On Darboux vector in Lorentzian 5-space
We introduce the Darboux vector in the Lorentzian 5-space. We give some characterizations of this vector in the space. In addition, we consider some special cases in the space.
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| Дата: | 2018 |
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Institute of Mathematics, NAS of Ukraine
2018
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Ukrains’kyi Matematychnyi Zhurnal| _version_ | 1860507388874850304 |
|---|---|
| author | Iyigün, E. Ійігун, Е. |
| author_facet | Iyigün, E. Ійігун, Е. |
| author_sort | Iyigün, E. |
| baseUrl_str | https://umj.imath.kiev.ua/index.php/umj/oai |
| collection | OJS |
| datestamp_date | 2019-12-05T09:19:33Z |
| description | We introduce the Darboux vector in the Lorentzian 5-space. We give some characterizations of this vector in the space. In
addition, we consider some special cases in the space. |
| first_indexed | 2026-03-24T02:08:32Z |
| format | Article |
| fulltext |
UDC 517.5
E. Iyigün (Uludag Univ., Bursa, Turkey)
ON DARBOUX VECTOR IN LORENTZIAN 5-SPACE
ПРО ВЕКТОР ДАРБУ В 5-ПРОСТОРI ЛОРЕНЦА
We introduce the Darboux vector in the Lorentzian 5-space. We give some characterizations of this vector in the space. In
addition, we consider some special cases in the space.
Введено вектор Дарбу в 5-просторi Лоренца. Наведено деякi харатеристики даного вектора в цьому просторi. Крiм
того, розглянуто деякi частиннi випадки в цьому ж просторi.
1. Introduction. Let X = (x1, x2, x3, x4, x5) and Y = (y1, y2, y3, y4, y5) be two non-zero vectors
in Lorentzian 5-space \BbbL 5. For X, Y \in \BbbL 5
g(X,Y ) = \langle X,Y \rangle = - x1y1 +
5\sum
i=2
xiyi,
is called Lorentzian inner product. The couple
\bigl\{
\BbbL 5, \langle , \rangle
\bigr\}
is called Lorentzian 5-space. Then the
vector X of \BbbL 5 is called: (i) time-like if \langle X,X\rangle < 0, (ii) space-like if \langle X,X\rangle > 0 or X = 0,
(iii) null (or light-like) vector if \langle X,X\rangle = 0 and X \not = 0. The set of a null vectors in Tp(\BbbL 5) is called
the nullcone at p \in \BbbL 5.
Similarly, an arbitrary curve \alpha = \alpha (s) in \BbbL 5 can be locally be space-like, time-like or null, if all
of its velocity vectors \alpha \prime (s) are respectively space-like, time-like or null. Also, recall the norm of
a vector X is given by \| X\| =
\sqrt{}
| \langle X,X\rangle | . Therefore, X is a unit vector if \langle X,X\rangle = \pm 1. Next,
vectors X, Y in \BbbL 5 are said to be orthogonal if \langle X,Y \rangle = 0. The velocity of the curve \alpha is given
by \| \alpha \prime (s)\| . Thus, a space-like or a time-like \alpha is said to be parametrized by arclength function s,
if \langle \alpha \prime (s), \alpha \prime (s)\rangle = \pm 1 [1].
2. Basic definitions of \BbbL 5 .
Definition 1. Let \alpha : I - \rightarrow \BbbL 5 be a curve in \BbbL 5 and k1, k2, k3, k4 be the Frenet curvatures of
\alpha . Then for the unit tangent vector V1 = \alpha \prime (s) over M the ith e-curvature function mi, 1 \leq i \leq 5,
is defined by
mi =
\left\{
0, i = 1,
\varepsilon 1\varepsilon 2
k1
, i = 2,\biggl[
d
dt
(mi - 1) + \varepsilon i - 2mi - 2ki - 2
\biggr]
\varepsilon i
ki - 1
, 2 < i \leq 5,
(1)
where \varepsilon i = \langle Vi, Vi\rangle = \pm 1.
