A characterization of totally umbilical hypersurfaces of a space form by geodesic mapping
The idea of considering the second fundamental form of a hypersurface as the first fundamental form of another hypersurface has found very useful applications in Riemannian and semi-Riemannian geometry, specially when trying to characterize extrinsic hyperspheres and ovaloids. Recently, T. Adachi...
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| author | Canfes, E. Ö. Özdeğer, A. Канфес, Е. О. Оздер, А. |
| author_facet | Canfes, E. Ö. Özdeğer, A. Канфес, Е. О. Оздер, А. |
| author_sort | Canfes, E. Ö. |
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| datestamp_date | 2020-03-18T19:15:36Z |
| description | The idea of considering the second fundamental form of a hypersurface as the first fundamental form of another hypersurface has found very useful applications in
Riemannian and semi-Riemannian geometry, specially when trying to characterize extrinsic hyperspheres and ovaloids.
Recently, T. Adachi and S. Maeda gave a characterization of totally umbilical hypersurfaces in a space form by circles.
In this paper, we give a characterization of totally umbilical hypersurfaces of a space form by means of geodesic mapping. |
| first_indexed | 2026-03-24T02:23:29Z |
| format | Article |
| fulltext |
К О Р О Т К I П О В I Д О М Л Е Н Н Я
UDC 517.9
E. Ö. Canfes, A. Özdeğer (Istanbul Techn. Univ., Kadir Has Univ., Istanbul, Turkey)
A CHARACTERIZATION OF TOTALLY UMBILICAL HYPERSURFACES
OF A SPACE FORM BY GEODESIC MAPPING
ХАРАКТЕРИСТИКА ТОТАЛЬНО ОМБIЛIЧНИХ ГIПЕРПОВЕРХОНЬ
ПРОСТОРОВОЇ ФОРМИ ЗА ДОПОМОГОЮ ГЕОДЕЗИЧНИХ ВIДОБРАЖЕНЬ
The idea of considering the second fundamental form of a hypersurface as the first fundamental form of another hypersurface
has found very useful applications in Riemannian and semi-Riemannian geometry, specially when trying to characteri-
ze extrinsic hyperspheres and ovaloids. Recently, T. Adachi and S. Maeda gave a characterization of totally umbilical
hypersurfaces in a space form by circles. In this paper, we give a characterization of totally umbilical hypersurfaces of a
space form by means of geodesic mapping.
Iдея використання другої фундаментальної форми гiперповерхнi як першої фундаментальної форми iншої гiперпо-
верхнi знайшла дуже важливi застосування у рiмановiй та напiврiмановiй геометрiї, зокрема при описi зовнiшнiх
гiперсфер та овалоїдiв. Нещодавно T. Adachi та S. Maeda навели характеристику тотально омбiлiчних гiперпо-
верхонь у просторовiй формi за допомогою кiл. У цiй роботi ми наводимо характеристику тотально омбiлiчних
гiперповерхонь просторової форми за допомогою геодезичних вiдображень.
1. Introduction. Let Mn and M ′
n be two hypersurfaces of the space form M̄n+1 [3 – 5] and let g, g′
and ḡ be the respective positive definite metric tensors. Denote by ∇, ∇′ and ∇̄ the corresponding
connections induced by g, g′ and ḡ.
In this paper, we choose the first fundamental form of M ′
n as
g′ = e2σω, (1.1)
where ω is the second fundamental form of Mn which is supposed to be positive definite and σ is a
differentiable function defined on Mn.
Let {xi}, {x′i} and {yα} be the respective coordinate systems in Mn, M ′
n and M̄n+1 and let f
be a one-to-one differentiable mapping of Mn upon M ′
n defined by
x′i = f i(x1, x2, . . . , xn), i = 1, 2, . . . , n, (1.2)
in which f i are smooth functions defined on Mn and have a non-vanishing Jacobian. Then, it is clear
that the corresponding points of Mn and M ′
n are represented by the same set of coordinates and that
the coordinate vectors correspond.
Let R̄, R and R′ be the covariant curvature tensors of M̄n+1, Mn and M ′
n respectively and let
K̄ be the Riemannian curvature of M̄n+1.
