On the energy and pseudo-angle of Frenet vector fields in Rⁿᵥ
In this paper, we compute the energy of a Frenet vector field and the pseudo-angle between Frenet vectors for a given non-null curve C in semi-Euclidean space of signature (n,v). It is shown that the energy and pseudo-angle can be expressed in terms of the curvature functions of C.
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irk-123456789-1662532020-02-19T01:27:06Z On the energy and pseudo-angle of Frenet vector fields in Rⁿᵥ Altin, A. Короткі повідомлення In this paper, we compute the energy of a Frenet vector field and the pseudo-angle between Frenet vectors for a given non-null curve C in semi-Euclidean space of signature (n,v). It is shown that the energy and pseudo-angle can be expressed in terms of the curvature functions of C. Обчислено енергiю векторного поля Френе та псевдокут мiж векторами Френе для заданої ненульової кривої C у напiвевклiдовому просторi сигнатури (n,v). Показано, що енергiя та псевдокут можуть бути вираженi через функцiї кривини C. 2011 Article On the energy and pseudo-angle of Frenet vector fields in Rⁿᵥ / A. Altin // Український математичний журнал. — 2011. — Т. 63, № 6. — С. 833–839. — Бібліогр.: 9 назв. — англ. 1027-3190 http://dspace.nbuv.gov.ua/handle/123456789/166253 517.9 en Український математичний журнал Інститут математики НАН України |
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Короткі повідомлення Короткі повідомлення |
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Короткі повідомлення Короткі повідомлення Altin, A. On the energy and pseudo-angle of Frenet vector fields in Rⁿᵥ Український математичний журнал |
| description |
In this paper, we compute the energy of a Frenet vector field and the pseudo-angle between Frenet vectors for a given non-null curve C in semi-Euclidean space of signature (n,v). It is shown that the energy and pseudo-angle can be expressed in terms of the curvature functions of C. |
| format |
Article |
| author |
Altin, A. |
| author_facet |
Altin, A. |
| author_sort |
Altin, A. |
| title |
On the energy and pseudo-angle of Frenet vector fields in Rⁿᵥ |
| title_short |
On the energy and pseudo-angle of Frenet vector fields in Rⁿᵥ |
| title_full |
On the energy and pseudo-angle of Frenet vector fields in Rⁿᵥ |
| title_fullStr |
On the energy and pseudo-angle of Frenet vector fields in Rⁿᵥ |
| title_full_unstemmed |
On the energy and pseudo-angle of Frenet vector fields in Rⁿᵥ |
| title_sort |
on the energy and pseudo-angle of frenet vector fields in rⁿᵥ |
| publisher |
Інститут математики НАН України |
| publishDate |
2011 |
| topic_facet |
Короткі повідомлення |
| url |
http://dspace.nbuv.gov.ua/handle/123456789/166253 |
| citation_txt |
On the energy and pseudo-angle of Frenet vector fields in Rⁿᵥ / A. Altin // Український математичний журнал. — 2011. — Т. 63, № 6. — С. 833–839. — Бібліогр.: 9 назв. — англ. |
| series |
Український математичний журнал |
| work_keys_str_mv |
AT altina ontheenergyandpseudoangleoffrenetvectorfieldsinrnv |
| first_indexed |
2025-07-14T21:04:15Z |
| last_indexed |
2025-07-14T21:04:15Z |
| _version_ |
1837657810929713152 |
| fulltext |
UDC 517.9
A. Altın (Hacettepe Univ., Ankara, Turkey)
ON THE ENERGY AND PSEUDO-ANGLE
OF FRENET VECTOR FIELDS IN Rn
v
ПРО ЕНЕРГIЮ ТА ПСЕВДОКУТ
ВЕКТОРНОГО ПОЛЯ ФРЕНЕ В Rn
v
In this paper, we compute the energy of a Frenet vector field and the pseudo-angle between Frenet vectors
for a given non-null curve C in semi-Euclidean space of signature (n, ν). It is shown that the energy and
pseudo-angle can be expressed in terms of the curvature functions of C.
