On geodesic bifurcations of product spaces
The bifurcation is described as a situation where there exist at least two different geodesics going through the given point in the given direction. In the previous works, the examples of local and closed bifurcations are constructed. This paper is devoted to the further study of these bifurcations....
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irk-123456789-1694022020-06-13T01:27:03Z On geodesic bifurcations of product spaces Ryparova, L. Mikes, J. Sabykanov, A. The bifurcation is described as a situation where there exist at least two different geodesics going through the given point in the given direction. In the previous works, the examples of local and closed bifurcations are constructed. This paper is devoted to the further study of these bifurcations. We construct an example of n-dimensional (pseudo-) Riemannian and Kahlerian spaces which are product ones that admit a local bifurcation of geodesics and also a closed geodesic. 2018 Article On geodesic bifurcations of product spaces / L. Ryparova, J. Mikes, A. Sabykanov // Український математичний вісник. — 2018. — Т. 15, № 2. — С. 264-271. — Бібліогр.: 6 назв. — англ. 1810-3200 2010 MSC. 53A05, 53B21, 53B30, 53B35, 53C22 http://dspace.nbuv.gov.ua/handle/123456789/169402 en Український математичний вісник Інститут прикладної математики і механіки НАН України |
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The bifurcation is described as a situation where there exist at least two different geodesics going through the given point in the given direction. In the previous works, the examples of local and closed bifurcations are constructed. This paper is devoted to the further study of these bifurcations. We construct an example of n-dimensional (pseudo-) Riemannian and Kahlerian spaces which are product ones that admit a local bifurcation of geodesics and also a closed geodesic. |
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Ryparova, L. Mikes, J. Sabykanov, A. |
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Ryparova, L. Mikes, J. Sabykanov, A. On geodesic bifurcations of product spaces Український математичний вісник |
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Ryparova, L. Mikes, J. Sabykanov, A. |
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Ryparova, L. |
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On geodesic bifurcations of product spaces |
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On geodesic bifurcations of product spaces |
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On geodesic bifurcations of product spaces |
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On geodesic bifurcations of product spaces |
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On geodesic bifurcations of product spaces |
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on geodesic bifurcations of product spaces |
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Інститут прикладної математики і механіки НАН України |
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On geodesic bifurcations of product spaces / L. Ryparova, J. Mikes, A. Sabykanov // Український математичний вісник. — 2018. — Т. 15, № 2. — С. 264-271. — Бібліогр.: 6 назв. — англ. |
| series |
Український математичний вісник |
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AT ryparoval ongeodesicbifurcationsofproductspaces AT mikesj ongeodesicbifurcationsofproductspaces AT sabykanova ongeodesicbifurcationsofproductspaces |
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2025-07-15T04:08:38Z |
| last_indexed |
2025-07-15T04:08:38Z |
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1837684509960568832 |
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Український математичний вiсник
Том 15 (2018), № 2, 264 – 271
On geodesic bifurcations of product spaces
Lenka Rýparová, Josef Mikeš and Almaz Sabykanov
(Presented by F. Abdullayev)
Abstract. The bifurcation is described as a situation where there
exist at least two different geodesics going through the given point in
the given direction. In the previous works the examples of a local bi-
furcation and closed bifurcation are constructed. This paper is devoted
to further study of these bifurcations. We construct an example of an
n-dimensional (pseudo-) Riemannian and Kählerian spaces which are
product that admit local bifurcation of geodesics and that also admit
closed geodesics.
2010 MSC. 53A05, 53B21, 53B30, 53B35, 53C22.
Key words and phrases. (pseudo-) Riemannian space, product space,
geodesic, geodesic bifurcation.
1. Introduction
It is well know that on a surface there exists one and the only one
geodesic going through given point in given direction. The proof of this
statement follows from the analysis of system of a ordinary differential
equations (ODE), and the statement is valid for surfaces for which the
Christoffel symbols are differentiable functions. Kuiper [2] and Borisov [1]
presented isometric imbending of class C1.
On the other hand, if the Christoffel symbols are continuous and are
not differentiable functions then the solution of above mentioned system
of ODE is not unique. Therefore, there exist more than one solution
of ODE and more geodesics going through given point which, at this
point, have the same tangent vector. This situation is called a geodesic
bifurcation.