Definition 2. Let \alpha : I - \rightarrow \BbbL 5 be a unit speed non-null curve. The curve \alpha is called Frenet
curve of osculating order d, d \leq 5, if its 5th order derivatives \alpha \prime (s), \alpha \prime \prime (s), . . . , \alpha v(s) are linearly
independent and \alpha \prime (s), \alpha \prime \prime (s), . . . , \alpha v\imath (s) are no longer linearly independent for all s \in I. For each
Frenet curve of order 5 one can associate an orthonormal 5-frame
\bigl\{
V1, V2, V3, V4, V5
\bigr\}
along \alpha (such
that \alpha \prime (s) = V1) called the Frenet frame k1, k2, k3, k4 : I - \rightarrow \BbbR called the Frenet curvatures, such
that the Frenet formulas is defined in the usual way;
c\bigcirc E. IYIGÜN, 2018
ISSN 1027-3190. Укр. мат. журн., 2018, т. 70, № 5 635
636 E. IYIGÜN
\nabla V1V1 = \varepsilon 2k1V2,
\nabla V1V2 = - \varepsilon 1k1V1 + \varepsilon 3k2V3,
\nabla V1V3 = - \varepsilon 2k2V2 + \varepsilon 4k3V4, (2)
\nabla V1V4 = - \varepsilon 3k3V3 + \varepsilon 5k4V5,
\nabla V1V5 = - \varepsilon 4k4V4,
where V1, V2, V3, V4, and V5 are orthogonal vectors satisfying equations
\langle V1, V1\rangle = - 1,
\langle Vi, Vi\rangle = 1, 2 \leq i \leq 5,
and \nabla is the Levi-Civita connection of \BbbL 5.
Definition 3. Let \alpha be a non-null curve of osculating order 5. The harmonic functions
Hj : I - \rightarrow \BbbR , 0 \leq j \leq 3,
defined by
H0 = 0,
H1 =
k1
k2
,\biggl[
d
dt
(Hj - 1) + \varepsilon j - 2Hj - 2kj
\biggr]
\varepsilon j
kj+1
, 2 \leq j \leq 3,
are called the harmonic curvatures of \alpha , where k1, k2, kj , kj+1 are Frenet curvatures of \alpha which
are not necessarily constant and \varepsilon j = \langle Vj , Vj\rangle = \pm 1 [2].
Definition 4. Let \alpha be a non-null curve of osculating order 5. Then \alpha is called a general helix
of rank 3 if
3\sum
i=1
(Hi)
2 = c,
holds, where c \not = 0 is a real constant.
Corollary 1. If \alpha is a general helix of rank 3, then
H2
1 +H2
2 +H2
3 = c.
Proof. By the use of above definition, we obtain the corollary.
3. The Darboux vector in \BbbL 5 .
Theorem 1. Let \alpha be a non-null curve of osculating order 5 in \BbbL 5, then
\nabla VjVi =
\varepsilon i
mi+1
Vi+1, i, j = 1,
\nabla VjVi+1 = - \varepsilon i+1
mi+1
Vi +
m\prime
i+1
mi+2
Vi+2, i, j = 1,
ISSN 1027-3190. Укр. мат. журн., 2018, т. 70, № 5
ON DARBOUX VECTOR IN LORENTZIAN 5-SPACE 637
\nabla VjVi+1 =
\biggl( - \varepsilon i\varepsilon i+1mim
\prime
i - \varepsilon i - 1\varepsilon i+1mi - 1m
\prime
i - 1
mimi+1
\biggr)
Vi+
+
\biggl(
mi+1m
\prime
i+1 + \varepsilon i\varepsilon i+1mim
\prime
i + \varepsilon i - 1\varepsilon i+1mi - 1m
\prime
i - 1
mi+1mi+2
\biggr)
Vi+2, j = 1, i = 2, 3,
\nabla VjVi+1 =
\biggl( - \varepsilon i\varepsilon i+1mim
\prime
i - \varepsilon i - 1\varepsilon i+1mi - 1m
\prime
i - 1 - \varepsilon i - 2\varepsilon i+1mi - 2m
\prime
i - 2
mimi+1
\biggr)
Vi, j = 1, i = 4,
where mi are the ith e-curvature functions and \varepsilon i = \langle Vi, Vi\rangle = \pm 1.
