Bicomplex number and tensor product surfaces in $\mathbb{R}^4_2$
We show that a hyperquadric $M$ in $\mathbb{R}^4_2$ is a Lie group by using the bicomplex number product. For our purpose, we change the definition of tensor product. We define a new tensor product by considering the tensor product surface in the hyperquadric $M$. By using this new tensor product,...
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Institute of Mathematics, NAS of Ukraine
2012
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| author | Karakus, S. O. Yayli, Y. Каракус, С. О. Яйлі, І. |
| author_facet | Karakus, S. O. Yayli, Y. Каракус, С. О. Яйлі, І. |
| author_sort | Karakus, S. O. |
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| datestamp_date | 2020-03-18T19:30:02Z |
| description | We show that a hyperquadric $M$ in $\mathbb{R}^4_2$ is a Lie group by using the bicomplex number product. For our purpose, we change the definition of tensor product.
We define a new tensor product by considering the tensor product surface in the hyperquadric $M$. By using this new tensor product,
we classify totally real tensor product surfaces and complex tensor product surfaces of a Lorentzian plane curve and a Euclidean plane curve.
By means of the tensor product surfaces of a Lorentzian plane curve and a Euclidean plane curve, we determine a special subgroup of the Lie group M. Thus, we obtain the Lie group structure of tensor product surfaces of a Lorentzian plane curve and a Euclidean plane curve. Morever, we obtain left invariant vector fields of these Lie groups. We consider the left invariant vector fields on these groups, which constitute a pseudo-Hermitian structure. By using this, we characterize these Lie groups as totally real or slant in $\mathbb{R}^4_2$. |
| first_indexed | 2026-03-24T02:26:09Z |
| format | Article |
| fulltext |
UDC 514.7
S. Ö. Karakuş (Bilecik Univ., Turkey),
Y. Yayli (Ankara Univ., Turkey)
BICOMPLEX NUMBER AND TENSOR PRODUCT SURFACES IN R4
2
ПОВЕРХНI ДОБУТКУ БIКОМПЛЕКСНИХ ЧИСЕЛ
ТА ТЕНЗОРНОГО ДОБУТКУ В R4
2
We show that a hyperquadric M in R4
2 is a Lie group by using the bicomplex number product. For our purpose, we
change the definition of tensor product. We define a new tensor product by considering the tensor product surface in the
hyperquadric M. By using this new tensor product, we classify totally real tensor product surfaces and complex tensor
product surfaces of a Lorentzian plane curve and a Euclidean plane curve.
By means of the tensor product surfaces of a Lorentzian plane curve and a Euclidean plane curve, we determine a
special subgroup of the Lie group M. Thus, we obtain the Lie group structure of tensor product surfaces of a Lorentzian
plane curve and a Euclidean plane curve. Morever, we obtain left invariant vector fields of these Lie groups. We consider
the left invariant vector fields on these groups, which constitute a pseudo-Hermitian structure. By using this, we characterize
these Lie groups as totally real or slant in R4
2.
Iз використанням добутку бiкомплексних чисел показано, що гiперквадрика M у R4
2 є групою Лi. Для досягнення
нашої мети модифiковано означення тензорного добутку. Новий тензорний добуток означено шляхом розгляду
поверхнi тензорного добутку в гiперквадрицi M. За допомогою цього нового добутку класифiковано тотально
дiйснi поверхнi тензорного добутку та комплекснi поверхнi тензорного добутку плоскої кривої Лоренца та евклiдової
плоскої кривої.
За допомогою поверхонь тензорного добутку плоскої кривої Лоренца та евклiдової плоскої кривої отримано
спецiальну пiдгрупу групи Лi M. Таким чином, отримано структуру групи Лi для поверхонь тензорного добутку
плоскої кривої Лоренца та евклiдової плоскої кривої. Крiм того, отримано лiвоiнварiантнi векторнi поля цих груп
Лi. Розглянуто лiвоiнварiантнi векторнi поля на цих групах, якi утворюють псевдоермiтову структуру. Це дає змогу
охарактеризувати групи Лi як тотально дiйснi або скiснi в R4
2.