In [4] we obtained a surface of revolution where geodesic bifurcation
exists. The construction of closed geodesic bifurcation on a surface is
Research supported by IGA PrF 2018012 at Palacky University Olomouc, and by spe-
cific university research at Brno University of Technology FAST-S-18-5184.
ISSN 1810 – 3200. c⃝ Iнститут прикладної математики i механiки НАН України
L. Rýparová, J. Mikeš, A. Sabykanov 265
published in [5]. Here, we include the previously obtained results [4,5] in a
short form. Furthermore, we state new results about geodesic bifurcation
on product spaces. Latest results prove that there exist Kähler and also
hyperbolical Kähler spaces which admit these geodesic bifurcations.
Let us note, that geodesic bifurcations were also studied in paper [6],
but in this case the term geodesic bifurcation does not have the same
meaning as we use in the presented work.
2. Geodesic bifurcation
Let M be a manifold with an affine connection ∇. In local chart the
components Γhij of the connection ∇ are given. A curve γ on the manifold
M is called geodesic if its tangent vector is recurrent along it.
Locally, on the geodesic γ there exists a canonical parameter (arc
length) s for which the geodesic satisfy the equation: ∇sγ̇ = 0. In this
case, vector field γ̇ is parallel along curve γ. Here “ · ” denotes derivative
with respect to arc length.
In chart (U, x) above mentioned equation has the following form
ẍh(s) + Γhij(x(s)) ẋ
i(s) ẋj(s) = 0, (2.1)
where γ: xh = xh(s) are equation of geodesic γ on chart (U, x), and Γhij
are components of ∇. Also, geodesics are often defined as curves that
satisfy the equations (2.1), see [3].
We can rewrite the equations (2.1) in the form of a system of ordinary
differential equations of the first order with respect to unknown functions
xh(s) and λh(s) like follows
ẋh(s) = λh(s), λ̇h(s) = −Γhij (x(s)) ẋ
i(s) ẋj(s), (2.2)
where λh(s) is a tangent vector of the curve γ(s) at the point xh(s). For
the initial conditions
xh(s) = xh0 , λh(s) = λh0 (2.3)
the solution of the system (2.2) is a geodesic going through the point xh0
in the direction λh0 (̸= 0). If the function Γhij(x) are continuous functions,
then from the general theory of differential equations follows that the
equations (2.2) with initial conditions (2.3) have solution. Furthermore,
if the functions Γhij(x) belong to differentiability class C1 then the solution
of such system is unique. It is worth to note that the last condition of
differentiability of the functions Γhij(x) can be replaced with the Lipschitz
condition.
266 On geodesic bifurcations of product spaces
In [3] there is given an example of the space with affine connection
where the components are only continuous functions. In this case, there
exist one and only one geodesic passing through given point xh0 in given
direction λh0 .
Example 2.1. Let us consider a space An with affine connection ∇
defined in a certain coordinate neighbourhood (U, x) by its continuous
components
Γhhh = fh(xh(s)), h = 1, . . . , n,
the other components are vanishing. The solution of the Cauchy problem
(2.2) and (2.3) was obtained in the following explicit form∫ xh(s)
xh0
exp
(∫ w
xh0
fh(τ) dτ
)
dw = λh0 s.
This solution is unique and exists in the whole neighbourhood U , that is
the geodesic passing through the point xh0 in direction λh0 is also unique.
Let us remind that in this example the components of the connection
were continuous but not differentiable functions.
It was shown [4], that there exist a surface of revolution which admits
geodesic bifurcation.
Example 2.2. Let us consider a surface of revolution S given by the
equations
x = r(u) cos v, y = r(u) sin v, z = z(u),
where
r(u) =
1√
1− u2α
, z(u) =
∫ u
0
√
1− α2t4α−2 · (1− t2α)−3 dt,
u ∈ (−1, 1), v ∈ (−π, π) are parameters.
For this surface the Christoffel symbols are continuous functions which
are evidently not differentiable and also do not satisfy Lipschitz condi-
tion. Then on the surface of revolution S there exist geodesic bifurcations
for α ∈ (0, 1).
The proof of this statement follows from the existence of two different
geodesics going through the point (0, 0) in the direction (0, 1). These
geodesics are given by the equations
I. u = 0, v = s;
II. u =
(
(1− α) s
) 1
1−α , v = s−
(
(1− α) s
) 1+α
1−α
1 + α
.