Corollary 2. Let \alpha be a non-null curve of osculating order 5 in \BbbL 5, then
\nabla V1V1 =
\varepsilon 1
m2
V2,
\nabla V1V2 = - \varepsilon 2
m2
V1 +
m\prime
2
m3
V3,
\nabla V1V3 = - \varepsilon 2\varepsilon 3
m\prime
2
m3
V2 +
\biggl(
m3m
\prime
3 + \varepsilon 2\varepsilon 3m2m
\prime
2
m3m4
\biggr)
V4,
\nabla V1V4 =
\biggl(
- \varepsilon 3\varepsilon 4m3m
\prime
3 - \varepsilon 2\varepsilon 4m2m
\prime
2
m3m4
\biggr)
V3 +
\biggl(
m4m
\prime
4 + \varepsilon 3\varepsilon 4m3m
\prime
3 + \varepsilon 2\varepsilon 4m2m
\prime
2
m4m5
\biggr)
V5,
\nabla V1V5 =
\biggl(
- \varepsilon 4\varepsilon 5m4m
\prime
4 - \varepsilon 3\varepsilon 5m3m
\prime
3 - \varepsilon 2\varepsilon 5m2m
\prime
2
m4m5
\biggr)
V4.
Proof. By the use of a previous theorem, we obtain the corollary.
Theorem 2. Let \alpha be a non-null curve of osculating order 5 in \BbbL 5, then
\nabla V1V1 = \varepsilon 2k2H1V2,
\nabla V1V2 = - \varepsilon 1k2H1V1 + \varepsilon 3
k1
H1
V3,
\nabla V1V3 = - \varepsilon 2
k1
H1
V2 + \varepsilon 2\varepsilon 4
H \prime
1
H2
V4,
\nabla V1V4 = - \varepsilon 2\varepsilon 3
H \prime
1
H2
V3 + \varepsilon 3\varepsilon 5
\biggl(
H2H
\prime
2 + \varepsilon 1\varepsilon 2H1H
\prime
1
H2H3
\biggr)
V5,
\nabla V1V5 = - \varepsilon 3\varepsilon 4
\biggl(
H2H
\prime
2 + \varepsilon 1\varepsilon 2H1H
\prime
1
H2H3
\biggr)
V4,
where H1, H2, H3 are harmonic curvature of \alpha and \varepsilon i = \langle Vi, Vi\rangle = \pm 1 for 1 \leq i \leq 5.
Proof. By using definition of harmonic curvatures, we obtain the theorem.
Theorem 3. Let \alpha : - \rightarrow \BbbL 5 be a non-null curve of osculating order 5 given over the Frenet
frame
\bigl\{
V1, V2, V3, V4, V5
\bigr\}
. If mi, 2 \leq i \leq 5, are the ith e-curvature functions, Hi, 1 \leq i \leq 3, are
the harmonic curvatures and m\prime
i =
dmi
ds
, 2 \leq i \leq 5, then the following relations hold:
\mathrm{d}\mathrm{e}\mathrm{t}(m\prime
2,m
\prime
3,m
\prime
4,m
\prime
5) = 0 \Leftarrow \Rightarrow
3\sum
i=1
(Hi)
2 = \mathrm{c}\mathrm{o}\mathrm{n}\mathrm{s}\mathrm{t}\mathrm{a}\mathrm{n}\mathrm{t},
ISSN 1027-3190. Укр. мат. журн., 2018, т. 70, № 5
638 E. IYIGÜN
\mathrm{d}\mathrm{e}\mathrm{t}(m\prime
2,m
\prime
3,m
\prime
4,m
\prime
5) = 0 \Leftarrow \Rightarrow
5\sum
i=2
m2
i = \mathrm{c}\mathrm{o}\mathrm{n}\mathrm{s}\mathrm{t}\mathrm{a}\mathrm{n}\mathrm{t}.
Proof. The proof can seen by using the definitions of ith e-curvature functions and harmonic
curvatures.
Definition 5. Let \alpha be a non-null curve of osculating order 5 in \BbbL 5, with Frenet curvatures
k1, k2, k3, k4. Let us denote
a0 = k2k4,
aj = \varepsilon j
k2j - 1
k2j
aj - 1, 1 \leq j \leq 2, k2j \not = 0.