1. Introduction. In the Euclidean space En, the tensor product immersion of two immersions of a
given Riemannian manifold was firstly defined and studied by Chen in [3]. In particular, the direct
sum and the tensor product maps of two immersions of two different Riemannian manifolds are
defined by Decruyenaere and coauthors in [4] in the following way:
Let M and N be two differentiable manifolds and assume that f : M → Em and h : N → En
are two immersions. The direct sum map and tensor product map are defined respectively by
f ⊕ h : M ×N → Em+n, (f ⊕ h) (p, q) =
(
f1(p), . . . , fm(p), h1(q), . . . , hn(q)
)
,
and
f ⊗ h : M ×N → Emn, (f ⊗ h) (p, q) =
(
f1(p)h1(q), . . . , f1(p)hn(q), . . . , fm(p)hn(q)
)
.
Under certain conditions obtained in [4], the tensor product map f ⊗ h is an immersion in the
space Emn.
The simplest examples of the tensor product immersions are tensor product surfaces. In the
Euclidean space En, the tensor product surfaces of two Euclidean planar curves, as well as of a
Euclidean space curve and a Euclidean plane curve are investigated in [8] and [1], respectively.
Morever, in the semi-Euclidean space Env , the tensor product surfaces of two Lorentzian planar
c© S. Ö. KARAKUŞ, Y. YAYLI, 2012
ISSN 1027-3190. Укр. мат. журн., 2012, т. 64, № 3 307
308 S. Ö. KARAKUŞ, Y. YAYLI
curves, as well as of a Lorentzian plane curve and a Euclidean plane curve are studied in [9] and
[10], respectively. Also, the tensor product surfaces of a Lorentzian space curve and a Euclidean
plane curve as well as of a Euclidean space curve and a Lorentzian plane curve are studied in [5] and
[6], respectively.
It is often quite diffucult to decide if a manifold is paralelizable. Sn is paralelizable if and only
if n = 1, 3, 7. Is it possible to make paralelization of any surface? The answer is yes. If M is a Lie
group then M is paralelizable.
In [12], the authors showed that a hyperquadric M in R4 is a Lie group by using bicomplex
number product. Also, in the same paper, Lie group structure of tensor product surfaces of Euclidean
planar curves was obtained .
In this paper, we obtain Lie group structure of some special hypersurface in R4
2. By changing the
tensor product rule given in R4
2 in [3] we give a new tensor product definition. As a result, the tensor
product surface is obtained as a subset of the hyperquadric M. Hence, we investigate the tensor
product surface as a Lie group. For our aim, if we change the definition of tensor product given in
above as;
(α⊗ β) (t, s) =
(
α1(t)β1(s), α1(t)β2(s),−α2(t)β2(s), α2(t)β1(s)
)
we can easily obtain the same results which given in [10]. By using the new tensor product, we
classify totally real tensor product surfaces and complex tensor product surfaces of a Lorentzian
plane curve and a Euclidean plane curve. Furthermore we give some theorems for tensor product
surfaces to be Lie groups and one parameter Lie subgroups. Finally, we give the necessary conditions
for Lie group structures of tensor product surfaces to be totally real or slant in R4
2, respectively.
At the beginning, we recall notions of bicomplex numbers.
2. Preliminary. A bicomplex number is defined by the basis {1, i, j, ij} where i, j, ij satisfy
i2 = −1, j2 = −1, ij = ji. Thus any bicomplex number x can be expressed as x = x11 +
+ x2i + x3j + x4ij ∀x1, x2, x3, x4 ∈ R. We denote the set of bicomplex numbers by C2. For any
x = x11 + x2i+ x3j + x4ij and y = y11 + y2i+ y3j + y4ij in C2, the bicomplex number addition
is defined as
x+ y = (x1 + y1) 1 + (x2 + y2) i+ (x3 + y3) j + (x4 + y4) ij.
The multiplication of a bicomplex number x = x11 + x2i + x3j + x4ij by a real scalar λ is
defined as
λx = λx11 + λx2i+ λx3j + λx4ij.
With this addition and scalar multiplication operations, C2 is a real vector space.
Bicomplex number product, denoted by ×, over the set of bicomplex numbers C2 is given by
the following table:
× 1 i j ij
1 1 i j ij
i i −1 ij −j
j j ij −1 −i
ij ij −j −i 1
ISSN 1027-3190. Укр. мат. журн., 2012, т. 64, № 3
BICOMPLEX NUMBER AND TENSOR PRODUCT SURFACES IN R4
2 309
The vector space C2 together with the bicomplex product × is an real algebra [13].