We then have1
R̄
βγδε
= K̄(ḡ
βδ
ḡγε − ḡβε ḡγδ). (1.3)
On the other hand, under the condition (1.3) the Codazzi equations
1In the sequel, Latin indices i, j, k, . . . run from 1 to n, while the Greek indices α, β, γ will run from 1 to n+ 1.
c© E. Ö. CANFES, A. ÖZDEĞER, 2013
ISSN 1027-3190. Укр. мат. журн., 2013, т. 65, № 4 583
584 E. Ö. CANFES, A. ÖZDEĞER
∇kωij −∇jωik + R̄
βγδε
Nβ ∂y
γ
∂xi
∂yδ
∂xj
∂yε
∂xk
= 0
and the Gauss equation
Rijkl = R̄
βγδε
∂yβ
∂xi
∂yγ
∂xj
∂yδ
∂xk
∂yε
∂xl
+ (ωikωjl − ωilωjk)
transform, respectively, into
∇kωij −∇jωik = 0 (1.4)
and
Rijkl = K̄(gikgjl − gilgjk) + (ωikωjl − ωilωjk) (1.5)
in which Nβ are the components of the unit normal vector field of Mn [4].
2. Relation between the connections ∇ and ∇′. It is well-known that the connection coeffi-
cients of a Riemannian space whose metric tensor is g are given by [5]
Γlij =
1
2
glh (∂igjh + ∂jgih − ∂hgij), ∂k =
∂
∂xk
. (2.1)
Replacing g in (2.1) by the metric tensor g′ of M ′
n given by (1.1) and doing the necessary
calculations we first find the connection coefficients Γ′l
ij of M ′
n as
Γ′l
ij =
1
2
e2σg′
lk
(∂jωik + ∂iωjk − ∂kωij) + (∂jσ)δli + (∂iσ)δlj − (∂kσ)g′
lk
g′ij . (2.2)
On the other hand, for the covariant derivative of the second fundamental tensor ω of Mn we
have [3, 4]
∇iωjk = ∂iωjk − Γhijωhk − Γhikωjh. (2.3)
Changing the indices i, j and k cyclically we obtain two more equations:
∇jωki = ∂jωki − Γhijωhk − Γhkjωih, (2.4)
∇kωij = ∂kωij − Γhkiωhj − Γhkjωih. (2.5)
Subtracting (2.5) from the sum of (2.3) and (2.4) and using the Codazzi equations (1.4), we
obtain
∇iωjk = ∂iωjk + ∂jωik − ∂kωij − 2ωhkΓ
h
ij . (2.6)
In view of (2.6), (2.2) becomes
Γ′l
ij = Γlij + δli∂jσ + δlj∂iσ − g′
lk
g′ij∂kσ +
1
2
e2σg′
lk∇iωjk. (2.7)
(2.7) is the desired relation connecting the connection coefficients of Mn and M ′
n.
3. Geodesic mappings of Mn upon M ′
n. If the map f defined by (1.2) transforms every
geodesic in Mn into a geodesic in M ′
n, f is called a geodesic mapping of Mn into M ′
n.
Mn and M ′
n will be in geodesic correspondence if and only if the respective connection coeffi-
cients Γhij and Γ′h
ij of Mn and M ′
n are related by [3]
Γ′i
jk = Γijk + δijψk + δikψj , (3.1)
where ψk are the components of some 1-form which is known to be a gradient.
We first prove the following lemma which will be needed in our subsequent work.
ISSN 1027-3190. Укр. мат. журн., 2013, т. 65, № 4
A CHARACTERIZATION OF TOTALLY UMBILICAL HYPERSURFACES OF A SPACE FORM . . . 585
Lemma 3.1. Let Mn and M ′
n be hypersurfaces of the space form M̄n+1 and let the metric
tensor of M ′
n be defined by (1.1). If Mn and M ′
n are in geodesic correspondence, then the 1-form ψk
is the gradient of 2σ.
Proof. Since ∇′ is a metric connection we have
0 = ∇′
kg
′
ij = ∂kg
′
ij − g′ljΓ′l
ik − g′liΓ′l
jk
so that with the help of (1.1) and (3.1) we obtain
0 = 2ωij∂kσ +∇kωij − 2ψkωij − ψiωkj − ψjωki. (3.2)
Interchanging the indices j and k in (3.2) we find
0 = 2ωik∂jσ +∇jωik − 2ψjωik − ψiωkj − ψkωji. (3.3)
Subtracting (3.3) from (3.2) and putting
φk = ψk − 2∂kσ (3.4)
in (3.3) we conclude that
ωijφk − ωikφj = 0 (3.5)
in which the Codazzi equations (1.4) have been used.
We note that, since ψk is a gradient, it follows from (3.4) that φk is also a gradient. Multiplying
(3.5) by e2σ and using (1.1) we obtain
φkg
′
ij − φjg′ik = 0 (3.6)
or, multiplying (3.6) by g′ij and summing with respect to i and j we find for n > 1 that
φk = 0. (3.7)
Combination of (3.4) and (3.7) yields ψk = 2∂kσ.
We next prove the following theorem.