Обчислено енергiю векторного поля Френе та псевдокут мiж векторами Френе для заданої ненульової
кривої C у напiвевклiдовому просторi сигнатури (n, ν). Показано, що енергiя та псевдокут можуть бути
вираженi через функцiї кривини C.
1. Introduction. It is well known that the studies on the energy of a unit vector
field on a compact oriented m-dimensional Riemannian manifold M basically consider
the equality M = S2n+1 (see [1 – 3]). Let C be a curve with a pair (I, α) of para-
metric unit speed in Rn. Let us take an initial point a ∈ I and the Frenet frames
{V1(α(a)), . . . , Vr(α(a))} and {V1(α(s)), . . . , Vr(α(s))} at the points α(a) and α(s),
respectively. In [4], we calculated the energy of a Frenet vector field and the angle
between each vector Vi(α(a)) and Vi(α(s)) where 1 ≤ i ≤ r. Further, we observed
that the energy and angle may be expressed in terms of the curvature functions of the
given curve C. In this paper, we consider the Frenet frame at the point α(a) for a given
non-null curve C in semi-Euclidean space
Rnν =
(
Rn, −
ν∑
i=1
dxi +
n∑
i=ν+1
dxi
)
.
Note that the Frenet vectors may not be the same type here, i.e., both of the spacelike
and timelike vectors are contained in the collection of Frenet vectors. It is well known
that the angle between timelike vectors is not defined. In [5], the integral
∫ s
a
∥∥∥∥dVidu
∥∥∥∥ du
is called the pseudo-angle between timelike vectors Vi(α(a)) and Vi(α(s)). Recall that
the angle between spacelike vectors is
∫ s
a
∥∥∥∥dVidu
∥∥∥∥ du [4]. In this work, the angle between
arbitrary vectors Vi(α(a)) and Vi(α(s)) will be called pseudo-angle.
Definition 1.1 [6]. Let α : I ⊂ R −→ Rnv be a regular curve in Rnv . Then α is
called spacelike (timelike, null) if for all t ∈ I the velocity vector α′(t) is spacelike
(timelike, null). If 〈α′(t), α′(t)〉 = 1 or −1, then α is called a unit speed curve where
〈 , 〉 denotes the scalar product of Rnv .
Definition 1.2. Let α be a regular curve in Rnv and ψ = {α′(t), α′′(t), . . . , αr(t)}
be a linear independent and non-null system. Further, let αm(t) ∈ Spψ for all αm(t)
where m > r ≥ 2. Then the orthonormal system {V1(t), V2(t), . . . , Vr(t)} obtained
from ψ is called r-Frenet frame at the point α(t).
In this paper, curve means that {α′, α′′, . . . , αr} is a set of derivatives of α such that
αl is non-null for 1 ≤ l ≤ r.
Lemma 1.1 [8]. For a curve α in Rnv , r-Frenet frame exists if and only if the space
Sp {α′(t), . . . , αr(t)} is non-degenerated for k = 1, . . . , r.
c© A. ALTıN, 2011
ISSN 1027-3190. Укр. мат. журн., 2011, т. 63, № 6 833
834 A. ALTıN
Definition 1.3. Let α be a regular curve in Rnv and {V1(t), V2(t), . . . , Vr(t)} be
the Frenet frame at the point α(t). If εi(t) = 〈Vi(t), Vi(t)〉 = 1 or −1, then the function
ki : I −→ R defined by
ki(t) = εi(t)εi+1(t)〈V ′i (t), Vi+1(t)〉, 1 ≤ i ≤ r − 1, (1)
is called curvature function on α. Moreover, the real number ki(t) is called the i th
curvature on α at the point α(t).
Theorem 1.1 [8]. Let α be a unit speed curve in Rnv and ki(t) be the i th curvature.