Since bicomplex number algebra is associative it can be considered in terms of matrices. Consider
the set of matrices
Q =
x1 −x2 −x3 x4
x2 x1 −x4 −x3
x3 −x4 x1 −x2
x4 x3 x2 x1
: xi ∈ R, 1 ≤ i ≤ 4
.
The set Q together with matrix addition and scalar matrix multiplication is a real vector space.
Furthermore, the vector space together with matrix product is an algebra.
The transformation
h : C2 → Q
given by
h (x = x11 + x2i+ x3j + x4ij) =
x1 −x2 −x3 x4
x2 x1 −x4 −x3
x3 −x4 x1 −x2
x4 x3 x2 x1
is one-to-one and onto. Moreover, ∀x, y ∈ C2 and ∀λ ∈ R, we have
h (x+ y) = h (x)⊕ h (y) ,
h (λx) = λh (x) ,
h (x× y) = h (x)h (y) .
Thus the algebras C2 and Q are isomorphic.
For further bicomplex number concepts see [13].
3. Tensor product surfaces of a Lorentzian plane curve and a Euclidean plane curve. In this
section, we change the definition of tensor product as follows:
Let α : R→ R2
1 (+−) and β : R→ R2 be respectively a Lorentzian planar curve and a Euclidean
planar curve. Put α(t) = (α1(t), α2(t)) and β(s) = (β1(s), β2(s)). Let us define their tensor prod-
uct as
f = α⊗ β : R2 → R4
2(+ +−−),
f (t, s) =
(
α1(t)β1(s), α1(t)β2(s),−α2(t)β2(s), α2(t)β1(s)
)
.
(3.1)
By using equation (3.1), the canonical tangent vectors of f (t, s) can be easily computed as
∂f
∂t
=
(
α′
1(t)β1(s), α
′
1(t)β2(s),−α′
2(t)β2(s), α
′
2(t)β1(s)
)
,
∂f
∂s
=
(
α1(t)β
′
1(s), α1(t)β
′
2(s),−α2(t)β
′
2(s), α2(t)β
′
1(s)
)
,
(3.2)
where α′ means the derivative of α.
ISSN 1027-3190. Укр. мат. журн., 2012, т. 64, № 3
310 S. Ö. KARAKUŞ, Y. YAYLI
In the following, we will assume that α is a spacelike or a timelike curve with spacelike or a
timelike position vector and we will assume that β is a regular curve. We shall also assume that the
tensor product surface f (t, s) is a regular surface, i.e., g11g22 − g212 6= 0.
Hence relations (3.1) and (3.2) imply that the coefficients of the pseudo-Riemannian metric,
induced on f (t, s) by the pseudo-Euclidian metric g of R4
2 is given g = dx21 + dx22− dx23− dx24, are
g11 = g
(
∂f
∂t
,
∂f
∂t
)
= g1
(
α′, α′ ) g2 (β, β) ,
g12 = g
(
∂f
∂t
,
∂f
∂s
)
= g1
(
α, α′) g2 (β, β′) ,
g22 = g
(
∂f
∂s
,
∂f
∂s
)
= g1 (α, α) g2
(
β′, β′
)
,
where g1 = dx21−dx22 and g2 = dx21 +dx22 are the metrics of R2
1 and R2, respectively. Consequently,
an orthonormal basis for the tangent space of f (t, s) is given by
e1 =
1√
|g11|
∂f
∂t
,
e2 =
1√∣∣g11 (g11g22 − g212)∣∣
(
g11
∂f
∂s
− g12
∂f
∂t
)
.
Recall that the mean curvature vector field H is defined by
H =
1
2
(
ε1h(e1, e1) + ε2h(e2, e2)
)
,
where h is a second fundamental form of α⊗β and εi = g(ei, ei), i = 1, 2. In particular by Beltrami’s
formula we have
H = −1
2
∆f.
Next, recall that a surface M in R4
2 is said to be minimal, if its mean curvature vector field H
vanishes identically.
A basis of the normal space of f(t, s) can ce calculated as follows. Let J1 : R2
1 → R2
1 and
J2 : R2 → R2 be the folowing maps:
J1(x, y) = (y, x),
J2(x, y) = (−y, x).
Observe that g1 (X, J1(X)) = 0 for X ∈ R2
1 and g2 (Y, J2(Y )) = 0 for Y ∈ R2.