Theorem 3.1. The hypersurface Mn of a space form M̄n+1 will be totally umbilical if and
only if Mn can be geodesically mapped upon M ′
n.
Proof. Sufficiency. Let γ be a geodesic through the point p ∈Mn which is defined by xi = xi(s),
s being the arc length of γ. Then, the normal curvature, say κn, of Mn in the direction of γ, i.e., in
the direction of
dxi
ds
, is [4]
κn = ωij
dxi
ds
dxj
ds
. (3.8)
Multiplying (3.2) by
dxi
ds
dxj
ds
dxk
ds
and summing with respect to i, j, k and using (3.8) we obtain
2κn(∂kσ)
dxk
ds
+ (∇kωij)
dxk
ds
dxi
ds
dxj
ds
− 2
(
ψk
dxk
ds
)
κn −
(
ψi
dxi
ds
)
κn −
(
ψj
dxj
ds
)
κn = 0.
(3.9)
ISSN 1027-3190. Укр. мат. журн., 2013, т. 65, № 4
586 E. Ö. CANFES, A. ÖZDEĞER
Since ψk is a gradient, there exists a differentiable function ψ such that ψk = ∂kψ. On the other
hand, differentiating (3.8) covariantly in the direction of γ and using the Frenet’s formula [3](
∇k
dxi
ds
)
dxk
ds
= κg
1
ηi,
where κg is the geodesic curvature and
1
η is the unit principal normal vector field of γ relative to Mn,
we find that
(∇kωij)
dxk
ds
dxi
ds
dxj
ds
=
dκn
ds
− 2κgωij
1
ηi
dxj
ds
. (3.10)
Using (3.10) in (3.9) and remembering that γ is a geodesic (κg = 0) in Mn, we get[
∂κn
∂xi
+
(
2
∂σ
∂xi
− 4
∂ψ
∂xi
)
κn
]
dxi
ds
= 0,
or [
∂
∂xi
(ln |κn|+ 2σ − 4ψ)
]
dxi
ds
= 0 (3.11)
along γ.
On the other hand, by (1.1) and (3.11), we find
ds′
2
= g′ijdx
idxj = e2σωijdx
idxj = e2σωij
dxi
ds
dxj
ds
ds2 = e2σκnds
2,
from which it follows that κn > 0. From (3.1) it follows that,
lnκn + 2σ − 4ψ = const = C1 (3.12)
along γ.
By Lemma 3.1, ψ = 2σ + C2, C2 = Const and therefore (3.12) gives
κn = ce6σ, (3.13)
where c is an arbitrary positive constant.
From (3.13) it follows that the lines of curvature of Mn are indeterminate at all points of Mn.
Consequently, Mn is totally umbilical.
Necessity. Assume that Mn is a totally umbilical hypersurface of M̄n+1 which means that ωij =
=
H
n
gij where H is the mean curvature of Mn. In this case, (1.1) becomes
g′ij = ρ2gij
(
ρ2 = e2σ
H
n
)
, (3.14)
so that Mn and M ′
n are conformal.
From (1.5) it follows that
Rijkl =
(
K̄ +
H2
n2
)
(gikgjl − gilgjk)
showing that Mn has the constant curvature K̄ +
H2
n2
. So H is constant.
ISSN 1027-3190. Укр. мат. журн., 2013, т. 65, № 4
A CHARACTERIZATION OF TOTALLY UMBILICAL HYPERSURFACES OF A SPACE FORM . . . 587
We will show that Mn can also be geodesically mapped upon M ′
n. Since Mn is conformal to
M ′
n, their connection coefficients are related by [6]
Γ′h
ij = Γhij + δhj ρi + δhi ρj − gijρh
(
ρi = ∇iρ, ρh = gthρt
)
. (3.15)
To show that this conformal mapping between Mn and M ′
n is also a geodesic mapping, according
to (3.15) and (3.1) we have to find a 1-form ψk such that
Γhij + δhj ψi + δhi ψj = Γhij + δhj ρi + δhi ρj − gijρh
or
δhj (ψi − ρi) + δhi (ψj − ρj) + gijρ
h = 0. (3.16)
Transvecting (3.16) by gij we get
gih(ψi − ρi) + gjh(ψj − ρj) + nρh = 0
or
2gih(ψi − ρi) + nρh = 0. (3.17)
Multiplying (3.17) by ghj and summing for h we obtain
2ψj + (n− 2)ρj = 0.
Then, by (3.14) we find that
ψj =
(
2− n
2
√
n
√
H
)
∂je
σ, H > 0.
With this choice of ψj the conformal mapping mentioned above becomes also a geodesic mapping.
Theorem 1.1 is proved.
In the special case where σ = 0 throughout Mn, i.e., when g′ = ω, we may mention below some
properties of Mn which is in geodesic correspondence with M ′
n.