The Darboux vector in \BbbL 5 is defined by
D(s) =
2\sum
j=0
ajV2j+1 = a0V1 + a1V3 + a2V5,
where V2j+1 are the Frenet vectors of \alpha .
Corollary 3. Let \alpha be a non-null curve of osculating order 5 in \BbbL 5, then
aj
aj - 1
= \varepsilon j
k2j - 1
k2j
,
where 1 \leq j \leq 2, k2j \not = 0.
Lemma 1. The derivative of the Darboux vector D(s) is [4].
D\prime (s) = a\prime 0V1 + a\prime 1V3 + a\prime 2V5.
Definition 6. The point \alpha (s0) is called Darboux vertex of \alpha if the first derivative of the Darboux
vector D(s) is vanishing at that point [3].
Theorem 4. Let \alpha be a non-null curve of osculating order 5 in \BbbL 5, with Frenet curvatures k1,
k2, k4 and harmonic curvatures H1, H2. Let us denote
a0 =
k1k4
H1
,
a1 = \varepsilon 1H1k2k4,
a2 = \varepsilon 1
H \prime
1
H2
k1,
where H1 \not = 0, H2 \not = 0, \varepsilon 1 = \langle V1, V1\rangle = \pm 1.
Proof. By using definition of harmonic curvature, we get the result.
The following theorem is true.
Theorem 5. Let \alpha : I - \rightarrow \BbbL 5 be a non-null curve of osculating order 5. Then
H1 = \varepsilon 1
a1
a0
, a0 \not = 0,
ISSN 1027-3190. Укр. мат. журн., 2018, т. 70, № 5
ON DARBOUX VECTOR IN LORENTZIAN 5-SPACE 639
H2 = \varepsilon 1
a0H1H
\prime
1
a2k4
, a2 \not = 0, k4 \not = 0,
where H1, H2 are harmonic curvatures of \alpha , k4 is Frenet curvature of \alpha , \varepsilon 1 = \langle V1, V1\rangle = \pm 1 and
a0, a1, a2 \in R.
Proof. By the use of a previous theorem, we obtain the theorem.
Corollary 4. (i) If the vector V1 is space-like, then
H1 =
a1
a0
, a0 \not = 0,
H2 =
a0H1H
\prime
1
a2k4
, a2 \not = 0, k4 \not = 0.
(ii) If the vector V1 is time-like, then
H1 = - a1
a0
, a0 \not = 0,
H2 = - a0H1H
\prime
1
a2k4
, a2 \not = 0, k4 \not = 0.
Theorem 6. Let \alpha be a non-null curve of osculating order 5 in \BbbL 5,with Frenet curvatures k1,
k2, k3, k4. Then the curve \alpha has a Darboux vertex at point \alpha (s) if and only if
\varepsilon i
\biggl(
ki
ki+1
\biggr) \prime
= 0, ki+1 \not = 0, 1 \leq i \leq 3,
where \varepsilon i = \langle Vi, Vi\rangle = \pm 1.
Corollary 5. (i) If the vector V1 is space-like, then
(H1)
\prime = 0.
(ii) If the vector V1 is time-like, then
- (H1)
\prime = 0.
Corollary 6. If \alpha : I - \rightarrow \BbbL 5 has a Darboux vertex at the point \alpha (s0), then \alpha is a general helix
of order 3 [3].
4. ccr-Curve in \BbbL 5 .
Definition 7. A curve \alpha : I - \rightarrow \BbbL 5 is said to have constant curvature ratios (that is to say, it
is a ccr-curve) if all the quotients \varepsilon i
\biggl(
ki+1
ki
\biggr)
are constant, ki \not = 0. Here, ki, ki+1, 1 \leq i \leq 3, are
Frenet curvatures of \alpha , and \varepsilon i = \langle Vi, Vi\rangle = \pm 1, 1 \leq i \leq 3.
Theorem 7. (i) For i = 1, the ccr-curve is
a1
a0(H1)2
.
(ii) For i = 2, the ccr-curve is \varepsilon 1
a2H1
(k1)2
.
(iii) For i = 3, the ccr-curve is \varepsilon 2\varepsilon 3
a0H2
k2H \prime
1
.
