A study on tensor product surfaces in low-dimensional Euclidean spaces
We consider a special case for curves in two-, three-, and four-dimensional Euclidean spaces and obtain a necessary and sufficient condition for the tensor product surfaces of the planar unit circle centered at the origin and these curves to have a harmonic Gauss map.
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2012
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| author | Etemad, Dehkordy A. Етемад, Дегкорді А. |
| author_facet | Etemad, Dehkordy A. Етемад, Дегкорді А. |
| author_sort | Etemad, Dehkordy A. |
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| description | We consider a special case for curves in two-, three-, and four-dimensional Euclidean spaces and obtain a necessary and sufficient condition for the tensor product surfaces of the planar unit circle centered at the origin and these curves to have
a harmonic Gauss map. |
| first_indexed | 2026-03-24T02:28:21Z |
| format | Article |
| fulltext |
UDC 515.14
A. Etemad Dehkordy (Isfahan Univ. Technology, Iran)
A STUDY ON TENSOR PRODUCT SURFACES
IN LOW-DIMENSIONAL EUCLIDEAN SPACES
ДОСЛIДЖЕННЯ ПОВЕРХОНЬ ТЕНЗОРНОГО ДОБУТКУ
В ЕВКЛIДОВИХ ПРОСТОРАХ МАЛОЇ РОЗМIРНОСТI
We consider a special case for curves in two-, three-, and four-dimensional Euclidean spaces and obtain a necessary and
sufficient condition for the tensor product surfaces of the planar unit circle centered at the origin and these curves to have
a harmonic Gauss map.
Розглянуто спецiальний випадок для кривих у дво-, три- та чотиривимiрних евклiдових просторах i отримано
необхiдну та достатню умову, за якої поверхнi тензорного добутку плоского одиничного кола з центром у початку
координат та цих кривих мають гармонiчне гауссове зображення.
1. Introduction. Tensor product of two immersions of a given Riemannian manifold is one of the
intersting topics in differential geometry. This notion is a generalization of the quadratic represen-
tation of a submanifold. In spatial case, a tensor product surface is obtained by taking the tensor
product of two curves. In [7], many properties such as minimality and totally reality are studied for
tensor product of two planar curves.
We also know that the Gauss map is one of the important topics in study of surfaces. In the other
hand harmonic functions have very properties in advanced analysis. Therefore we want search for
tensor product surfaces that have harmonic Gauss map in special cases. In this paper, since the case of
general dimension involves rather tedious calculations, we will restrict ourselves to low dimensions.
2. Preliminaries. In this section we recall some standard definitions and results from Riemannian
geometry. Let M be an n-dimensional Riemannian manifold, Em be an m-dimensional Euclidean
space and ϕ : M → Em be an isometric immersion. Let 5 be the Levi-Civita connection of Em
and 5 the induced connection on M. We denote the second fundamental form of M in Em by II,
normal connection in the normal bundle of M by 5⊥ and the shape operator in the direction of
normal vector field n by An. It is well known that the two later notions are related by
〈II(X,Y ), n〉 = 〈An X,Y 〉, (1)
where X, Y are vector fields tangent to M. For an n-dimensional submanifold M in Em, M is
said to be totally geodesic if II ≡ 0. Furthermore, the Gaussian and Weingarten formula are given
respectively by
5XY = 5XY + II(X,Y ), (2)
5Xn = −AnX +5⊥
Xn. (3)
Let G(n,m) be the Grassmannian consisting of all oriented n-planes through the origin of Em [1].
For an isometric immersion ϕ : M → Em, the (generalized) Gauss map Γ: M → G(n,m) of ϕ is a
smooth map which carries p ∈ M into the oriented n-plane in Em which obtained from the parallel
translation of the tangent space of M at p in Em [6]. It is well known that G(n,m) is canonically
c© A. ETEMAD DEHKORDY, 2012
1630 ISSN 1027-3190. Укр. мат. журн., 2012, т. 64, № 12
A STUDY ON TENSOR PRODUCT SURFACES IN LOW-DIMENSIONAL . . . 1631
imbedded in ∧nEm, the vector space obtained by the exterior product of n vectors in Em. It must
be said that we can assume ∧nEm as Euclidean space EN , where N =
(
m
n
)
. So the Gauss map at
p ∈M can be write as Γ(p) = (e1∧e2∧ . . .∧en)(p), where {e1, . . . , en, en+1, . . . , em} is an adapted
local orthonormal frame field in Em such that e1, e2, . . . , en are tangent to M and en+1, . . . , em are
normal to M [3].
