The Energy of a Domain on the Surface
We compute the energy of a unit normal vector field on a Riemannian surface M. It is shown that the energy of the unit normal vector field is independent of the choice of an orthogonal basis in the tangent space. We also define the energy of the surface. Moreover, we compute the energy of spheres, d...
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nasplib_isofts_kiev_ua-123456789-1655212025-02-09T23:37:21Z The Energy of a Domain on the Surface Енергія області на поверхні Altın, A. Короткі повідомлення We compute the energy of a unit normal vector field on a Riemannian surface M. It is shown that the energy of the unit normal vector field is independent of the choice of an orthogonal basis in the tangent space. We also define the energy of the surface. Moreover, we compute the energy of spheres, domains on a right circular cylinder and torus, and of the general surfaces of revolution. Розраховано енергію одиничного нормального векторного поля на рiмановiй поверхні M. Показано, що енергія одиничного нормального векторного поля не залежить від вибору ортогонального базиса в дотичному просторі. Визначено енергію поверхні. Більш того, розраховано енергію сфер, областей на прямому круговому циліндрі та торі і, більш загально, поверхонь обертання. 2015 Article The Energy of a Domain on the Surface / A. Altın // Український математичний журнал. — 2015. — Т. 67, № 4. — С. 568–573. — Бібліогр.: 7 назв. — англ. 1027-3190 https://nasplib.isofts.kiev.ua/handle/123456789/165521 517.2 en Український математичний журнал application/pdf Інститут математики НАН України |
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Короткі повідомлення Короткі повідомлення |
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Короткі повідомлення Короткі повідомлення Altın, A. The Energy of a Domain on the Surface Український математичний журнал |
| description |
We compute the energy of a unit normal vector field on a Riemannian surface M. It is shown that the energy of the unit normal vector field is independent of the choice of an orthogonal basis in the tangent space. We also define the energy of the surface. Moreover, we compute the energy of spheres, domains on a right circular cylinder and torus, and of the general surfaces of revolution. |
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Article |
| author |
Altın, A. |
| author_facet |
Altın, A. |
| author_sort |
Altın, A. |
| title |
The Energy of a Domain on the Surface |
| title_short |
The Energy of a Domain on the Surface |
| title_full |
The Energy of a Domain on the Surface |
| title_fullStr |
The Energy of a Domain on the Surface |
| title_full_unstemmed |
The Energy of a Domain on the Surface |
| title_sort |
energy of a domain on the surface |
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Інститут математики НАН України |
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2015 |
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Короткі повідомлення |
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https://nasplib.isofts.kiev.ua/handle/123456789/165521 |
| citation_txt |
The Energy of a Domain on the Surface / A. Altın // Український математичний журнал. — 2015. — Т. 67, № 4. — С. 568–573. — Бібліогр.: 7 назв. — англ. |
| series |
Український математичний журнал |
| work_keys_str_mv |
AT altına theenergyofadomainonthesurface AT altına energíâoblastínapoverhní AT altına energyofadomainonthesurface |
| first_indexed |
2025-12-01T20:07:27Z |
| last_indexed |
2025-12-01T20:07:27Z |
| _version_ |
1850337814348038144 |
| fulltext |
К О Р О Т К I П О В I Д О М Л Е Н Н Я
UDC 517.2
A. Altın (Hacettepe Univ., Ankara, Turkey)
THE ENERGY OF A DOMAIN ON THE SURFACE
ЕНЕРГIЯ ОБЛАСТI НА ПОВЕРХНI
We compute the energy of a unit normal vector field on a Riemannian surface M. It is shown that the energy of the unit
normal vector field is independent of the choice of an orthogonal basis of the tangent space. We also define the energy of
the surface. Moreover, we compute the energy of spheres, domains on a right circular cylinder, torus, and more generally,
of the surfaces of revolution.
Розраховано енергiю одиничного нормального векторного поля на рiмановiй поверхнi M . Показано, що енергiя
одиничного нормального векторного поля не залежить вiд вибору ортогонального базиса в дотичному просторi.
Визначено енергiю поверхнi. Бiльш того, розраховано енергiю сфер, областей на прямому круговому цилiндрi та
торi i, бiльш загально, поверхонь обертання.
1. Introduction. The energy of a unit vector field X on a Riemannian manifold M is defined as the
energy of the section into the unit tangent bundle T 1M determined by X. In this respect, the energy
of distributions of Riemannian manifolds and the energy of unit vector fields on the sphere S3 were
considered in papers by P. M. Chacon, A. M. Naveira [1] and A. Higuichi, B. S. Kay and C. M.
