On the Group of Foliation Isometries
The purpose of our paper is to introduce some topology on the group GrF(M) of all Cr-isometries of foliated manifold (M, F), which depends on a foliation F and coincides with compact-open topology when F is an n-dimensional foliation. If the codimension of F is equal to n, convergence in our topolog...
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| Cite this: | On the Group of Foliation Isometries / A.Ya. Narmanov, A.S. Sharipov // Methods of Functional Analysis and Topology. — 2009. — Т. 15, № 2. — С. 195-200. — Бібліогр.: 10 назв. — англ. |
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| author_facet | Narmanov, A.Yu. Sharipov, A.S. |
| citation_txt | On the Group of Foliation Isometries / A.Ya. Narmanov, A.S. Sharipov // Methods of Functional Analysis and Topology. — 2009. — Т. 15, № 2. — С. 195-200. — Бібліогр.: 10 назв. — англ. |
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| description | The purpose of our paper is to introduce some topology on the group GrF(M) of all Cr-isometries of foliated manifold (M, F), which depends on a foliation F and coincides with compact-open topology when F is an n-dimensional foliation. If the codimension of F is equal to n, convergence in our topology coincides with pointwise convergence, where n = dimM. It is proved that the group GrF(M) is a topological group with compact-open topology, where r ≥ 0. In addition it is showed some properties of F-compact-open topology.
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Methods of Functional Analysis and Topology
Vol. 15 (2009), no. 2, pp. 195–200
ON THE GROUP OF FOLIATION ISOMETRIES
A. YA. NARMANOV AND A. S. SHARIPOV
Abstract. The purpose of our paper is to introduce some topology on the group
Gr
F
(M) of all Cr-isometries of foliated manifold (M, F ), which depends on a fo-
liation F and coincides with compact-open topology when F is an n-dimensional
foliation. If the codimension of F is equal to n, convergence in our topology coin-
cides with pointwise convergence, where n = dimM . It is proved that the group
Gr
F
(M) is a topological group with compact-open topology, where r ≥ 0. In addition
it is showed some properties of F-compact-open topology.
Let M, N be n-dimensional smooth manifolds on which there are given k-dimensional
smooth foliations F1, F2, respectively (where 0 < k < n).
If for the some Cr-diffeomorphism f : M → N the image f(Lα) of any leaf Lα of the
foliation F1 is a leaf of the foliation F2, we say that the pairs (M, F1) and (N, F2) are
Cr-diffeomorphic. In this case the mapping f is called Cr-diffeomorphism, preserving
foliation and is written as
f : (M, F1) → (N, F2).
In the case where M = N , f is said to be a diffeomorphism of the foliated manifold
(M, F ).
Diffeomorphisms, preserving foliation, are investigated in [1], [2].
Definition. A diffeomorphism ϕ : M → M of the class Cr(r ≥ 0), preserving foliation,
is called a foliation isometry F (an isometry of the foliated manifold (M, F )) if it is an
isometry on each leaf of the foliation F , i.e. for each leaf Lα of the foliation, ϕ : Lα →
ϕ(Lα) is an isometry.
Papers [3], [4] are devoted to isometric mappings of foliations. In these papers it
is investigated the question under what conditions any isometry of the foliation is an
isometry of the manifold and it is proved the existence of a diffeomorphism of a foliated
manifold onto itself which is an isometry of the foliation, but it is not an isometry of the
manifold. There is constructed an example of a diffeomorphism of a three-dimensional
sphere which is an isometry of the Hopf fibration but is not an isometry of the three-
dimensional sphere.
Let M be a n-dimensional smooth connected Riemannian manifold with a Riemann-
ian metric g, F a smooth k-dimensional foliation on M . (In this paper manifolds and
foliations have C∞-smoothness). We denote by L(p) a leaf of the foliation F passing
through point the p, by TpF the tangent space to the the leaf L(p) at p, and by HpF its
orthogonal complement of TpF in TpM , p ∈ M . We get two subbundles (smooth distrib-
utions) TF = {TpF : p ∈ M}, HF = {HpF : p ∈ M} of the tangent bundle TM of the
manifold M and, as a result, the tangent bundle TM of the manifold M decomposing
into the sum of two orthogonal bundles, i.e., TM = TF ⊕ HF. The restriction of the
Riemannian metric g to TpF for all p induces a Riemannian metric on the leaves. The in-
duced Riemannian metric defines a distance function on every leaf. Further, everywhere
2000 Mathematics Subject Classification. 57R30, 57M07.
Key words and phrases. Manifold, foliation, leaf, isometry, group, compact open topology.