We denote by C∞(TM) the space of all sections of TM. The (rough) Laplacian of f in
C∞(TM) is defined by
∆f = −
∑
i
(
5ei5eif −55ei
eif
)
. (4)
Note that in this paper smooth can be replace by second differentiability. We also assume that an
Euclidean smooth curve c : R → En with parametrization c(t) =
(
α1(t), α2(t), . . . , αn(t)
)
is called
i-th suitable and denoted by i-s, 1 ≤ i ≤ n, if αi(t) 6= 0 and α′
i(t) 6= 0 for t ∈ R, where from now
on we use prime to denote the differential respect to t.
3. A tensor product surface in E4 with harmonic Gauss map. This section is similar to a
part of Section 3 in [2], but because of deficiency in essential assumptions for i-s curve and for
development of results to other dimensions in Sections 4 and 5, we state this section with another
discipline on normal vector basis.
Let c1 : R → E2 be the unit planar circle centered at origin of E2 with parametrization c1(t) =
= (cos s, sin s) and c2 : R→ E2 be a unit speed i-s smooth curve in E2 . Without loss of generality
let c2(t) =
(
α(t), β(t)
)
with α(t) 6= 0, α′(t) 6= 0 for t ∈ R, i.e., 1-s curve. The tensor product
surface M of two curves c1 and c2 is given by
f = c1 ⊗ c2 : R2 → E4,
f(s, t) =
(
α(t) cos s, β(t) cos s, α(t) sin s, β(t) sin s
)
.
Assume that f(s, t) = c1(s)⊗ c2(t) defines an isometric immersion of R2 into R4. It follows directly
that
e1 =
1
‖c2‖
(
− α(t) sin s,−β(t) sin s, α(t) cos s, β(t) cos s
)
and
e2 =
(
α′(t) cos s, β′(t) cos s, α′(t) sin s, β′(t) sin s
)
are form an orthonormal basis for tangent space to M. Moreover, an orthonormal basis for normal
space to M is given by
e3 =
1
‖c2‖
(
β(t) sin s,−α(t) sin s,−β(t) cos s, α(t) cos s
)
,
e4 =
(
β′(t) cos s,−α′(t) cos s, β′(t) sin s,−α′(t) sin s
)
.
In this section, for simplification of relations, we have to introduce following abbreviations:
ISSN 1027-3190. Укр. мат. журн., 2012, т. 64, № 12
1632 A. ETEMAD DEHKORDY
A = −α(t)α′(t) + β(t)β′(t)
‖c2‖2
, D =
α(t)β′(t)− α′(t)β(t)
‖c2‖
, E = α′′(t)β′(t)− α′(t)β′′(t).
Now covariant differentiation with respect to e1 and e2 give us
5e1e1 = (A)e2 +
(
− D
‖c2‖
)
e4, 5e1e2 = (−A)e1 +
(
D
‖c2‖
)
e3,
(5)
5e2e1 =
(
D
‖c2‖
)
e3, 5e2e2 = (E)e4
and
5e1e3 =
(
− D
‖c2‖
)
e2 + (−A)e4, 5e1e4 =
(
D
‖c2‖
)
e1 + (A)e3,
(6)
5e2e3 =
(
− D
‖c2‖
)
e1, 5e2e4 = (−E)e2.
The first result of above relations follows from (3) and (6) as follows.
Lemma 1. Let M be the tensor product surface f = c1 ⊗ c2 of unite circle c1 centered at
origin in Euclidean plane E2 and a unite speed 1-s smooth curve c2(t) =
(
α(t), β(t)
)
in E2, then
Ae3 =
0
D
‖c2‖
D
‖c2‖
0
, Ae4 =
−D‖c2‖ 0
0 E
. (7)
It must be said that in Lemma 1 the similar result is true, if we replace 1-s by 2-s. The same note
is correct for all future subjects about 1-s.
Theorem 1. LetM be a tensor product surface of Euclidean planar circle c1(s) = (cos s, sin s)
and a unit speed, 1-s smooth curve c2(t) =
(
α(t), β(t)
)
in E2. The Gauss map of M is harmonic, if
and only if M is part of a plane.