1. From Lemma 3.1 and the relation (3.1) we conclude that any geodesic mapping of Mn upon
M ′
n is connection preserving.
2. By (3.13) it follows that Mn has constant normal curvature along each geodesic through a
point p ∈Mn.
3. The underlying geodesic mapping is a homothety.
1. Verpoort S. The geometry of the second fundamental form: curvature properties and variational aspects: Ph. D Thesis.
– Katholieke Univ. Leuven, 2008.
2. Adachi T., Maeda S. Characterization of totally umbilic hypersurfaces in a space form by circles // Czechoslovak
Math. J. – 2005. – 55, № 1. – P. 203 – 207.
3. Gerretsen J. Lectures on tensor calculus and differential geometry. – Groningen: P. Noordhoff, 1962.
4. Weatherburn C. E. An Introduction to Riemannian geometry and the tensor calculus. – Cambridge Univ. Press, 1966.
5. Yano K., Kon M. Structures on manifolds. – Worldsci., 1984.
6. Chen B. Y. Geometry of submanifolds. – New York: Marcel Dekker Inc., 1973.
Received 02.02.11
ISSN 1027-3190. Укр. мат. журн., 2013, т. 65, № 4
|
| id | umjimathkievua-article-2441 |
| institution | Ukrains’kyi Matematychnyi Zhurnal |
| keywords_txt_mv | keywords |
| language | English |
| last_indexed | 2026-03-24T02:23:29Z |
| publishDate | 2013 |
| publisher | Institute of Mathematics, NAS of Ukraine |
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| resource_txt_mv | umjimathkievua/a8/b60ca13d703d90ab61fc3495589073a8.pdf |
| spelling | umjimathkievua-article-24412020-03-18T19:15:36Z A characterization of totally umbilical hypersurfaces of a space form by geodesic mapping Характеристика тотально омбiлiчних гiперповерхонь просторової форми за допомогою геодезичних вiдображень Canfes, E. Ö. Özdeğer, A. Канфес, Е. О. Оздер, А. The idea of considering the second fundamental form of a hypersurface as the first fundamental form of another hypersurface has found very useful applications in Riemannian and semi-Riemannian geometry, specially when trying to characterize extrinsic hyperspheres and ovaloids. Recently, T. Adachi and S. Maeda gave a characterization of totally umbilical hypersurfaces in a space form by circles. In this paper, we give a characterization of totally umbilical hypersurfaces of a space form by means of geodesic mapping. Iдея використання другої фундаментальної форми гiперповерхнi як першої фундаментальної форми iншої гiперповерхнi знайшла дуже важливi застосування у рiмановiй та напiврiмановiй геометрiї, зокрема при описi зовнiшнiх гiперсфер та овалоїдiв. Нещодавно T. Adachi та S. Maeda навели характеристику тотально омбiлiчних гiперповерхонь у просторовiй формi за допомогою кiл. У цiй роботi ми наводимо характеристику тотально омбiлiчних гiперповерхонь просторової форми за допомогою геодезичних вiдображень. Institute of Mathematics, NAS of Ukraine 2013-04-25 Article Article application/pdf https://umj.imath.kiev.ua/index.php/umj/article/view/2441 Ukrains’kyi Matematychnyi Zhurnal; Vol. 65 No. 4 (2013); 583-587 Український математичний журнал; Том 65 № 4 (2013); 583-587 1027-3190 en https://umj.imath.kiev.ua/index.php/umj/article/view/2441/1648 https://umj.imath.kiev.ua/index.php/umj/article/view/2441/1649 Copyright (c) 2013 Canfes E. Ö.; Özdeğer A. |
| spellingShingle | Canfes, E. Ö. Özdeğer, A. Канфес, Е. О. Оздер, А. A characterization of totally umbilical hypersurfaces of a space form by geodesic mapping |
| title | A characterization of totally umbilical hypersurfaces of a space form by geodesic mapping |
| title_alt | Характеристика тотально омбiлiчних гiперповерхонь просторової форми за допомогою геодезичних вiдображень |
| title_full | A characterization of totally umbilical hypersurfaces of a space form by geodesic mapping |
| title_fullStr | A characterization of totally umbilical hypersurfaces of a space form by geodesic mapping |
| title_full_unstemmed | A characterization of totally umbilical hypersurfaces of a space form by geodesic mapping |
| title_short | A characterization of totally umbilical hypersurfaces of a space form by geodesic mapping |
| title_sort | characterization of totally umbilical hypersurfaces of a space form by geodesic mapping |
| url | https://umj.imath.kiev.ua/index.php/umj/article/view/2441 |
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