If {V1(t), V2(t), . . . , Vr(t)} is the Frenet frame at the point α(t), then the following
equalities hold:
V ′1(t) = ε1(t)k1(t)V2(t), (2)
V ′i (t) = −εi(t)ki−1(t)Vi−1(t) + εi(t)ki(t)Vi+1(t), 1 < i < r, (3)
V ′r (t) = −εr(t)kr−1(t)Vr−1(t). (4)
Definition 1.4. The energy of a differentiable map f : (M, 〈, 〉)→ (N,h) between
Riemannian manifolds is given by
E(f) = 1
2
∫
M
n∑
a=1
h
(
df(ea), df(ea)
)
υ, (5)
where υ is the canonical volume form in M and {ea} is a local basis of the tangent
space (see for example [1, 3]).
The energy of a unit vector field X is defined to be the energy of the section
X : M → T 1M, where T 1M is the unit tangent bundle equipped with the restriction
of the Sasaki metric on TM. Now let π : T 1M → M be the bundle projection, and
let T (T 1M) = V ⊕ H denote the vertical/horizontal splitting induced by the Levi-
Civita connection. Further, let us write TM = F⊕G where F denotes the line bundle
generated by X, and G is the orthogonal complement [3].
Proposition 1.1 [9]. The connection map K : T (T 1M) → T 1M verifies the fol-
lowing conditions:
1) π◦K = π◦dπ and π◦K = π◦π̃, where π̃ : T (T 1M) → T 1M is the tangent
bundle projection;
2) for ω ∈ TxM and a section ξ : M → T 1M, we have
K(dξ(ω)) = ∇ωξ,
where ∇ is the Levi-Civita covariant derivative.
Definition 1.5 [9]. For η1, η2 ∈ Tξ(T 1M) define
gS(η1, η2) = 〈dπ(η1), dπ(η2)〉+ 〈K(η1),K(η2)〉. (6)
This gives a Riemannian metric on TM. Recall that gS is called the Sasaki metric. The
metric gs makes the projection π : T 1M →M a Riemannian submersion.
ISSN 1027-3190. Укр. мат. журн., 2011, т. 63, № 6
ON THE ENERGY AND PSEUDO-ANGLE OF FRENET VECTOR FIELDS IN Rn
v 835
2. The energy and pseudo-angle of a Frenet vector field in Rn
v .
Definition 2.1. Let C be a non-null curve where with a pair (I, α) of parametric
unit speeds in a space Rnv at a initial point a ∈ I. Further, let
{V1(α(a)), . . . , Vr(α(a))} and {V1(α(s)) . . . , Vr(α(s))}
be the Frenet frames at the points α(a) and α(s), respectively. The integral
∫ s
a
∥∥∥∥dVidu
∥∥∥∥ du
is called the pseudo-angle between the vectors Vi(α(a)) and Vi(α(s)).
Theorem 2.1. Let C be a spacelike curve. Then we have the following conditions.
(i) If the energy of Vi is E(Vi(s)), i.e., let the function E(Vi) be defined as
E(Vi) : I → R, 1 ≤ i ≤ r.
Then the following relations are valid:
E(V1)(s) =
1
2
s∫
a
ε2k
2
1(α(u)) du+
1
2
(s− a),
E(Vi)(s) =
1
2
s∫
a
(εi−1k
2
i−1(α(u)) + εi+1k
2
i (α(u))) du+
1
2
(s− a), 2 ≤ i ≤ r − 1,
E(Vr)(s) =
1
2
s∫
a
εr−1k
2
r−1(α(u)) du+
1
2
(s− a).
(ii) If the pseudo-angle between vectors Vi(α(a)) and Vi(α(s)) is θi(s), i.e., let the
function θi be defined as
θi : I → R, 1 ≤ i ≤ r.