Proof. If we apply (4) for the Gauss map Γ = e1 ∧ e2, then by a direct calculation and use of
(5) and (6), we get the following expression for Laplacian of the Gauss map:
−4Γ =
2∑
i=1
5ei(5eiΓ)−55ei
eiΓ =
=
2∑
i=1
5ei(5ei(e1 ∧ e2))−55ei
ei(e1 ∧ e2) =
=
2∑
i=1
5ei(5eie1 ∧ e2 + e1 ∧5eie2)− (55ei
eie1 ∧ e2)− (e1 ∧55ei
eie2) =
= −
[
3D2
‖c2‖2
+ E2
]
(e1 ∧ e2) + [. . .] (e1 ∧ e4)− [. . .](e2 ∧ e3) + [. . .] (e3 ∧ e4). (8)
ISSN 1027-3190. Укр. мат. журн., 2012, т. 64, № 12
A STUDY ON TENSOR PRODUCT SURFACES IN LOW-DIMENSIONAL . . . 1633
If the Gauss map of M is harmonic, i.e., 4Γ = 0, then (8) implies that
3D2
‖c2‖2
+ E2 = 0. (9)
Since all terms in the right – hand side of (10) are non-negative, so we have
D ≡ E ≡ 0.
This implies that M is a totally geodesic surface in E4 by (7), therefore M is part of a plane. The
converse is obvious.
Theorem 1 is proved.
4. A tensor product surface in E6 with harmonic Gauss map. Let c1 : R → E2 be the unit
planar circle centered at origin of E2 with parametrization c1(t) = (cos s, sin s) and c2 : R→ E3 ba
a unit speed, i-s smooth curves in E3 . Without loss of generality let c2(t) =
(
α(t), β(t), γ(t)
)
be
1-s, then the tensor product surface M of two curves c1 and c2 is given by
f = c1 ⊗ c2 : R2 → E6,
f(s, t) =
(
α(t) cos s, β(t) cos s, γ(t) cos s, α(t) sin s, β(t) sin s, γ(t) sin s
)
.
Assume that f(s, t) = c1(s)⊗ c2(t) defines an isometric immersion of R2 into R6. It follows directly
that
e1 =
1
‖c2‖
(
−α(t) sin s,−β(t) sin s,−γ(t) sin s, α(t) cos s, β(t) cos s, γ(t) cos s
)
and
e2 =
(
α′(t) cos s, β′(t) cos s, γ′(t) cos s, α′(t) sin s, β′(t) sin s, γ′(t) sin s
)
are form an orthonormal frame for tangent space to M. Moreover, an orthonormal basis for normal
space to M is given by
e3 =
1√
α2 + β2
(
β(t) sin s,−α(t) sin s, 0,−β(t) cos s, α(t) cos s, 0
)
,
e4 =
1√
α′2 + β′2
(
β′(t) cos s,−α′(t) cos s, 0, β′(t) sin s,−α′(t) sin s, 0
)
,
e5 =
1√
α2 + γ2
(
γ(t) sin s, 0,−α(t) sin s,−γ(t) cos s, 0, α(t) cos
)
,
e6 =
1√
α′2 + γ′2
(
γ′(t) cos s, 0,−α′(t) cos s, γ′(t) sin s, 0,−α′(t) sin s
)
.
ISSN 1027-3190. Укр. мат. журн., 2012, т. 64, № 12
1634 A. ETEMAD DEHKORDY
In this section, for simplification of relations, we have to introduce following abbreviations:
A = −α(t)α′(t) + β(t)β′(t) + γ(t)γ′(t)
‖c2‖2
, B1 = −α(t)α′(t) + β(t)β′(t)
α2(t) + β2(t)
B2 = −α(t)α′(t) + γ(t)γ′(t)
α2(t) + γ2(t)
, C1 = −α
′(t)α′′(t) + β′(t)β′′(t)
α′2(t) + β′2(t)
,
C2 = −α
′(t)α′′(t) + γ′(t)γ′′(t)
α′2(t) + γ′2(t)
, D1 =
α(t)β′(t)− α′(t)β(t)√
α2(t) + β2(t)
,
D2 =
α(t)γ′(t)− α′(t)γ(t)√
α2(t) + γ2(t)
, E1 =
α′′(t)β′(t)− α′(t)β′′(t)√
α′2(t) + β′2(t)
,
E2 =
α′′(t)γ′(t)− α′(t)γ′′(t)√
α′2(t) + γ′2(t)
, F1 =
α′(t)β(t)− α(t)β′(t)√
α′2(t) + β′2(t)
,
F2 =
α′(t)γ(t)− α(t)γ′(t)√
α′2(t) + γ′2(t)
, L1 =
√
α2(t) + β2(t)
α′2(t) + β′2(t)
,
G23 =
β(t)γ′(t)√
(α2(t) + β2(t))(α′2(t) + γ′2(t))
, G23 =
−β′(t)γ(t)√
(α′2(t) + β′2(t))(α2(t) + γ2(t))
,
H23 =
B2βγ + β(t)γ′(t)√
(α2(t) + β2(t))(α2(t) + γ2(t))
, H23 =
B1βγ + β′(t)γ(t)√
(α2(t) + β2(t))(α2(t) + γ2(t))
,
K23 =
C2β
′γ′ + β′(t)γ′′(t)√
(α′2(t) + β′2(t))(α′2(t) + γ′2(t))
, K23 =
C1β
′γ′ + β′′(t)γ′(t)√
(α′2(t) + β′2(t))(α′2(t) + γ′2(t))
,
L2 =
√
α2(t) + γ2(t)
α′2(t) + γ′2(t)
.