Then the normal space is spanned by
n1(t, s) = J1(α(t))⊗ J2 (β(s)) = (α2(t), α1(t))⊗ (−β2(s), β1(s)) =
=
(
− α2(t)β2(s), α2(t)β1(s),−α1(t)β1(s),−α1(t)β2(s)
)
,
ISSN 1027-3190. Укр. мат. журн., 2012, т. 64, № 3
BICOMPLEX NUMBER AND TENSOR PRODUCT SURFACES IN R4
2 311
n2(t, s) = J1(α
′(t))⊗ J2(β′(s)) = (α′
2(t), α
′
1(t))⊗ (−β′2(s), β′1(s)) =
=
(
− α′
2(t)β
′
2(s), α
′
2(t)β
′
1(s),−α′
1(t)β
′
1(s),−α′
1(t)β
′
2(s)
)
.
4. Totally real and complex Lorentzian immersion and slant tensor product surface. Let
α : R→ R2
1 (+−) and β : R→ R2 be respectively a Lorentzian planar curve and a Euclidean planar
curve and let f = α⊗β be their tensor product. We consider the pseudo-Hermitian structure J given
by
J (u, v, z, w) = (−v, u,−w, z) , u, v, z, w ∈ R.
In the next theorem by using the new product we classify totally real tensor product surface in the
semi-Euclidean space R4
2, i.e., the pseudo-Hermitian structure J at each point transforms the tangent
space to the surface into the normal space.
Theorem 1. The tensor product immersion f = α ⊗ β of a Lorentzian plane curve and a
Euclidean plane curve is a totally real Lorentzian immersion with respect to the pseudo-Hermitian
structure J on R4
2 if and only if α is a Lorentzian circle centered at 0 or β is a straight line passing
through origin.
Proof. Im f is a totally real surface if and only if J
(
∂f
∂t
)
is orthogonal to
∂f
∂s
and J
(
∂f
∂s
)
is
orthogonal to
∂f
∂t
.
We have
J
(
∂f
∂t
)
=
(
−α′
1(t)β2(s), α
′
1(t)β1(s),−α′
2(t)β1(s),−α′
2(t)β2(s)
)
.
By a straightforward calculation, we obtain
g
(
J
(
∂f
∂t
)
,
∂f
∂s
)
= −g
(
J
(
∂f
∂s
)
,
∂f
∂t
)
,
g
(
J
(
∂f
∂t
)
,
∂f
∂s
)
= 0
if and only if (α1α
′
1 − α2α
′
2) = 0 or (β1β
′
2 − β′1β2) = 0. Integrating these equations, we find that
either β is a straight line passing through origin, or α is a Lorentzian circle centered at 0.
Theorem 1 is proved.
Theorem 2. The tensor product immersion f = α ⊗ β of a Lorentzian plane curve and a
Euclidean plane curve is a complex Lorentzian immersion with respect to the pseudo-Hermitian
structure J on R4
2 if and only if α is a straight line passing through origin and β is an Euclidean
planar curve.
Proof. By definition, the following equations are satisfied:
g
(
J
(
∂f
∂t
)
, ni
)
= 0, g
(
J
(
∂f
∂s
)
, ni
)
= 0, i = 1, 2. (4.1)
We have
ISSN 1027-3190. Укр. мат. журн., 2012, т. 64, № 3
312 S. Ö. KARAKUŞ, Y. YAYLI
J
(
∂f
∂t
)
=
(
−α′
1(t)β2(s), α
′
1(t)β1(s),−α′
2(t)β1(s),−α′
2(t)β2(s)
)
,
J
(
∂f
∂s
)
=
(
−α1(t)β
′
2(s), α1(t)β
′
1(s),−α2(t)β
′
1(s),−α2(t)β
′
2(s)
)
.
By a straightforward calculation, we obtain
g
(
J
(
∂f
∂t
)
, n2
)
= g
(
J
(
∂f
∂s
)
, n1
)
= 0,
g
(
J
(
∂f
∂t
)
, n1
)
=
(
α1(t)α
′
2(t)− α′
1(t)α2(t)
) (
β21(s) + β22(s)
)
,
g
(
J
(
∂f
∂s
)
, n2
)
=
(
α′
1(t)α2(t)− α1(t)α
′
2(t)
) (
β′21 (s) + β′22 (s)
)
.