Wood [2]. Further, the energy of differentiable maps has been also studied by C. M. Wood [3].
Generally, every geometric problem about curves and surfaces can be solved by means of the
Frenet vectors field of the curve and the normal vector field of the surface. Therefore, in [4], we
focus on the curve C instead of the manifold M. For a given curve C with a pair (I, α) of parametric
unit speeds in a space Rnv on which we take a fixed point a ∈ I, we denote Frenet frames at
the points α(a) and α(s) by {V1(α(a)), . . . , Vr(α(a))} and {V1(α(s)), . . . , Vr(α(s))}, respectively.
We calculate the energy of a Frenet vectors fields as well as the pseudo-angle between the vectors
Vi(α(a)) and Vi(α(s)), where 1 ≤ i ≤ r. We observed that both energy and pseudo-angle depend on
the curvature functions of the curve C.
In this paper, we calculate the energy of a unit normal vector field on a Riemannian surface M.
We achieve that energy of a unit normal vector field depends on the norm of the matrix of the shape
operator and the area AS(M) of M. We also prove that the energy can be expressed in terms of the
Gaussian and mean curvatures of the surface. Hence, the energy of the unit normal vector fields of
a Riemannian surface M can be expressed via principal curvatures of M, which are independent of
the choice of the orthogonal basis of the tangent space of M. Taking into account the above fact
we define the energy of the surface by ignoring the constant term obtained from area of the surface.
We prove that the energy of a sphere is independent of radius and is equal to 4π. Furthermore, we
calculate the energy of the domain on right circular cylinder of height h and radius r as
1
r
πh. We
also compute the energy of the torus. Finally, we establish a formula for the energy of a surface of
revolution, which verifies the above computations.
c© A. ALTIN, 2015
568 ISSN 1027-3190. Укр. мат. журн., 2015, т. 67, № 4
THE ENERGY OF A DOMAIN ON THE SURFACE 569
Definition 1. Let M is a Riemannian surface in R3. If p is a point of M, then for each tangent
vector v to M at p, let
Sp(v) = −∇vpN,
where N is a unit normal vector field on a neighborhood of p in M. Sp is called the shape operator
of M at p (derived from N ) (see [6]). The Gaussian curvature of M is the real-valued function
G = detS on M . Explicitly, for each point p of M, the Gaussian curvature G(p) of M at p
is the determinant of the shape operator S of M at p. The mean curvature of M is the function
H =
1
2
traceS. Let u be a unit vector tangent to M at a point p. Then the number k(u) = 〈S(u), u〉
is called the normal curvature of M in the u direction. The maximum and minimum values of the
normal curvature k(u) of M at p are called the principal curvatures of M at p, and are denoted
by k1 and k2 respectively. The directions in which these extreme values occur are called principal
directions of M at p. Unit vectors in these directions are called principal vectors of M at p.
Lemma 1. If k1 and k2 are principal curvatures at a point p, then we have
Sp =
[
k1(p) 0
0 k2(p)
]
.
Proposition 1. The connection map K : T (T 1M)→ T 1M satisfies the following:
1) π◦K = π◦dπ and π◦K = π◦π̃ . Here π̃ : T (T 1M)→ T 1M is the tangent bundle projection.
2) For ω∈TxM and a section ξ : M → T 1M, we get
K(dξ(ω)) = ∇ωξ.
Here ∇ is the Levi – Civita covariant derivative (see [5]).
Definition 2. For η1, η2∈Tξ(T 1M) define
gS(η1, η2) = 〈dπ(η1), dπ(η2)〉+ 〈K(η1),K(η2)〉. (1)
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 (see [5]).
We use the classical notation of surface theory; for this purpose we can give [7] as a general
reference. Let ϕ : U → R3, ϕ(U) = M, ϕ(U) = (ϕ1(u, v), ϕ2(u, v), ϕ3(u, v)) and ϕ(u, v) be a
local parametrization of surface M in R3.
Let < be a domain on surface M. Area of < is
As(<) =
∫ ∫
ϕ−1(<)
√
EG− F 2dudv,
where υ =
√
EG− F 2dudv is the area form in M and E = 〈ϕu, ϕu〉, F = 〈ϕu, ϕv〉, G = 〈ϕv, ϕv〉
(see [6]).
ISSN 1027-3190. Укр. мат. журн., 2015, т. 67, № 4
570 A. ALTIN
Definition 3. 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, (2)
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 (see [3]).
Since we provide some background material in the previous section, we are in a position to
calculate the energy of a unit normal vector field on the Riemannian surface M .