So covariant differentiation with respect to e1 and e2 give us
5e1e1 =
(
A‖c2‖
)
e2 +
(
F1
‖c2‖
)
e4 +
(
F2
‖c2‖
)
e6, 5e2e1 =
(
D1
‖c2‖
)
e3 +
(
D2
‖c2‖
)
e5,
(10)
5e1e2 =
(
−A‖c2‖
)
e1 + (D1)e3 + (D2)e5, 5e2e2 = (E1)e4 + (E2)e6
ISSN 1027-3190. Укр. мат. журн., 2012, т. 64, № 12
A STUDY ON TENSOR PRODUCT SURFACES IN LOW-DIMENSIONAL . . . 1635
and
5e1e3 = (−D1)e2 + (−L1B1)e4 + (G23)e6, 5e2e3 =
(
− D1
‖c2‖
)
e1 + (H23)e5,
5e1e4 =
(
−F1
‖c2‖
)
e1 + (B1L1)e3 + (G23)e5, 5e2e4 = (−E1)e2 + (K23)e6,
(11)
5e1e5 = (−D2)e2 + (−G23)e4 + (−L2B2)e6, 5e2e5 =
(
−D2
‖c2‖
)
e1 + (H23)e3,
5e1e6 =
(
−F2
‖c2‖
)
e1 + (−G23)e3 + (B2L2)e5, 5e2e6 = (−E2)e2 + (K23)e4.
The first result of above relations follows from (3) and (11) as follows.
Lemma 2. Let f = c1 ⊗ c2 be the tensor product of unite circle with parametrization c1 =
= (cos s, sin s) in Euclidean plane E2 and a unite speed 1-s smooth curve c2(t) =
(
α(t), β(t), γ(t)
)
in E3, then
Ae3 =
0 D1
D1
‖c2‖
0
, Ae4 =
F1
‖c2‖
0
0 E1
,
(12)
Ae5 =
0 D2
D2
‖c2‖
0
, Ae6 =
d F2
‖c2‖
0
0 E2
.
The following theorem states a necessary and sufficient condition for the Gauss map of a tensor
product surface with same conditions in Lemma 2 to be a harmonic function.
Theorem 2. LetM be a tensor product surface of Euclidean planar circle c1(s) = (cos s, sin s)
and unit speed, 1-s smooth curve c2(t) =
(
α(t), β(t), γ(t)
)
in E3. The Gauss map of M is harmonic,
if and only if M is part of a plane.
Proof. Proof is exactly the same as proof of Theorem 1. If we apply (4), (10) and (11), then a
tedious computation implies that the 4Γ, (rough) Laplacian of the Gauss map, is given by
−4Γ =
2∑
i=1
5ei
(
5eiΓ)−55ei
eiΓ
)
=
= −
[
F 2
1
‖c2‖2
+
F 2
2
‖c2‖2
+
(
D2
1 +D2
2
)( 1
‖c2‖2
+ 1
)
+ E2
1 + E2
2
]
(e1 ∧ e2)+
+[. . .](e1 ∧ e4) + [. . .](e1 ∧ e6) + [. . .](e2 ∧ e3) + [. . .](e2 ∧ e5) + [. . .](e3 ∧ e4)+
+[. . .](e3 ∧ e6) + [. . .](e4 ∧ e5) + [. . .](e5 ∧ e6). (13)
ISSN 1027-3190. Укр. мат. журн., 2012, т. 64, № 12
1636 A. ETEMAD DEHKORDY
If the Gauss map of M is harmonic, i.e., 4Γ = 0, then (13) implies that
F 2
1
‖c2‖2
+
F 2
2
‖c2‖2
+
(
D2
1 +D2
2
)( 1
‖c2‖2
+ 1
)
+ E2
1 + E2
2 = 0. (14)
Since all terms in the right-hand side of (14) are non-negative, so we have,
D1 ≡ D2 ≡ E1 ≡ E2 ≡ F1 ≡ F2 ≡ 0.