Here H1, H2 are harmonic curvatures of \alpha ; k1, k2 are Frenet curvatures of \alpha ; \varepsilon i = \langle Vi, Vi\rangle =
= \pm 1, 1 \leq i \leq 3, and a0, a1, a2 \in R.
ISSN 1027-3190. Укр. мат. журн., 2018, т. 70, № 5
640 E. IYIGÜN
Proof. The proof can be easily seen by using the definitions of the harmonic curvature and
ccr-curve.
Corollary 7. (i) If the vector V1 is space-like, then the ccr-curve is
a2H1
(k1)2
, where \varepsilon 1 = 1.
(ii) If the vector V1 is time-like, then the ccr-curve is - a2H1
(k1)2
, where \varepsilon 1 = - 1.
(iii) If the vectors V2, V3 are space-like, then the ccr-curve is
a0H2
k2H \prime
1
, where \varepsilon 2 = \varepsilon 3 = 1.
(iv) If the vector V2 is time-like, then the ccr-curve is - a0H2
k2H \prime
1
, where \varepsilon 2 = - 1 and \varepsilon 3 = 1.
(v) If the vector V3 is time-like, then the ccr-curve is - a0H2
k2H \prime
1
, where \varepsilon 3 = - 1 and \varepsilon 2 = 1.
Corollary 8. \alpha is a ccr-curve in \BbbL 5 \leftrightarrow \varepsilon 1 (H1)
- 1 = \mathrm{c}\mathrm{o}\mathrm{n}\mathrm{s}\mathrm{t}\mathrm{a}\mathrm{n}\mathrm{t}.
Corollary 9. For \alpha : I - \rightarrow \BbbL 5 curve, we have
\varepsilon i
\biggl(
ki
ki+1
\biggr) \prime
= \varepsilon i
\biggl(
ki+1
ki
\biggr)
, 1 \leq i \leq 3,
that is the Darboux vertex is equal to the constant curvature ratio.
Now, we will calculate Darboux vector and Darboux vertex of the unit speed time-like curve in
\BbbL 5 studied in [5].
5. An example.
Example 1. Let us consider the following curve:
\alpha (s) =
\Bigl( \surd
3 \mathrm{s}\mathrm{i}\mathrm{n}\mathrm{h} s,
\surd
3 \mathrm{c}\mathrm{o}\mathrm{s}\mathrm{h} s, \mathrm{s}\mathrm{i}\mathrm{n} s, s, \mathrm{c}\mathrm{o}\mathrm{s} s
\Bigr)
,
V1(s) = \alpha \prime (s) =
\Bigl( \surd
3 \mathrm{c}\mathrm{o}\mathrm{s}\mathrm{h} s,
\surd
3 \mathrm{s}\mathrm{i}\mathrm{n}\mathrm{h} s, \mathrm{c}\mathrm{o}\mathrm{s} s, 1, - \mathrm{s}\mathrm{i}\mathrm{n} s
\Bigr)
,
where \langle \alpha \prime (s), \alpha \prime (s)\rangle = - 1. One can easily see that \alpha (s) is an unit speed time-like curve. We express
the following differentiations:
\alpha \prime \prime (s) =
\Bigl( \surd
3 \mathrm{s}\mathrm{i}\mathrm{n}\mathrm{h} s,
\surd
3 \mathrm{c}\mathrm{o}\mathrm{s}\mathrm{h} s, - \mathrm{s}\mathrm{i}\mathrm{n} s, 0, - \mathrm{c}\mathrm{o}\mathrm{s} s
\Bigr)
\Rightarrow
\Rightarrow \alpha \prime \prime \prime (s) =
\Bigl( \surd
3 \mathrm{c}\mathrm{o}\mathrm{s}\mathrm{h} s,
\surd
3 \mathrm{s}\mathrm{i}\mathrm{n}\mathrm{h} s, - \mathrm{c}\mathrm{o}\mathrm{s} s, 0, \mathrm{s}\mathrm{i}\mathrm{n} s
\Bigr)
\Rightarrow
\Rightarrow \alpha \imath v(s) =
\Bigl( \surd
3 \mathrm{s}\mathrm{i}\mathrm{n}\mathrm{h} s,
\surd
3 \mathrm{c}\mathrm{o}\mathrm{s}\mathrm{h} s, \mathrm{s}\mathrm{i}\mathrm{n} s, 0, \mathrm{c}\mathrm{o}\mathrm{s} s
\Bigr)
\Rightarrow
\Rightarrow \alpha v(s) =
\Bigl( \surd
3 \mathrm{c}\mathrm{o}\mathrm{s}\mathrm{h} s,
\surd
3 \mathrm{s}\mathrm{i}\mathrm{n}\mathrm{h} s, \mathrm{c}\mathrm{o}\mathrm{s} s, 0, - \mathrm{s}\mathrm{i}\mathrm{n} s
\Bigr)
.