(4.2)
From equations (4.1) and (4.2) we have
α′
1(t)α2(t)− α1(t)α
′
2(t) = 0.
It follows that α is straight line passing through the origin.
Theorem 2 is proved.
Recall the definition of a slant surface with respect to pseudo-Hermitian structure J on R4
2. Let
M be a surface with respect to the pseudo-Hermitian structure J on R4
2. Then M is said to be a
proper slant if
g (J (e1), e2) = λ, λ ∈ R,
along M for a given orthonormal basis {e1, e2} of TpM (p ∈M) which is independent of the choice
of {e1, e2} [3].
Let α : R→ R2
1 (+−) and β : R→ R2 be respectively a Lorentzian planar curve and a Euclidean
planar curve. We consider polar coordinates on α and β. Then,
α(t) = ρ1(t) (cosh t, sinh t) ,
β (s) = ρ2 (s) (cos s, sin s) .
A straightforward computation leads to
g (J (e1) , e2) =
ρ′1ρ2√∣∣(ρ′21 − ρ21) (ρ′22 + ρ22
)
− ρ′21 ρ′22
∣∣ .
If ρ2 = constant, it follows that ρ1 = a1e
b1t, a1 ∈ R, b1 ∈ R. Hence α is a hyperbolic spiral and β
is a circle centered at the origin. If ρ2 6=constant, let us put
ρk
ρ′k
= ck, k = 1, 2. Then
g (J (e1) , e2) =
c2√∣∣(c22 + 1
) (
1− c21
)
− 1
∣∣ .
ISSN 1027-3190. Укр. мат. журн., 2012, т. 64, № 3
BICOMPLEX NUMBER AND TENSOR PRODUCT SURFACES IN R4
2 313
Therefore Im f is a proper slant surface if and only if
c22(
c22 + 1
) (
1− c21
)
− 1
= λ2.
It follows that
c22 + λ2
c22 + 1
= λ2
(
1− c21
)
.
This means that c1(t) and c2(s) must be constant functions, which implies that ρ1(t) = a1e
b1t,
ρ2(s) = a2e
b2s, a1, b1, a2, b2 ∈ R. Consequently, α is a hyperbolic spiral and β is a logarithmic
spiral. In this way, we proved the following theorem.
Theorem 3. The tensor product immersion f = α ⊗ β of a Lorentzian plane curve and a
Euclidean plane curve is a slant surface with respect to pseudo-Hermitian structure J on R4
2 if and
only if α is a hyperbolic spiral and β is either a circle centered at O or a spiral curve.
5. Lie groups and some special subgroup. In this section, we deal with the hyperquadric
M =
{
x =
(
x1, x2, x3, x4
)
: x1x3 + x2x4 = 0, g (x, x) 6= 0
}
,
M =
{
x =
(
x1, x2, x3, x4
)
: x1x3 + x2x4 = 0, x21 + x22 − x23 − x24 6= 0
}
.
We consider M as the set of bicomplex numbers,
M =
{
x = x11 + x2i+ x3j + x4ij ∈ C2 : x1x3 + x2x4 = 0, g (x, x) 6= 0
}
.
The components of M are easily obtained by representing bicomplex number multiplication in
matrix form:
M̃ =
x =
x1 −x2 −x3 x4
x2 x1 −x4 −x3
x3 −x4 x1 −x2
x4 x3 x2 x1
: x1x3 + x2x4 = 0, g (x, x) 6= 0
.
Theorem 4. The set of M together with the bicomplex number product is a Lie group.
Proof. M̃ is a differentiable manifold and at the same time a group with group operation given
by matrix multiplication. The group function
. : M̃ × M̃ → M̃
defined by (x, y) → x.y is differentiable. So, (M,×) can be made a Lie group so that h is a
isomorphism.
Theorem 4 is proved.
Consider the group M1 of all unit bicomplex numbers x = x11 + x2i + x3j + x4ij on M with
the group operation of bicomplex multiplication. That is
M1 = {x ∈M : g (x, x) = 1} ,
M1 =
{
x ∈M : x21 + x22 − x23 − x24 = 1
}
.
ISSN 1027-3190. Укр. мат. журн., 2012, т. 64, № 3
314 S. Ö. KARAKUŞ, Y. YAYLI
Lemma 1. M1 is 2-dimensional Lie subgroup of M.