Then the following relations are valid:
θ1(s) =
s∫
a
k1(α(u)) du,
θi(s) =
s∫
a
√∣∣εi−1k2i−1(α(u)) + εi+1k2i (α(u))
∣∣ du, 2 ≤ i ≤ r − 1,
θr(s) =
s∫
a
∣∣kr−1(α(u))∣∣du.
Proof. Note that by Lemma 1.1, the definitions of the energy and Sasaki metric in
Rnv can be defined in the same manner in Riemannian manifolds.
(i) Let TC be the tangent bundle and let {V1, V2, . . . , Vr} be the Frenet vector field
of the curve C. Then we have V1 : C → TC =
⋃
t∈I Tα(t)C. Let π : TC → C be the
bundle projection and T (TC) = V ⊕ H be the vertical/horizontal splitting induced by
the Levi-Civita connection. Let us write TC = F ⊕ G where F denotes the line bundle
ISSN 1027-3190. Укр. мат. журн., 2011, т. 63, № 6
836 A. ALTıN
generated by V1. Consider the Levi-Civita connection map K : T (TC)→ TC. By using
equation (5), we obtain that the energy of V1 is
E(V1)(s) =
1
2
s∫
a
gS(dV1(V1(α(u)), dV1(V1α(u)))du, (7)
where du is an arc length element. From (6) we have
gS(dV1(V1), dV1(V1)) =
= 〈dπ(V1(V1)), dπ(V1(V1))〉+ 〈K(V1(V1)),K(V1(V1))〉.
Since V1 is a section, then we obtain
d(π)◦d(V1) = d(π◦V1) = d(idC) = idTC .
On the other hand, by Proposition 1.1, we may write that
K(V1(V1)) = ∇V1V1 = V ′1 .
Then we obtain
gS(dV1(V1), dV1(V1)) = 〈V1, V1〉+ 〈V ′1 , V ′1〉 .
Since C is a spacelike curve, from (2) we get
gS(dV1(V1), dV1(V1)) = 1 + ε2k
2
1 (8)
putting (8) in (7), and this yields that
E(V1(s)) =
1
2
s∫
a
ε2k
2
1(α(u)) du+
1
2
(s− a).
Let NiC be the i th normal bundle. Thus we have Vi : C → NiC where NiC =
=
⋃
t∈I Niα(t)C. Here Niα(t)C is generated by Vi. Now let πi : NiC → C be the i th
bundle projection and T (NiC) = Vi⊕Hi be the vertical/horizontal splitting induced by
the Levi-Civita connection. Take the Levi-Civita connection map Ki : T (NiC)→ NiC.
By using equation (5), we obtain that the energy of Vi is
E(Vi)(s) =
1
2
s∫
a
gS(dVi(V1(α(u)), dVi(V1α(u)))du, (9)
where 2 ≤ i ≤ r. From (6) we have
gS(dVi(V1), dVi(V1)) =
= 〈dπi(Vi(V1)), dπi(Vi(V1))〉+ 〈K(Vi(V1)),K(Vi(V1))〉 =
= 〈d(πi◦Vi)(V1), d(πi◦Vi)(V1)〉+ 〈∇V1Vi,∇V1Vi〉 =
= 〈V1, V1〉+ 〈V ′i , V ′i 〉 .
By (3) we obtain
ISSN 1027-3190. Укр. мат. журн., 2011, т. 63, № 6
ON THE ENERGY AND PSEUDO-ANGLE OF FRENET VECTOR FIELDS IN Rn
v 837
gS(dVi(V1), dVi(V1)) = 1 + εi−1k
2
i−1 + εi+1k
2
i .
Using (9), we have
E(Vi)(s) =
1
2
s∫
a
(
εi−1k
2
i−1(α(u)) + εi+1k
2
i (α(u))
)
du+
1
2
(s− a),
2 ≤ i ≤ r − 1.
So, (4) gives us that
gS(dVr(V1), dVr(V1)) = 1 + εr−1k
2
r−1.
Then (9) yields that
E(Vr)(s) =
1
2
s∫
a
εr−1k
2
r−1(α(u)) du+
1
2
(s− a).
