the hypersurface
f
if the function
F
(
t
) =
f
(
c
(
t
))
satisfies
F
(
t
0
) =
. . .
=
F
(
r
)
(
t
0
) = 0
, F
(
r
+1)
(
t
0
) = 0
.
Definition 2.
Let
c
be an admissible
C
r
curve in
I
k
n
,
r
≥
n
,
P
0
a point of
c
.
A hypersphere which has contact of
n
th
order with the curve
c
in
P
0
is called the
osculating hypersphere.
We can write equation (1) of a hypersphere in the following form.
Let
P
0
=
(
p
1
, . . . , p
n
) be a point of hypersphere (1). Then
n
−
k
i
=1
p
2
i
+ 2
n
i
=1
α
i
p
i
+
α
0
= 0
.
(2)
Subtracting (2) from (1) we get
n
−
k
i
=1
(
x
2
i
−
p
2
i
) + 2
n
i
=1
α
i
(
x
i
−
p
i
) = 0
,
which can be written as
n
−
k
i
=1
(
x
i
−
p
i
)
2
+ 2
n
−
k
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