L. Rýparová, J. Mikeš, A. Sabykanov 267
The first one we shall call a trivial geodesic - also called gorge circle -
and the second one nontrivial geodesic passing through the point (0, 0)
in the direction (0, 1).
We note a remarkable fact that the consequence of this statement is
that geodesic bifurcation exists in each point of the gorge circle since the
given surface of revolution is symmetric. Furthermore, there exists an
infinite number of geodesics going through the point (0, 0) in direction
(0, 1). Moreover, the surface S has the metric ds2 = du2 + r2(u) dv2
and because any surface with this metric admits a nontrivial geodesic
mapping, then projective corresponding spaces preserve above mentioned
geodesic bifurcations.
Detailed description of this problem can be found in [4].
The previously obtained results are further developed in [5]. In this
paper there is constructed a surface on which bifurcation of closed geo-
desics exists.
Example 2.3. Let us consider a certain neighbourhood of the gorge
circle of the surface S mentioned in the previous example, where the
geodesic bifurcations exists. Let us suppose that the geodesic curve γ
starts at the point (0, 0) and has a tangent vector (0, 1), the “end” lies
also on the gorge circle, and is part of another nontrivial geodesic.
To construct a surface where bifurcation of closed geodesic exists,
the goal is to connect the ends of the geodesic γ with the smooth curve
which would form the surface. In this construction is used a fact that
the surface of revolution is symmetric and also results obtained in [4].
For further information and more detailed description of the above
mentioned construction of this example see [5].
3. Product manifolds
A Riemannian manifold Vn is called a product manifold of Rieman-
nian manifolds
1
Vn1 ,
2
Vn2 , . . .,
m
Vnm (n1 + n2 + · · · + nm = n)
Vn =
1
Vn1 ⊗
2
Vn2 ⊗ · · · ⊗
m
Vnm (3.1)
if the metrics are related by
g = g1 ⊗ g2 ⊗ · · · ⊗ gm.
Locally this means that there exists a coordinate system (xi) such that
the metric forms of these Riemannian manifolds satisfy
ds2 = ds21 + ds22 + . . .+ ds2m,
268 On geodesic bifurcations of product spaces
where
ds2 = gij (x
k) dxi dxj and ds2σ =
σ
giσjσ(x
kσ) dxiσ dxjσ ,
i, j, k = ⟨1, n⟩; iσ, jσ, kσ = ⟨pσ, rσ⟩;
1 = p1 ≤ r1 < p2 ≤ r2 < · · · < pm ≤ rm = n.
It is know [3, pp. 192] that a product manifold Vn with metric
ḡ = (α1g1)⊗ (α2g2)⊗ · · · ⊗ (αmgm), where constants ασ ̸= 0
admit affine mappings f : Vn(M, g) −→ Vn(M, ḡ).
By analysing equations (2.1) we can verify that the geodesic γ in
product manifold Vn = (M, g) can be generated by geodesic i
γ ⊂
i
Vn for
i = 1, 2, . . .m like follows
γ =
1
γ ⊗ 2
γ ⊗ · · · ⊗ m
γ,
while among these geodesics i
γ there can exist “trivial geodesics" which
are points in the space, i.e. they are defined by the equations xki = x
0
ki =
const.
Kähler manifold Kn is a Riemannian space with a metric g and a
structure F that satisfies following conditions
F 2 = −Id, g(X,FX) = 0, ∇F = 0,
for any vector X, where ∇ is Levi-Civita connection of the space.
It can be easily verified that the product space of the Kählerian spaces
Kn =
1
K ⊗
2
K ⊗ · · · ⊗
m
K
is also Kählerian space with structure which have analogical construction
F =
1
F ⊗
2
F ⊗ · · · ⊗
m
F , i.e. F hσiσ =
σ
F hσiσ ,
for σ = 1, . . . ,m, and the other components of the structure F are van-
ishing.
From the other side, two dimensional Riemannian manifold is always
Kähler, see [3, pp. 130], where the following holds
F hi = εijg
jh, εij =
√
g11g22 − g212 ·
(
0 1
−1 0
)
.
L. Rýparová, J. Mikeš, A. Sabykanov 269
4. Geodesic bifurcation of product spaces
The above mentioned construction of the product spaces can be now
used for a similar construction of geodesic bifurcation in n-dimensional
Riemannian spaces and also in Kähler spaces.