195
196 A. YA. NARMANOV AND A. S. SHARIPOV
in this paper, under the distance on a leaf, we understand this distance. This distance
on a leaf is different from the distance induced by the distance on M.
Let us denote by Gr
F (M) the set of all Cr isometries of a foliated manifold (M, F ),
where r ≥ 0. The following remarks show that the notion of an isometry of a foliated
manifold is correctly defined.
Remark 1. If r ≥ 1, for each element ϕ ∈ Gr
F (M) the differential dϕ preserves the length
of each tangent vector ν ∈ TpF , i.e., | dϕp(ν) |=| ν | at any p ∈ M .
Remark 2. If r = 0, each element ϕ from Gr
F (M) is a homeomorphism of the manifold
M . A Riemannian metric on the manifold M induces a Riemannian metric on each leaf
Lα which defines a distance on it. In this case, ϕ is an isometry between the metric
spaces Lα and ϕ(Lα). Then, according to the known theorem, ϕ is a diffeomorphism of
Lα onto ϕ(Lα) for each leaf Lα and its differential preserves the length of each tangent
vector ν ∈ TpF , i. e., | dϕp(ν) |=| ν | at any p ∈ M [5, page 74]. But as shown by a
simple example, from differentiability of a mapping on each leaf does one can not infer
its differentiability on the entire manifold M .
Example 1. Let M = R2(x, y) be the Euclidean plane with the Cartesian coordinates
(x, y). Leaves Lα of foliation a F are given by the equations y = α = const . Then the
plane homeomorphism ϕ : R2 → R2 determined by the formula
ϕ(x, y) = (x + y, y
1
3 )
is an isometry of the foliation F , but is not a diffeomorphism of the plane.
The set Diffr(M) of all diffeomorphisms of a manifold M onto itself is a group with
the operations of composition and taking the inverse. The set Gr
F (M) is a subgroup of
the group Diffr(M).
It is known that if a manifold M is compact, the group Diffr(M) is a topological group
in the compact-open topology. This fact follows from Proposition 2 on page 269 and from
Propositions 3, 4 on page 270 of [6]. It is also known that the group of isometries I(M)
is always a topological group with compact-open topology [5, page 192].
The purpose of this paper is to introduce some topology on the set Gr
F (M), depending
on the foliation F , such that it coincides with the compact-open topology if F is an n-
dimensional foliation. If the codimension of the foliation F is equal to n, convergence in
our topology coincides with the pointwise convergence.
Let {Kλ} be a family of all compact sets where each Kλ is a subset of some leaf of
the foliation F and let {Uβ} be a family of all open sets on M . We consider, for each
pair Kλ and Uβ , the set of all mappings f ∈ Gr
F (M) such that f(Kλ) ⊂ Uβ. This set of
mappings is denoted by [Kλ, Uβ] = {f : M → M |f(Kλ) ⊂ Uβ}.
It is not difficult to show that every possible finite intersections of sets of the form
[Kλ, Uβ] forms a base for some topology. We will call this topology the foliated compact-
open topology or, briefly, the F -compact-open topology.
The following proposition can be proved using a standard argument.
Proposition. The set Gr
F (M) with the F -compact open topology is a Hausdorff space.
Theorem 1. Let M be a smooth, connected and finite-dimensional manifold. Then the
group of homeomorphisms Homeo(M) is a topological group with compact-open topology.
In particular, the subgroups Diffr(M), Gr
F (M) are topological groups in this topology.
Proof. It is known that the mapping (g, h) → g ◦h is continuous for every Hausdorff and
locally compact space [6, page 269].
By d(x, y) we denote the distance between the points x and y, determined by some
complete Riemannian metric. It is known that a smooth manifold M possesses a complete
Riemannian metric [7, page 186].