This implies that M is a totally geodesic surface in E6 by (12), therefore M is part of plane. The
converse is obvious.
Theorem 2 is proved.
5. A tensor product surface in E8 with harmonic Gauss map. Let c1 : R → E2 be the unit
planar circle centered at origin in E2 with parametrization c1(t) = (cos s, sin s) and c2 : R→ E4 ba
a unit speed, i-s smooth curves in E4 . Without loss of generality let c2(t) = (α(t), β(t), γ(t), δ(t))
with α(t) 6= 0, α′(t) 6= 0 for every t ∈ R, then the tensor product surface M of two curves c1 and c2
is given by
f = c1 ⊗ c2 : R2 → E8,
f(s, t) =
(
α(t) cos s, β(t) cos s, γ(t) cos s, δ(t) cos s, α(t) sin s, β(t) sin s, γ(t) sin s, δ(t) sin s
)
.
Assume that f(s, t) = c1(s)⊗ c2(t) defines an isometric immersion of R2 into R8. It follows directly
that
e1 =
=
1
‖c2‖
(
− α(t) sin s,−β(t) sin s,−γ(t) sin s,−δ(t) sin s, α(t) cos s, β(t) cos s, γ(t) cos s, δ(t) cos s
)
and
e2 =
(
α′(t) cos s, β′(t) cos s, γ′(t) cos s, δ′(t) cos s, α′(t) sin s, β′(t) sin s, γ′(t) sin s, δ′(t) sin s
)
are form an orthonormal basis for tangent space of M. Moreover, an orthonormal basis for normal
space to M is given by
e3 =
1√
α2 + β2
(
β(t) sin s,−α(t) sin s, 0, 0,−β(t) cos s, α(t) cos s, 0, 0
)
,
e4 =
1√
α′2 + β′2
(
β′(t) cos s,−α′(t) cos s, 0, 0, β′(t) sin s,−α′(t) sin s, 0, 0
)
,
e5 =
1√
α2 + γ2
(
γ(t) sin s, 0,−α(t) sin s, 0,−γ(t) cos s, 0, α(t) cos
)
, 0,
e6 =
1√
α′2 + γ′2
(
γ′(t) cos s, 0,−α′(t) cos s, 0, γ′(t) sin s, 0,−α′(t) sin s, 0
)
,
ISSN 1027-3190. Укр. мат. журн., 2012, т. 64, № 12
A STUDY ON TENSOR PRODUCT SURFACES IN LOW-DIMENSIONAL . . . 1637
e7 =
1√
α2 + δ2
(
δ(t) sin s, 0, 0,−α(t) sin s,−δ(t) cos s, 0, 0, α(t) cos
)
,
e8 =
1√
α′2 + δ′2
(
δ′(t) cos s, 0, 0,−α′(t) cos s, γ′(t) sin s, 0, 0,−α′(t) sin s
)
.