Evidently, it is sufficient to take take one from the components
i
V
of the above described product space (3.1) as the space with metric de-
scribed in examples 2.2 and 2.3. For example, if
1
V is the space with the
metric described in examples 2.2 and 2.3 and this space has a geodesic
bifurcation at the point 1
γ(0) then it is obvious that geodesic
γ =
1
γ ⊗ 2
γ ⊗ · · · ⊗ m
γ
has a geodesic bifurcation at the point γ(0).
From the above mentioned follows that in this manner it is possible to
construct an n-dimensional Riemannian spaces where the geodesic bifur-
cations exist. Analogically, we can contruct an n-dimensional Kählerian
spaces which also admit geodesic bifurcations.
Let us note that the geodesic bifurcations do not have to exist in
other spaces
2
V, . . . ,
m
V. In case there exists a geodesic bifurcation at the
point i
γ(0) of the space
i
V then the image of the geodesic in the product
space Vn would be much more complicated.
Using global results about geodesic bifurcations obtained and de-
scribed in previous paper [5] where the bifurcations of closed geodesic are
constructed, it is also possible to construct bifurcation of closed geodesics
in n-dimensional Riemannian and Kähler manifolds.
In addition, the construction of the product spaces makes it possible
to construct also pseudo-Riemannian spaces where there exist geodesic
bifurcations of isotropic geodesics, i.e. geodesic which have vanishing
length.
Example 4.1. Let V =
1
V ⊗
2
V be product space of the spaces
1
V and
2
V and let gi1j1 , i1, j1 = 1, 2; gi2j2 , i2, j2 = 1, 2 be positive and negative
metric forms of spaces
1
V,
2
V respectively.
In Riemannian space
1
V the geodesic
1
γ has a bifurcation at point
1
γ(0) (space
1
V can be the space from the examples 2.2 and 2.3). Then
270 On geodesic bifurcations of product spaces
the length of the vector is |1γ(0)| = 1. Let us suppose that in Rieman-
nian space
2
V there exists a geodesic
2
γ for which |2γ(0)| = 1. Then the
geodesic γ for which
xi1 = x
1
i1 and xi2 = x
2
i2
has tangent vectors λi for which
λi1 = dx
1
i1/dt and λi2 = dx
2
i2/dt.
Since gi1j1 is a positive form and gi2j2 is a negative form, it is evident
that |λ| = 0 because
|λ| =
√
gijλiλj =
√
gi1j1λ
i1λj1 + gi2j2λ
i2λj2 =
√
1− 1 = 0.
Conclusion
The work develops the results and ideas from the previously published
papers [4, 5]. In presented work we have constructed an n-dimensional
Riemannian product spaces and also Kählerian product spaces which
admit local geodesic bifurcations and also bifurcations of closed geodesic.
References
[1] Y. F. Borisov, Irregular C1,β-surfaces with an analytic metric // Siberian Math.
J., 45 (2004), N. 1, 19–52.
[2] N. H. Kuiper, On C1-isometric imbeddings. I // Proc. Koninkl. nederl. akad.
wetensch. A, 58 (1955), 545–556.
[3] J. Mikeš et al., Differential geometry of special mappings, Palacky Univ. Press,
Olomouc, 2015.
[4] L. Rýparová, J. Mikeš, On geodesic bifurcations // Geometry, integrability and
quantization, 18 (2017), 217–224.
[5] L. Rýparová, J. Mikeš, Bifurcation of closed geodesics // Geometry, integrability
and quantization, 19 (2018), 188–192.
[6] H. Thielhelm, A. Vais, F. E. Wolter, Geodesic bifurcation on smooth surfaces //
The Visual Computer, 31 (2015), N. 2, 187–204.
L. Rýparová, J. Mikeš, A. Sabykanov 271
Contact information
Lenka Rýparová Department of Algebra and Geometry,
Palacky University Olomouc,
Olomouc, Czech Republic
E-Mail: lenka.ryparova01@upol.cz
Josef Mikeš Department of Algebra and Geometry,
Palacky University Olomouc,
Olomouc, Czech Republic
E-Mail: josef.mikes@upol.cz
Almaz Sabykanov Department of Geometry,
Kyrgyz National University,
Bishkek, Kyrgyzstan
E-Mail: almazbek.asanovich@mail.ru
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