ON THE GROUP OF FOLIATION ISOMETRIES 197
Using a complete Riemannian metric on M we will prove that the mapping χ : f →
f−1 is continuous. For this purpose, we shall prove that the full preimage χ−1(A) of the
open set A ⊂ Homeo(M) is an open set. Indeed, it is enough to show this fact if A is
an element of the prebase, i.e., A = {f ∈ Homeo(M) : f(K) ⊂ V }, where K is compact
and V is an open set. In this case, χ−1(A) = {f ∈ Homeo(M) : f−1(K) ⊂ V }. We will
show that χ−1(A) is an open set in the compact-open topology. Let g ∈ χ−1(A), U be
a neighborhood of g−1(K) with compact closure such that U ⊂ V . We put U(g) = {g ∈
Diffr(M) : d(g(x), h(x)) < ε
2
, ∀x ∈ U}, where ε = d(K, M\g(U)) = inf{d(x, y) : x ∈
K, y ∈ (M\g(U))}.
Let us show that, if h ∈ U(g), then h−1(K) ⊂ V , i.e., U(g) ⊂ χ−1(A). We will show
that h−1(K) ⊂ U. Assume it is not true. Let, for the some h ∈ U(g), there exists a
point y ∈ K such that h−1(y) ∈ M\U, i.e., y ∈ M\h(U). Then, since g−1(y) ∈ U,
we have d(y = g(g−1(y)), h(g−1(y))) < ε
2
. Let γ be the shortest geodesic (by virtue
of completeness of M there exists a shortest geodesic between any two points) going
from the point y to the point h(g−1(y)), and z ∈ γ ∩ ∂(h(U)). Then h−1(z) ∈ U and
besides d(g(h−1(z)), h(h−1(z)))) < ε
2
. In addition d(y, z) < ε
2
. Hence, d(y, g(h−1(z))) ≤
d(y, z) + d(z, g(h−1(z))) < ε. But, on the other hand, since z /∈ h(U), we have g(h−1(z)) ∈
M\g(U) and d(y, g(h−1(z))) ≥ ε. This contradiction shows that h−1(K) ⊂ U. Hence,
U(g) ⊂ χ−1(A).
Now we will show that U(g) is an open set in the compact-open topology. Let f ∈
U(g), 0 < r < ε
2
− a, where a = max
x∈U
{d(f(x), g(x))}. We will show that f is an
interior point of the set U(g). For this purpose, we cover the compact set f(U) with a
finite number of balls Bi with the centers in the points xi ∈ U and radius 0 < δ < r
2
.
If we put Ki = f−1(Bi), then
m⋃
i=1
Ki ⊂ U. Let h ∈
m⋂
i=1
[Ki, B̃i] , where B̃i is a ball of
radius r
2
with the center in the point xi. If x ∈ U , then it is not difficult show that
d(h(x), f(x)) < r + a < ε
2
, i.e., h ∈ U(g). Thus f , together with the neighborhood
m⋂
i=1
[Ki, B̃i], lies inside of U(g). Hence, the set U(g) is an open set in the compact-open
topology. Theorem 1 is proved. �
The following theorem will be used later and deals with the theory of foliation.
Theorem 2. Let M be a smooth complete Riemannian manifold of dimension n with a
smooth foliation of dimension k, where 0 < k < n.
1) Each leaf with the induced Riemannian metric is a complete Riemannian manifold.
2) Let γm : (a, b) → Lm be a sequence of geodesics (determined by the induced Rie-
mannian metrics) on leaves Lm. If γm(s0) → p, γ̇m(s0) → v for m → ∞ for the some
s0 ∈ (a, b), then the sequence γm pointwise converges to the geodesic γ : (a, b) → L(p) of
the leaf L(p) that passes through the point p at s = s0 in the direction of the vector v.
Proof. 1) It is known that for a connected manifold M , the following conditions are
equivalent [8, page 167]:
a) M is a complete Riemannian manifold;
b) M is a complete metric space with distance which is defined by a Riemannian
metric. That is why we can prove the first part of Theorem 2 by using sequences.
Let Lα be some leaf of the foliation F , dα the distance on Lα determined by the
induced Riemannian metric. Let {pm} be a Cauchy sequence in Lα, i.e., for every ε > 0
there exists N such that dα(pm, pi) < ε for m, i ≥ N . Since d(pm, pi) ≤ dα(pm, pi),
where d is the distance on M , the sequence {pm} is a Cauchy sequence in M . Since M
is complete, this sequence converges in M .