6. Lie algebra of Lie group M and M1. M is a Lie group of dimension three. Let us find its
Lie algebra. Thus, let
α(t) = α1(t)1 + α2(t)i+ α3(t)j + α4(t)ij
be a curve on M such that α (0) = 1, i.e., α1 (0) = 1, αm (0) = 0 for m = 2, 3, 4. Differentiation of
the equation α1(t)α3(t)+α2(t)α4(t) = 0 yields the equation α′
1(t)α3(t)+α1(t)α
′
3 (t)+α′
2(t)α4(t)+
+ α2(t)α
′
4(t) = 0. Substituting t = 0, we obtain α′
3 (0) = 0. The Lie algebra is thus constituted by
vectors of the form ζ = ζm
(
∂
∂αm
)
|α=1, where m = 1, 2, 4. The vector ζ is formally written in the
form ζ = ζ1+ζ2i+ζ4ij. Let us find the left invariant vector field X on M for which X |α=1= ζ. Let
β (t) be a curve on M such that β (0) = 1, β′ (0) = ζ. Then Lx (β(t)) = xβ(t) is the left translation
of the curve β(t) by the bicomplex number x, its tangent vector is xβ′ (0) = xζ. In particular, denote
by Xm those left invariant vector fields on M for which,
Xm |α=1=
∂
∂αm
∣∣∣∣
α=1
,
where m = 1, 2, 4. These three vector fields are represented at the point α = 1, in bicomplex
notation, by the bicomplex units 1, i, ij. For the components of these vector fields at the point
x = x11 + x2i+ x3j + x4ij we have (X1)x = x1, (X2)x = xi, (X3)x = xij:
X1 = (x1, x2, x3, x4),
X2 = (−x2, x1,−x4, x3),
X4 = (x4,−x3,−x2, x1),
where all the partial derivaties are at the point x.
M1 is a Lie group of dimension two. Its Lie algebra can be easily found that
X2 = (−x2, x1,−x4, x3),
X4 = (x4,−x3,−x2, x1).
Theorem 5. M is paralelizable.
Proof. If we put
x1 = ρ1 cosφ,
x2 = ρ1 sinφ,
x3 = ρ2 cos θ,
x4 = ρ2 sin θ,
then from x1x3 + x2x4 = 0 we have
ISSN 1027-3190. Укр. мат. журн., 2012, т. 64, № 3
BICOMPLEX NUMBER AND TENSOR PRODUCT SURFACES IN R4
2 315
cosφ = sin θ, sinφ = − cos θ,
or
cosφ = − sin θ, sinφ = cos θ.
Besides, the condition x21+x22−x23−x24 6= 0 gives ρ21−ρ22 6= 0. We have the parametric representation
of one component of M with vector position r(ρ1, ρ2, φ)
r = (ρ1 cosφ, ρ1 sinφ,−ρ2 sinφ, ρ2 cosφ) .
Hence, we have three vectors tangent to coordinate curves
rρ1 = (cosφ, sinφ, 0, 0) ,
rρ2 = (0, 0,− sinφ, cosφ) ,
rφ = (−ρ1 sinφ, ρ1 cosφ,−ρ2 cosφ,−ρ2 sinφ) .
Evidently these vectors are ortogonal each other and put together a paralelization of M.
Theorem 5 is proved.
7. Tensor product surfaces and Lie groups. In this section, by using the tensor product surfaces
of a Lorentzian plane curve and a Euclidean plane curve, we determine some special subgroup of this
Lie group M. Thus, we obtain Lie group structure of tensor product surfaces of a Lorentzian plane
curve and a Euclidean plane curve. Also, we obtain left invariant vector fields of these Lie groups.
Theorem 6. Let α : R→ R2
1 be a hyperbolic spiral, and β : R→ R2 be a spiral with the same
parameter, i.e., α(t) = eat(cosh t, sinh t) and β(t) = ebt(cos t, sin t), a, b ∈ R. Then their tensor
product is a one-parameter subgroup in a Lie group M.
Proof. We obtain
γ(t) = α(t)⊗ β(t) = e(a+b)t (cosh t cos t, cosh t sin t,− sinh t sin t, sinh t cos t) .
It can be easily seen that
γ (t1)× γ (t2) = γ (t1 + t2)
for all t1, t2. Hence, (γ(t),×) is a one-parameter Lie subgroup of (M,×) .