So, we have the first curvature as \bigm\| \bigm\| \alpha \prime \prime (s)
\bigm\| \bigm\| = k1(s) = 2.
Moreover we can write second, third, fourth and fifth Frenet vectors of the curve, respectively,
V2(s) = \varepsilon 2
\Biggl( \surd
3
2
\mathrm{s}\mathrm{i}\mathrm{n}\mathrm{h} s,
\surd
3
2
\mathrm{c}\mathrm{o}\mathrm{s}\mathrm{h} s, - 1
2
\mathrm{s}\mathrm{i}\mathrm{n} s, 0, - 1
2
\mathrm{c}\mathrm{o}\mathrm{s} s
\Biggr)
,
ISSN 1027-3190. Укр. мат. журн., 2018, т. 70, № 5
ON DARBOUX VECTOR IN LORENTZIAN 5-SPACE 641
V3(s) =
1\surd
14
\Bigl(
- 3
\surd
3 \mathrm{c}\mathrm{o}\mathrm{s}\mathrm{h} s, - 3
\surd
3 \mathrm{s}\mathrm{i}\mathrm{n}\mathrm{h} s, - 5 \mathrm{c}\mathrm{o}\mathrm{s} s, - 4, 5 \mathrm{s}\mathrm{i}\mathrm{n} s
\Bigr)
,
V4(s) = \mu
\biggl(
- 1
2
\mathrm{s}\mathrm{i}\mathrm{n}\mathrm{h} s, - 1
2
\mathrm{c}\mathrm{o}\mathrm{s}\mathrm{h} s, - 3
2
\mathrm{s}\mathrm{i}\mathrm{n} s, 0, - 3
2
\mathrm{c}\mathrm{o}\mathrm{s} s
\biggr)
and
V5(s) = \mu
\Biggl(
- 1\surd
14
\mathrm{c}\mathrm{o}\mathrm{s}\mathrm{h} s, - 1\surd
14
\mathrm{s}\mathrm{i}\mathrm{n}\mathrm{h} s,
\sqrt{}
3
14
\mathrm{c}\mathrm{o}\mathrm{s} s, -
\sqrt{}
2
7
, -
\sqrt{}
3
14
\mathrm{s}\mathrm{i}\mathrm{n} s
\Biggr)
,
where \mu is taken \mp 1 to make +1 determimant of \{ V1(s), V2(s), V3(s), V4(s), V5(s)\} matrix. In ad-
dition to, we can write second, third, fourth curvatures and harmonic curvature of \alpha (s), respectively,
k2(s) =
\surd
14, k3(s) =
\sqrt{}
3
14
, k4(s) =
\sqrt{}
2
7
, H1 =
2\surd
14
,
a0 = 2, a1 = \varepsilon 1
\Biggl(
2
\surd
2\surd
7
\Biggr)
, a2 = 0.
Now, we will calculate ccr-curves of \alpha (s) in \BbbL 5. If the vector V1 is time-like, then \mu = 1, \varepsilon 1 = - 1
and \varepsilon 2 = \varepsilon 3 = 1
\varepsilon 1
k2
k1
= -
\surd
14
2
= \mathrm{c}\mathrm{o}\mathrm{n}\mathrm{s}\mathrm{t}\mathrm{a}\mathrm{n}\mathrm{t}, \varepsilon 2
\mathrm{k}3
\mathrm{k}2
=
\surd
3
14
= \mathrm{c}\mathrm{o}\mathrm{n}\mathrm{s}\mathrm{t}\mathrm{a}\mathrm{n}\mathrm{t} and \varepsilon 3
\mathrm{k}4
\mathrm{k}3
=
2\surd
3
= \mathrm{c}\mathrm{o}\mathrm{n}\mathrm{s}\mathrm{t}\mathrm{a}\mathrm{n}\mathrm{t}.