2. The energy of the unit normal vector field of a surface.
Theorem 1. Let N be unit normal vector field of M. Then for the energy of N the following
formula hold:
E(N) =
∫
M
(2H2 −G)υ +As(M),
where υ is the area form in M, G and H are Gaussian and mean curvature of M respectively, As(M)
is area of M.
Proof. Let {eu, ev} be an local orthonormal basis of the tangent space, N be unit normal vector
field ofM andNM be normal bundle. Thus, we haveN : M → NM, whereNM =
⋃
q∈U Nϕ(q)M,
and Nϕ(q)M is the straight line through the point ϕ(q) in the N direction. By using equation (2), we
obtain
E(N) =
1
2
∫
M
(gS(dN(eu), dN(eu)) + gS(dN(ev), dN(ev)))υ. (3)
From (1) it follows that
E(N) =
1
2
∫
M
[〈dπ(dN(eu)), dπ(dN(eu))〉+ 〈K(dN(eu)),K(dN(eu))〉+
+〈dπ(dN(ev)), dπ(dN(ev))〉+ 〈K(dN(ev)),K(dN(ev))〉] υ,
where π : NM → M be the bundle projection and K : T (NM) → NM. Since N is a section, we
have d(π)◦d(N) = d(π◦N) = d(idM ) = idTM . We also have by Proposition 1 that K(dN(eu)) =
= ∇euN = −S(eu) ∀p ∈M. Combining all these, we get
E(N) =
1
2
∫
M
[〈eu, eu〉+ 〈S(eu), S(eu)〉+ 〈ev, ev〉+ 〈S(ev), S(ev)〉] υ. (4)
On the other hand, we have that
ISSN 1027-3190. Укр. мат. журн., 2015, т. 67, № 4
THE ENERGY OF A DOMAIN ON THE SURFACE 571
S(eu) = −∇euN = aeu + bev, S(ev) = −∇evN = ceu + dev ∀p ∈M, (5)
where a, b, c, d are real-valued functions.
Therefore, the matrix which corresponds to the shape operator of M is S =
[
a c
b d
]
.
Using equalities in (5) and putting them in equation (4), we obtain
E(N) =
1
2
∫
M
[a2 + b2 + c2 + d2 + 〈eu, eu〉+ 〈ev, ev〉]υ. (6)
Since S is a symmetric matrix, then the Gaussian curvature of M is G = detS = ad − b2 and
the mean curvature of M is H =
1
2
(a+ d).
Hence,
a2 + b2 + c2 + d2 = a2 + 2b2 + d2 = 4H2 − 2G. (7)
Putting (7) in (6), we get
E(N) =
1
2
∫
M
[4H2 − 2G+ 〈eu, eu〉+ 〈ev, ev〉]υ. (8)
Since {eu, ev} is orthonormal basis of the tangent space of M, then (8) becomes
E(N) =
∫
M
(2H2 −G)υ +
∫
M
υ =
∫
M
(2H2 −G)υ +As(M).
Theorem 1 is proved.
Consequently, combining Lemma 1 and equation (6) we can give the following corollary.
Corollary 1. If k1 and k2 are principal curvatures of M, then we have
E(N) =
1
2
∫
M
(k21 + k22)υ +As(M).
The Gaussian curvature and mean curvature of M are independent from the choice of the basis
{eu, ev}, thus the energy of a unit normal vector field is independent of the choice of the orthogonal
basis of the tangent space of M. We may ignore the constant term of As(M) and we can give the
following definition.
Definition 4. The integral
1
2
∫
M
(a2 + b2 + c2 + d2)υ =
1
2
∫
M
(4H2 − 2G)υ =
∫
M
(2H2 −G)υ
is called the energy of surface M and is denoted by E(M).
ISSN 1027-3190. Укр. мат. журн., 2015, т. 67, № 4
572 A. ALTIN
Once we have the Definition 4, we may be able to compute the energy of the surface of a sphere
but not compute the surface of a cylinder. However, we can only calculate the energy of a region of
cylinder (see Example 2). By using Definition 4, we can calculate the energy of a domain on surface
as follow.
Let ϕ : U → R3, ϕ(U) =M, ϕ(u, v) be a local parametrization of surface M in R3, and < be a
domain on surface M .
E(<) =
∫ ∫
ϕ−1(<)
(2H2 −G)
√
EG− F 2dudv,
where υ =
√
EG− F 2dudv is the area form in M .