198 A. YA. NARMANOV AND A. S. SHARIPOV
By the definition of a foliation, for each point p ∈ M there is a neighborhood U of the
point p and a local system of coordinates (x1, x2, . . . , xk, yk+1, yk+2, . . . , yn) on U such
that the set { ∂
∂x1 , ∂
∂x2 , . . . , ∂
∂xk } is basis for smooth sections TF |U (the restriction TF to
U). Such a neighborhood is called a foliated neighborhood of the point p [1, page 122].
Let U be a foliated neighborhood of the point p with local coordinates (x1, x2, . . . , xk,
yk+1, yk+2, . . . , yn), where p = lim
m→∞
pm. Then connected components of the intersection
U
⋂
Lβ for any leaf Lβ are given by the equations yk+1 = const, yk+2 = const, . . . , yn =
const .
Let ε > 0 be a small number such that the open ball Bε(p) = {q ∈ M : d(p, q) < ε} is
contained in U and m0 be an integer such that d(p, pl) < ε
4
for l ≥ m0 and dα(pl
′ , pl
′′ ) < ε
2
at l
′
, l
′′
≥ m0. If two points pl, pm belong to different connected components of Lα ∩ U
then any curve in Lα which goes from pl to pm leaves the ball Bε(p) and returns into
this ball before coming to the point pm. From here it follows that dα(pl, pm) ≥ ε, where
dα(pl, pm) is the distance between the points pl and pm on the leaf Lα. This contradiction
shows that all points pm for m ≥ m0 belong to the same connected component L0 of the
intersection Lα∩U . The component L0 is defined by the equations yk+1 = const, yk+2 =
const, . . . , yn = const and, therefore, the limit point p also belongs to L0. From here it
follows that p ∈ Lα and pm converges to p in Lα.
2) Let π : TM → TF be an orthogonal projection, V (M), V (F ), V (H) a set of smooth
sections of the bundles TM, TF, HF , respectively. We set ▽̃XY = π(▽XY ) for vector
fields X ∈ V (M), Y ∈ V (F ), where ▽ is a Levi-Civita connection determined by the
Riemannian metric g on M. It is known that ▽̃XY is a connection on TF , and its
restriction to each leaf Lα coincides with a connection on Lα, determined by the induced
Riemannian metric on Lα from M ([9, page 20], [10, page 59]). Therefore, the smooth
parametric curve µ : (a, b) → M lying on a leaf Lα of the foliation F is geodesic on Lα
(determined by the induced Riemannian metric) if and only if
(1) ▽̃µ̇µ̇ = 0.
If µ lies in the foliated neighborhood U , its equations have the form
{
xi = xi(s)
yα = const
,
where 1 ≤ i ≤ k, k + 1 ≤ α ≤ n. So, for ▽, we have
(2) ▽ ∂
∂xi
∂
∂xj
= Γl
i,j
∂
∂xl
+ Γα
i,j
∂
∂yα
,
hence,
(3) ▽̃ ∂
∂xi
= Γl
i,j
∂
∂xl
,
where 1 ≤ i, j, l ≤ k, k + 1 ≤ α ≤ n, Γβ
i,j are Christoffel symbols. From here using
properties of the operator ▽̃ it follows that equation (1) is equivalent to the following
system of differential equations of the 2nd order:
(4)
d2xi
ds
+ Γi
l,j
dxl
ds
dxj
ds
= 0.
If we put ui = dxi
ds
, then it is possible to write this system as
(5)
{
dxi
ds
= ui
dui
ds
= −Γi
l,ju
luj
.
Let us consider a geodesic γ : (a, b) → L(p), leaving a point p at s = s0 in the
direction of a vector v. This curve satisfies equation (1). Let K0 ⊂ (a, b) be a compact
ON THE GROUP OF FOLIATION ISOMETRIES 199
set containing s0 such that γ(K0) ⊂ V, {xi(s)}, {ui(s)} are the first k coordinates of the
point γ(s) and the velocity vector γ̇(s), respectively, where s ∈ K0, 1 ≤ i ≤ k. Then
these functions satisfy a system of differential equations (5) with the initial conditions:
xi(s0) = pi, ui(s0) = vi, where i = 1, 2, . . . , k, p = (p1, p2, . . . , pn), v = (v1, v2, . . . , vn).