In this section, for simplification of relations, we have to introduce following abbreviations:
A = −α(t)α′(t) + β(t)β′(t) + γ(t)γ′(t) + δ(t)δ′(t)
‖c2‖2
, B1 = −α(t)α′(t) + β(t)β′(t)
α2(t) + β2(t)
,
B2 = −α(t)α′(t) + γ(t)γ′(t)
α2(t) + γ2(t)
, B3 = −α(t)α′(t) + δ(t)δ′(t)
α2(t) + δ2(t)
,
C1 = −α
′(t)α′′(t) + β′(t)β′′(t)
α′2(t) + β′2(t)
, C2 = −α
′(t)α′′(t) + γ′(t)γ′′(t)
α′2(t) + γ′2(t)
,
C3 = −α
′(t)α′′(t) + δ′(t)δ′′(t)
α′2(t) + δ′2(t)
, D1 =
α(t)β′(t)− α′(t)β(t)√
α2(t) + β2(t)
,
D2 =
α(t)γ′(t)− α′(t)γ(t)√
α2(t) + γ2(t)
, D3 =
α(t)δ′(t)− α′(t)δ(t)√
α2(t) + δ2(t)
,
E1 =
α′′(t)β′(t)− α′(t)β′′(t)√
α′2(t) + β′2(t)
, E2 =
α′′(t)γ′(t)− α′(t)γ′′(t)√
α′2(t) + γ′2(t)
,
E3 =
α′′(t)δ′(t)− α′(t)δ′′(t)√
α′2(t) + δ′2(t)
, F1 =
α′(t)β(t)− α(t)β′(t)√
α′2(t) + β′2(t)
,
F2 =
α′(t)γ(t)− α(t)γ′(t)√
α′2(t) + γ′2(t)
, F3 =
α′(t)δ(t)− α(t)δ′(t)√
α′2(t) + δ′2(t)
,
G23 =
β(t)γ′(t)√
(α2(t) + β2(t))(α′2(t) + γ′2(t))
, G23 =
−β′(t)γ(t)√
(α′2(t) + β′2(t))(α2(t) + γ2(t))
,
G24 =
β(t)δ′(t)√
(α2(t) + δ2(t))(α′2(t) + δ′2(t))
, G24 =
−β′(t)δ(t)√
(α′2(t) + δ′2(t))(α2(t) + δ2(t))
,
G34 =
γ(t)δ′(t)√
(α2(t) + γ2(t))(α′2(t) + δ′2(t))
, G34 =
−γ′(t)δ(t)√
(α′2(t) + γ′2(t))(α2(t) + δ2(t))
,
H23 =
B2β(t)γ(t) + β(t)γ′(t)√
(α2(t) + β2(t))(α2(t) + γ2(t))
, H23 =
B1β(t)γ(t) + β′(t)γ(t)√
(α2(t) + β2(t))(α2(t) + γ2(t))
,
H24 =
B3β(t)δ(t) + β(t)δ′(t)√
(α2(t) + β2(t))(α2(t) + δ2(t))
, H24 =
B1β(t)δ(t) + β′(t)δ(t)√
(α2(t) + β2(t))(α2(t) + δ2(t))
,
ISSN 1027-3190. Укр. мат. журн., 2012, т. 64, № 12
1638 A. ETEMAD DEHKORDY
H34 =
B3γ(t)δ(t) + γ(t)δ′(t)√
(α2(t) + γ2(t))(α2(t) + δ2(t))
, H34 =
B2γ(t)δ(t) + γ′(t)δ(t)√
(α2(t) + γ2(t))(α2(t) + δ2(t))
,
K23 =
C2β
′(t)γ′(t) + β′(t)γ′′(t)√
(α′2(t) + β′2(t))(α′2(t) + γ′2(t))
, K23 =
C1β
′(t)γ′(t) + β′′(t)γ′(t)√
(α′2(t) + β′2(t))(α′2(t) + γ′2(t))
,
K24 =
C3β
′(t)δ′(t) + β′(t)δ′′(t)√
(α′2(t) + β′2(t))(α′2(t) + δ′2(t))
, K24 =
C1β
′(t)δ′(t) + β′′(t)δ′(t)√
(α′2(t) + β′2(t))(α′2(t) + δ′2(t))
,
K34 =
C3γ
′(t)δ′(t) + γ′(t)δ′′(t)√
(α′2(t) + γ′2(t))(α′2(t) + δ′2(t))
, K34 =
C2γ
′(t)δ′(t) + γ′′(t)δ′(t)√
(α′2(t) + γ′2(t))(α′2(t) + δ′2(t))
,
L1 =
√
α2(t) + β2(t)
α′2(t) + β′2(t)
, L2 =
√
α2(t) + γ2(t)
α′2(t) + γ′2(t)
, L3 =
√
α2(t) + δ2(t)
α′2(t) + δ′2(t)
.