Theorem 6 is proved.
Corollary 1. Let α : R→ R2
1 be a hyperbolic spiral and β : R→ R2 be a circle centered at O
with the same parameter, i.e., α(t) = eat(cosh t, sinh t), a ∈ R, and β(t) = (cos t, sin t). Then their
tensor product is a one-parameter subgroup in a Lie group M.
Proof. In Theorem 6 taking b = 0, we find that β is a circle centered at O. Then their tensor
product is a one-parameter subgroup in a Lie group M.
Corollary 2. Let α : R → R2
1 be a Lorentzian circle centered at O and β : R → R2 be circle
centered at O with the same parameter, i.e., α(t) = (cosh t, sinh t) and β(t) = (cos t, sin t). Then
their tensor product is a one-parameter subgroup in a Lie group M1.
ISSN 1027-3190. Укр. мат. журн., 2012, т. 64, № 3
316 S. Ö. KARAKUŞ, Y. YAYLI
Proof. Since ‖α(t)⊗ β(t)‖L = 1, it follows that α(t) ⊗ β(t) ⊂ M1. By taking a = b = 0, in
Theorem 6, we find that α is a Lorentzian circle centered at O and β is a circle centered at O. Then
their tensor product is a one-parameter subgroup in a Lie group M1.
Theorem 7. Let α : R → R2
1 be a Lorentzian circle centered at O and β : R → R2 be circle
centered at O with the same parameter, i.e., α(t) = (cosh t, sinh t), β(t) = (cos t, sin t), and γ(t) =
= α(t)⊗ β(t) be their tensor product. Then, the left invariant vector field on γ(t) is X = X2 +X4,
where X2 and X4 are left invariant vector fields on M1.
Proof. Let us find the left invariant vector field on γ(t) to the vector,
u =
d
dt
∣∣∣∣
e=0
η(t) = (1, t, 0, t) is a curve with tangent vector u. Its image under Lg is the curve,
Lg(η(t)) = gη(t) = (x11 + x2i+ x3j + x4ij)× (1 + ti+ tij) =
= (x1 − x2t+ x4t) + i (x1t+ x2 − x3t) + j (−x2t+ x3 − x4t) + ij (x1t+ x3t+ x4) .
Its tangent vector is,
Lg(η(t))(t) = (−x2 + x4) + i (x1 − x3) + j (−x2 − x4) + ij (x1 + x3) .
For the left invariant vector X we have
X = (−x2 + x4)
∂
∂x1
+ (x1 − x3)
∂
∂x2
+ (−x2 − x4)
∂
∂x3
+ (x1 + x3)
∂
∂x4
.
Theorem 7 is proved.
Conclusion 1. Let α : R → R2
1 be a hyperbolic spiral (or a Lorentzian circle centered at O)
and β : R → R2 be a spiral (or circle centered at O) with the same parameter. Then their tensor
product is the maximal integral curve.
Now, we want to classify these Lie groups as totally real or slant in semi-Euclidean space R4
2.
In order to do so, consider the left invariant vector field on these groups which constitute pseudo-
Hermitian structure which is given by J = X2.
Corollary 3. Let α : R → R2
1 be a Lorentzian circle centered at O, β : R → R2 be either a
spiral or a circle centered at O, and f = α ⊗ β be their tensor product immersion. Then the Lie
group f(t, s) is totally real Lorentzian immersion with respect to the pseudo-Hermitian structure J.
Proof. From Theorem 1 we know that, if α is a Lorentzian circle centered at O then f = α⊗ β
is totally real surface with respect to the pseudo-Hermitian structure J.
Corollary 4. Let α : R→ R2
1 be a hyperbolic spiral and β : R→ R2 be either a circle centered
at O or a spiral and f = α⊗ β be their tensor product. Then the Lie group f (t, s) is a proper slant
surface with respect to pseudo-Hermitian structure J on R4
2.
Proof. From Theorem 3 we know that, if α is a hyperbolic spiral and β is either a circle centered
at O or a spiral curve then f = α ⊗ β is proper slant surface with respect to the pseudo-Hermitian
structure J.