This completes the proof of (i).
(ii) From Definition 2.1 we have
θi(s) =
s∫
a
∥∥∥∥dVidu
∥∥∥∥ du.
Since α is spacelike, by using (2), we obtain
θ1(s) =
s∫
a
√
|ε2k21(α(u))|du =
s∫
a
k1(α(u)) du.
From (3) we have
θi(s) =
s∫
a
√∣∣εi−1k2i−1(α(u)) + εi+1k2i (α(u))
∣∣ du, 2 ≤ i ≤ r − 1.
By (4) we find
θr(s) =
s∫
a
√∣∣εr−1k2r−1(α(u))∣∣du =
s∫
a
∣∣kr−1(α(u))∣∣du.
Theorem 2.2. Let C be a timelike curve.
(i) The energy of Vi may be given by the following equalities:
E(V1)(s) =
1
2
s∫
a
ε2k
2
1(α(u)) du−
1
2
(s− a),
E(Vi)(s) =
1
2
s∫
a
(
εi−1k
2
i−1(α(u)) + εi+1k
2
i (α(u))
)
du− 1
2
(s− a), 2 ≤ i ≤ r − 1,
E(Vr)(s) =
1
2
s∫
a
εr−1k
2
r−1(α(u)) du−
1
2
(s− a).
ISSN 1027-3190. Укр. мат. журн., 2011, т. 63, № 6
838 A. ALTıN
(ii) The pseudo-angle between vectors Vi(α(a)) and Vi(α(s)) are
θ1(s) =
s∫
a
∣∣k1(α(u))∣∣ du,
θi(s) =
s∫
a
√∣∣εi−1k2i−1(α(u)) + εi+1k2i (α(u))
∣∣ du, 2 ≤ i ≤ r − 1,
θr(s) =
s∫
a
∣∣kr−1(α(u))∣∣du.
Proof. (i) As in the proof of Theorem 2.1, we may write that
gS(dV1(V1), dV1(V1)) = 〈V1, V1〉+ 〈V ′1 , V ′1〉 .
Since C is a timelike curve, from (2) we get
gS(dV1(V1), dV1(V1)) = −1 + ε2k
2
1 (10)
and
E(V1(s)) =
1
2
s∫
a
ε2k
2
1(α(u)) du−
1
2
(s− a).
By using (6) and (3) we have
gS(dVi(V1), dVi(V1)) = −1 + εi−1k
2
i−1 + εi+1k
2
i .
From (9), we obtain
E(Vi)(s) =
1
2
s∫
a
(
εi−1k
2
i−1(α(u)) + εi+1k
2
i (α(u))
)
du− 1
2
(s− a),
2 ≤ i ≤ r − 1.
Therefore, (4) and (9) gives us that
E(Vr)(s) =
1
2
s∫
a
εr−1k
2
r−1(α(u)) du−
1
2
(s− a).
(ii) From Definition 2.1 we have
θi(s) =
s∫
a
∥∥∥∥dVidu
∥∥∥∥ du.
Using (2), we obtain
θ1(s) =
s∫
a
√∣∣ε2k21(α(u))∣∣du =
s∫
a
∣∣k1(α(u))∣∣ du.
ISSN 1027-3190. Укр. мат. журн., 2011, т. 63, № 6
ON THE ENERGY AND PSEUDO-ANGLE OF FRENET VECTOR FIELDS IN Rn
v 839
From (3) we have
θi(s) =
s∫
a
√∣∣εi−1k2i−1(α(u)) + εi+1k2i (α(u))
∣∣ du, 2 ≤ i ≤ r − 1.
By (4) we obtain
θr(s) =
s∫
a
√∣∣εr−1k2r−1(α(u))∣∣du =
s∫
a
∣∣kr−1(α(u))∣∣du.
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Received 20.07.10
ISSN 1027-3190. Укр. мат. журн., 2011, т. 63, № 6
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