Example 1. Let ϕ : U → R3, U = ]− π, π[ ×
]
−π
2
,
π
2
[
, ϕ(u, v) = (rcosucosv, rcosvsinu,
rsinv) be a local parametrization of the sphere S2 of radius r. The matrix of the shape operator of
sphere is
S =
−1
r
0
0 −1
r
and
√
EG− F 2 = r2cosv.
Then
E(S2) =
1
2
π∫
−π
π
2∫
−π
2
2
(
−1
r
)2√
EG− F 2dudv = 4π.
Example 2. Let ϕ(u, v) = (rcosu, rsinu, v) be a local parametrization of a right circular cylin-
der, ϕ : U → R3, U = ]0, 2π[×]0, h[, ϕ(U) = < ⊂M. The matrix of the shape operator of cylinder
is
S =
[
−1
r
0
0 0
]
and
√
EG− F 2 = r2cosv.
Therefore,
E(<) = 1
2
2π∫
0
h∫
0
(
−1
r
)2√
EG− F 2dudv =
1
r
πh.
Example 3. Let ϕ : R × R → R3, ϕ(u, v) = (u, v, au + bv) be a local parametrization of a
plane M. The matrix of the plane is S = 0 and E(M) = 0.
Example 4. The energy of a minimal surface M is E(M) = −
∫
M
(G)υ =
∫
M
(a2 + b2)υ.
Example 5. Torus of revolution T. Suppose that α is the circle in the xz plane with radius r > 0
and center (R, 0, 0). We shall rotate about the z axis; hence we must require R > r to keep α from
meeting the axis of revolution. A natural parametrization of α is
α(u) = (R+ rcos(u), 0, rsin(u)), u ∈ I.
A local parametrization of the torus T is then given by
ISSN 1027-3190. Укр. мат. журн., 2015, т. 67, № 4
THE ENERGY OF A DOMAIN ON THE SURFACE 573
ϕ(u, v) = ((R+ rcosu)cosv, (R+ rcosu)sinv, rsin(u)), 0 < u < 2π, 0 < v < 2π.
The matrix of the shape operator of torus is
S =
1
r
0
0
cosu
R+ rcosu
and
√
EG− F 2 = r(R+ rcosu).
Hence,
E(T ) = 1
2
2π∫
0
2π∫
0
((
1
r
)2
+
(
cosu
R+ rcosu
)2
)
r(R+ rcosu)dudv
and
E(T ) = 1
2
2π∫
0
2π∫
0
1 +
cosu
R
r
+ cosu
2(R
r
+ cosu
)
dudv.
Example 6. Let M be a surfaces of revolution. We suppose the axis of revolution of M is the
z-axis of our coordinate system. We denote the profile curve of M by α. We can suppose that α is
represented by
α(u) = (f(u), 0, g(u)), u ∈ I,
where f, g are real functions on the open interval I, we suppose that α has arc length parametrization.
A local parametrization of the surface M is then given by
ϕ(u, v) = (f(u)cosv, f(u)sinv, g(u)), u ∈ I, 0 < v < 2π.
The matrix of the shape operator of surfaces of revolution is
S =
g
′′
(u)
f ′(u)
0
0
g
′
(u)
f(u)
and
√
EG− F 2 = f(u).
Let < be a domain on surface M, then
E(<) = 1
2
∫ ∫
ϕ−1(<)
(g′′(u)
f ′(u)
)2
+
(
g
′
(u)
f(u)
)2
f(u)dudv.
1. Chacón P. M., Naveira A. M. Corrected energy of distributions on Riemannian manifold // Osaka J. Math. – 2004. –
41. – P. 97 – 105.
2. Higuchi A., Kay B. S., Wood C. M. The energy of unit vector fields on the 3-sphere // J. Geom. and Phys. – 2001. –
37. – P. 137 – 155.
3. Wood C. M. On the energy of a unit vector field // Geom. dedic. – 1997. – 64. – P. 319 – 330.
4. Altın A. On the energy and pseudo-angle of a Frenet vector fields in Rnv // Ukr. Math. J. – 2011. – 63, № 6. –
P. 833 – 839.
5. Chacón P. M., Naveira A. M., Weston J. M. On the energy of distributions, with application to the quaternionic Hopf
fibration // Monatsh. Math. – 2001. – 133. – S. 281 – 294.
6. O’Neill B. Elementary differential geometry. – Acad. Press Inc., 1966.
7. Struik D. J. Differential geometry. – Reading MA: Addison-Wesley, 1961.
Received 07.10.11,
after revision — 13.03.14
ISSN 1027-3190. Укр. мат. журн., 2015, т. 67, № 4
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