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Received 15.10.09,
after revision — 24.02.12
ISSN 1027-3190. Укр. мат. журн., 2012, т. 64, № 3
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| id | umjimathkievua-article-2578 |
| institution | Ukrains’kyi Matematychnyi Zhurnal |
| keywords_txt_mv | keywords |
| language | English |
| last_indexed | 2026-03-24T02:26:09Z |
| publishDate | 2012 |
| publisher | Institute of Mathematics, NAS of Ukraine |
| record_format | ojs |
| resource_txt_mv | umjimathkievua/3e/16f77e8c7a516b125d4f7077abdd9d3e.pdf |
| spelling | umjimathkievua-article-25782020-03-18T19:30:02Z Bicomplex number and tensor product surfaces in $\mathbb{R}^4_2$ Поверхнi добутку бiкомплексних чисел та тензорного добутку в$\mathbb{R}^4_2$ Karakus, S. O. Yayli, Y. Каракус, С. О. Яйлі, І. We show that a hyperquadric $M$ in $\mathbb{R}^4_2$ is a Lie group by using the bicomplex number product. For our purpose, we change the definition of tensor product. We define a new tensor product by considering the tensor product surface in the hyperquadric $M$. By using this new tensor product, we classify totally real tensor product surfaces and complex tensor product surfaces of a Lorentzian plane curve and a Euclidean plane curve. By means of the tensor product surfaces of a Lorentzian plane curve and a Euclidean plane curve, we determine a special subgroup of the Lie group M. Thus, we obtain the Lie group structure of tensor product surfaces of a Lorentzian plane curve and a Euclidean plane curve. Morever, we obtain left invariant vector fields of these Lie groups. We consider the left invariant vector fields on these groups, which constitute a pseudo-Hermitian structure. By using this, we characterize these Lie groups as totally real or slant in $\mathbb{R}^4_2$. Iз використанням добутку бiкомплексних чисел показано, що гiперквадрика $M$ у $\mathbb{R}^4_2$ є групою Лi. Для досягнення нашої мети модифiковано означення тензорного добутку. Новий тензорний добуток означено шляхом розгляду поверхнi тензорного добутку в гiперквадрицi $M$. За допомогою цього нового добутку класифiковано тотально дiйснi поверхнi тензорного добутку та комплекснi поверхнi тензорного добутку плоскої кривої Лоренца та евклiдової плоскої кривої. За допомогою поверхонь тензорного добутку плоскої кривої Лоренца та евклiдової плоскої кривої отримано спецiальну пiдгрупу групи Лi $M$. Таким чином, отримано структуру групи Лi для поверхонь тензорного добутку плоскої кривої Лоренца та евклiдової плоскої кривої. Крiм того, отримано лiвоiнварiантнi векторнi поля цих груп Лi. Розглянуто лiвоiнварiантнi векторнi поля на цих групах, якi утворюють псевдоермiтову структуру. Це дає змогу охарактеризувати групи Лi як тотально дiйснi або скiснi в $M$. Institute of Mathematics, NAS of Ukraine 2012-03-25 Article Article application/pdf https://umj.imath.kiev.ua/index.php/umj/article/view/2578 Ukrains’kyi Matematychnyi Zhurnal; Vol. 64 No. 3 (2012); 307-317 Український математичний журнал; Том 64 № 3 (2012); 307-317 1027-3190 en https://umj.imath.kiev.ua/index.php/umj/article/view/2578/1915 https://umj.imath.kiev.ua/index.php/umj/article/view/2578/1916 Copyright (c) 2012 Karakus S. O.; Yayli Y. |
| spellingShingle | Karakus, S. O. Yayli, Y. Каракус, С. О. Яйлі, І. Bicomplex number and tensor product surfaces in $\mathbb{R}^4_2$ |
| title | Bicomplex number and tensor product surfaces in $\mathbb{R}^4_2$ |
| title_alt | Поверхнi добутку бiкомплексних чисел та тензорного добутку в$\mathbb{R}^4_2$ |
| title_full | Bicomplex number and tensor product surfaces in $\mathbb{R}^4_2$ |
| title_fullStr | Bicomplex number and tensor product surfaces in $\mathbb{R}^4_2$ |
| title_full_unstemmed | Bicomplex number and tensor product surfaces in $\mathbb{R}^4_2$ |
| title_short | Bicomplex number and tensor product surfaces in $\mathbb{R}^4_2$ |
| title_sort | bicomplex number and tensor product surfaces in $\mathbb{r}^4_2$ |
| url | https://umj.imath.kiev.ua/index.php/umj/article/view/2578 |
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