Thus, \alpha (s) is a ccr-curve in \BbbL 5. Also, we obtain, respectively, Darboux vector and Darboux vertex
of \alpha (s) in \BbbL 5,
(i) If the vector V1 is time-like, then \mu = 1, \varepsilon 1 = - 1 and \varepsilon 2 = \varepsilon 3 = 1,
D(s) =
\Biggl(
20
\surd
3
7
\mathrm{c}\mathrm{o}\mathrm{s}\mathrm{h} s,
20
\surd
3
7
\mathrm{s}\mathrm{i}\mathrm{n}\mathrm{h} s,
24
7
\mathrm{c}\mathrm{o}\mathrm{s} s,
22
7
, - 24
7
\mathrm{s}\mathrm{i}\mathrm{n} s
\Biggr)
and
D\prime (s) =
\Biggl(
20
\surd
3
7
\mathrm{s}\mathrm{i}\mathrm{n}\mathrm{h} s,
20
\surd
3
7
\mathrm{c}\mathrm{o}\mathrm{s}\mathrm{h} s, - 24
7
\mathrm{s}\mathrm{i}\mathrm{n} s, 0, - 24
7
\mathrm{c}\mathrm{o}\mathrm{s} s
\Biggr)
.
(ii) If the vector V1 is time-like, then Darboux vector and Darboux vertex are as in (i) when
\mu = - 1, \varepsilon 1 = - 1 and \varepsilon 2 = \varepsilon 3 = 1, since a2 = 0.
References
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Received 09.11.16
ISSN 1027-3190. Укр. мат. журн., 2018, т. 70, № 5
|
| id | umjimathkievua-article-1583 |
| institution | Ukrains’kyi Matematychnyi Zhurnal |
| keywords_txt_mv | keywords |
| language | English |
| last_indexed | 2026-03-24T02:08:32Z |
| publishDate | 2018 |
| publisher | Institute of Mathematics, NAS of Ukraine |
| record_format | ojs |
| resource_txt_mv | umjimathkievua/9c/793600d3079f5585200c8504f169669c.pdf |
| spelling | umjimathkievua-article-15832019-12-05T09:19:33Z On Darboux vector in Lorentzian 5-space Про вектор дарбу в 5-просторi Лоренца Iyigün, E. Ійігун, Е. We introduce the Darboux vector in the Lorentzian 5-space. We give some characterizations of this vector in the space. In addition, we consider some special cases in the space. Введено вектор Дарбу в 5-просторi Лоренца. Наведено деякi харатеристики даного вектора в цьому просторi. Крiм того, розглянуто деякi частиннi випадки в цьому ж просторi. Institute of Mathematics, NAS of Ukraine 2018-05-25 Article Article application/pdf https://umj.imath.kiev.ua/index.php/umj/article/view/1583 Ukrains’kyi Matematychnyi Zhurnal; Vol. 70 No. 5 (2018); 635-641 Український математичний журнал; Том 70 № 5 (2018); 635-641 1027-3190 en https://umj.imath.kiev.ua/index.php/umj/article/view/1583/565 Copyright (c) 2018 Iyigün E. |
| spellingShingle | Iyigün, E. Ійігун, Е. On Darboux vector in Lorentzian 5-space |
| title | On Darboux vector in Lorentzian 5-space |
| title_alt | Про вектор дарбу в 5-просторi Лоренца |
| title_full | On Darboux vector in Lorentzian 5-space |
| title_fullStr | On Darboux vector in Lorentzian 5-space |
| title_full_unstemmed | On Darboux vector in Lorentzian 5-space |
| title_short | On Darboux vector in Lorentzian 5-space |
| title_sort | on darboux vector in lorentzian 5-space |
| url | https://umj.imath.kiev.ua/index.php/umj/article/view/1583 |
| work_keys_str_mv | AT iyigune ondarbouxvectorinlorentzian5space AT íjígune ondarbouxvectorinlorentzian5space AT iyigune provektordarbuv5prostorilorenca AT íjígune provektordarbuv5prostorilorenca |