Now covariant differentiation with respect to e1 and e2 give us,
5e1e1 =
(
A‖c2‖
)
e2 +
(
F1
‖c2‖
)
e4 +
(
F2
‖c2‖
)
e6 +
(
F3
‖c2‖
)
e8,
5e2e1 =
(
D1
‖c2‖
)
e3 +
(
D2
‖c2‖
)
e5 +
(
D3
‖c2‖
)
e7,
(15)
5e1e2 =
(
−A‖c2‖
)
e1 + (D1)e3 + (D2)e5 + (D3)e7,
5e2e2 = (E1)e4 + (E2)e6 + (E3)e8,
and
5e1e3 = (−D1)e2 + (−B1L1)e4 + (G23)e6 + (G24)e8,
5e2e3 =
(
− D1
‖c2‖
)
e1 + (H23)e5 + (H24)e7,
5e1e4 =
(
−F1
‖c2‖
)
e1 + (B1L1)e3 + (G23)e5 + (G24)e7,
5e2e4 = (−E1)e2 + (K23)e6 + (K24)e8,
5e1e5 = (−D2)e2 + (−G23)e4 + (−L2B2)e6 + (G34)e8,
5e2e5 =
(
−D2
‖c2‖
)
e1 + (H23)e3 + (H34)e7,
(16)
5e1e6 =
(
−F2
‖c2‖
)
e1 + (−G23)e3 + (B2L2)e5 + (G34)e7,
ISSN 1027-3190. Укр. мат. журн., 2012, т. 64, № 12
A STUDY ON TENSOR PRODUCT SURFACES IN LOW-DIMENSIONAL . . . 1639
5e2e6 = (−E2)e2 + (K23)e4 + (K34)e8,
5e1e7 = (−D3)e2 + (−G24)e4 + (−G34)e6 + (−B3L3)e8,
5e2e7 =
(
−D3
‖c2‖
)
e1 + (H24)e3 + (H34)e5,
5e1e8 =
(
−F3
‖c2‖
)
e1 + (−G24)e3 + (G34)e5 + (B3L3)e7,
5e2e8 = (−E3)e2 + (K24)e4 + (K34)e6.
The first result of above relations yields from (3) and (16) as follows.
Lemma 3. Let f = c1 ⊗ c2 be the tensor product surface of unite circle c1(s) = (cos s, sin s)
in Euclidean plane E2 and a unite speed, 1-s smooth curve c2(t) =
(
α(t), β(t), γ(t)δ(t)
)
in E4, then
Ae3 =
0 D1
D1
‖c2‖
0
, Ae4 =
F1
‖c2‖
0
0 E1
,
Ae5 =
0 D2
D2
‖c2‖
0
, Ae6 =
F2
‖c2‖
0
0 E2
, (17)
Ae7 =
0 D3
D3
‖c2‖
0
, Ae8 =
F3
‖c2‖
0
0 E3
.
Let c1(s) = (cos s, sin s) and c2(t) =
(
α(t), β(t), γ(t), δ(t)
)
be the unit circle in E2 and a unit
speed, 1-s smooth curve in E4 respectively, then following theorem, gives us similar result to the
Theorems 1 and 2 in E8.
Theorem 3. Let M be the tensor product surface f = c1 ⊗ c2, The Gauss map of M is
harmonic, if and only if M is part of a plane.
Proof. Similar to proof of Theorems 1 and 2, we apply (4), (15) and (16), then a tedious
computation implies that the 4Γ, (rough) Laplacian of the Gauss map, is given by
−4Γ =
2∑
i=1
5ei(5eiΓ)−55ei
eiΓ) =
= −
[
F 2
1
‖c2‖2
+
F 2
2
‖c2‖2
+
F 2
3
‖c2‖2
+ (D2
1 +D2
2 +D2
3)
(
1
‖c2‖2
+ 1
)
+ E2
1 + E2
2 + E2
3
]
(e1 ∧ e2)+
+[. . .](e1 ∧ e4) + [. . .](e1 ∧ e6) + [. . .](e1 ∧ e8) + [. . .](e2 ∧ e3) + [. . .](e2 ∧ e5) + [. . .](e2 ∧ e7)+
ISSN 1027-3190. Укр. мат. журн., 2012, т. 64, № 12
1640 A. ETEMAD DEHKORDY
+[. . .](e3 ∧ e4) + [. . .](e3 ∧ e6) + [. . .](e3 ∧ e8) + [. . .](e4 ∧ e5) + [. . .](e4 ∧ e7) + [. . .](e5 ∧ e6)+
+[. . .](e5 ∧ e8) + [. . .](e6 ∧ e7) + [. . .](e7 ∧ e8). (18)
If the Gauss map of M is harmonic, then (18) implies that
F 2
1
‖c2‖2
+
F 2
2
‖c2‖2
+
F 2
3
‖c2‖2
+
(
D2
1 +D2
2 +D2
3
)( 1
‖c2‖2
+ 1
)
+ E2
1 + E2
2 + E2
3 = 0. (19)
Since all terms in the right-hand side of (19) are non-negative, so we have
D1 ≡ D2 ≡ D3 ≡ E1 ≡ E2 ≡ E3 ≡ F1 ≡ F2 ≡ F3 ≡ 0.
This implies that M is a totally geodesic surface in E8 by (17), therefore M is part of a plane. The
converse is obvious.
Theorem 3 is proved.
1. Aminov Yu. A. The geometry of submanifolds. – Amsterdam: Gordon and Breach Sci. Publ., 2001.
2. Arslan K., Bulca B., Kilic B., Kim Y. H., Murathan C., Ozturk G Tensor product surfaces with pointwise 1-type Gauss
map // Bull. Korean Math. Soc. – 2011. – 48. – P. 601 – 607.
3. Chen B. Y., Piccinni P. Submanifolds with finit type Gauss map // Bull. Austral. Math. Soc. – 1987. – 35, № 2. –
P. 161 – 186.
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Belg. Math. Soc. Simon Stevin. – 1994. – 1, № 5. – P. 643 – 648.
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und Geometrie. – 1993. – 34, № 2. – S. 209 – 215.
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– 35, № 5. – P. 1555 – 1581.
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mat. Messina. Ser. II. – 1994/1995. – 18, № 3. – P. 173 – 184.
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Received 20.09.11,
after revision — 21.10.12
ISSN 1027-3190. Укр. мат. журн., 2012, т. 64, № 12
|
| id | umjimathkievua-article-2687 |
| institution | Ukrains’kyi Matematychnyi Zhurnal |
| keywords_txt_mv | keywords |
| language | English |
| last_indexed | 2026-03-24T02:28:21Z |
| publishDate | 2012 |
| publisher | Institute of Mathematics, NAS of Ukraine |
| record_format | ojs |
| resource_txt_mv | umjimathkievua/79/906f390c6cf88ddf101eb5aa74cbeb79.pdf |
| spelling | umjimathkievua-article-26872020-03-18T19:32:55Z A study on tensor product surfaces in low-dimensional Euclidean spaces Дослiдження поверхонь тензорного добутку в евклiдових просторах малої розмiрностi Etemad, Dehkordy A. Етемад, Дегкорді А. We consider a special case for curves in two-, three-, and four-dimensional Euclidean spaces and obtain a necessary and sufficient condition for the tensor product surfaces of the planar unit circle centered at the origin and these curves to have a harmonic Gauss map. Розглянуто спецiальний випадок для кривих у дво-, три- та чотиривимiрних евклiдових просторах i отримано необхiдну та достатню умову, за якої поверхнi тензорного добутку плоского одиничного кола з центром у початку координат та цих кривих мають гармонiчне гауссове зображення. Institute of Mathematics, NAS of Ukraine 2012-12-25 Article Article application/pdf https://umj.imath.kiev.ua/index.php/umj/article/view/2687 Ukrains’kyi Matematychnyi Zhurnal; Vol. 64 No. 12 (2012); 1630-1640 Український математичний журнал; Том 64 № 12 (2012); 1630-1640 1027-3190 en https://umj.imath.kiev.ua/index.php/umj/article/view/2687/2132 https://umj.imath.kiev.ua/index.php/umj/article/view/2687/2133 Copyright (c) 2012 Etemad Dehkordy A. |
| spellingShingle | Etemad, Dehkordy A. Етемад, Дегкорді А. A study on tensor product surfaces in low-dimensional Euclidean spaces |
| title | A study on tensor product surfaces in low-dimensional Euclidean spaces |
| title_alt | Дослiдження поверхонь тензорного добутку в евклiдових просторах малої розмiрностi |
| title_full | A study on tensor product surfaces in low-dimensional Euclidean spaces |
| title_fullStr | A study on tensor product surfaces in low-dimensional Euclidean spaces |
| title_full_unstemmed | A study on tensor product surfaces in low-dimensional Euclidean spaces |
| title_short | A study on tensor product surfaces in low-dimensional Euclidean spaces |
| title_sort | study on tensor product surfaces in low-dimensional euclidean spaces |
| url | https://umj.imath.kiev.ua/index.php/umj/article/view/2687 |
| work_keys_str_mv | AT etemaddehkordya astudyontensorproductsurfacesinlowdimensionaleuclideanspaces AT etemaddegkordía astudyontensorproductsurfacesinlowdimensionaleuclideanspaces AT etemaddehkordya doslidžennâpoverhonʹtenzornogodobutkuvevklidovihprostorahmaloírozmirnosti AT etemaddegkordía doslidžennâpoverhonʹtenzornogodobutkuvevklidovihprostorahmaloírozmirnosti AT etemaddehkordya studyontensorproductsurfacesinlowdimensionaleuclideanspaces AT etemaddegkordía studyontensorproductsurfacesinlowdimensionaleuclideanspaces |