Since γm(s0) → p, γ̇m(s0) → v for m → ∞ under the theorem of continuous dependence
of a solution of the differential equation on the initial data, the sequence γm converges
to γ uniformly on the compact K0 ⊂ (a, b). Further for every compact set K ⊂ (a, b)
containing K0, and covering γ(K) with foliated neighborhoods, we will obtain that γm
converges to γ uniformly on the compact set K. Theorem 2 is proved. �
Remark. With a view of simplification of designations, in terms of a kind (2), (3) which
have a summation with a repeating index, the symbol of summation is omitted.
The following theorem shows some property of the group Gr
F (M) with the F -compact-
open topology.
Theorem 3. Let M be a complete smooth n-dimensional Riemannian manifold with a
smooth k-dimensional foliation F , fm ∈ Gr
F (M), r ≥ 1. Suppose that for each leaf Lα
there exists a point oα ∈ Lα such that fm(oα) → f(oα), d(fm(oα)) → d(f(oα)), where
f ∈ Gr
F (M). Then the sequence fm converges to f in F -compact-open topology.
Proof. Let p ∈ Lα for some leaf Lα. Under the conditions of the theorem there exists a
point oα ∈ Lα such that fm(oα) → f(oα). Let γ : [0; l] → Lα be a geodesics determined
by the induced Riemannian metric on Lα such that γ(0) = oα, γ(l) = p. By virtue of
completeness of the leaf Lα, without loss of generality, we can assume that γ realizes the
distance d0 = dα(oα, p) on Lα, and let γ be parametrized by the length of the arc. We
put γ
′
(s) = f(γ(s)) for all s ∈ [0; l]. Since f is an isometry, γ
′
is a geodesic on f(Lα),
moreover its length is equal to the distance between the points f(oα) and f(p). If we
consider γm = fm(γ), than they are geodesics on fm(Lα). From the conditions of the
theorem, we have γm(0) → γ
′
(0), γ̇m(0) → γ̇
′
(0) for at m → ∞, where γ̇m(0), γ̇
′
(0) are
tangential vectors. Then from Theorem 2 we get that the sequence γm(s) converges to
γ
′
(s) for each s ∈ [0; l]. Therefore it follows that lim
m→∞
fm(p) = f(p).
Now we show that fm → f uniformly on each compact set lying in a leaf of the
foliation F . We denote by d(x, y) the distance between points x and y, determined by
the Riemannian metric. Let K be a compact set in a leaf L and ε > 0. Since K is
compact there exists a finite number of points p1, p2, . . . , pm in L such that each point
p ∈ K has distance less than ε from some pi .
For each point pi there is a number Ni such that d(fm(pi), f(pi)) < ε
3
for any m ≥ Ni.
Besides, for each point p ∈ K there exists pi such that dL(p, pi) < ε
3
where dL(p, pi) is
the distance between the points p and pi determined by the induced Riemannian metric
on L. Therefore it follows that
d(fm(p), f(p)) ≤ d(fm(p), fm(pi)) + d(fm(pi), f(pi)) + d(f(pi), f(p))
≤ dm(fm(p), fm(pi)) + d(fm(pi), f(pi)) + d0(f(pi), f(p))
≤
ε
3
+
ε
3
+
ε
3
= ε
for m > N = max
1≤i≤m
{Ni}, where dm− is the distance on the leaf fm(L), d0 is the distance
on the leaf f(L). From here it follows that fm → f in the F -compact-open topology.
Theorem 3 is proved. �
Acknowledgments. The authors thank the anonymous reviewer, due to whom the
formulation of Theorem 1 and its proof has undergone essential changes.
200 A. YA. NARMANOV AND A. S. SHARIPOV
References
1. I. Tamura, Topology of Foliations, Mir, Moscow, 1979. (Russian)
2. S. Kh. Aranson, Topology of vector fields, foliations with singularities, and homeomorphisms
with invariant foliations on closed surfaces, Trudy Mat. Inst. Steklov 193 (1992), 15–21.
(Russian)
3. A. Narmanov, D. Skorobogatov, Isometric mappings of foliations, Dokl. Akad. Nauk Republic
of Uzbekistan 4 (2004), 12–16. (Russian)
4. D. Skorobogatov, On isometries of codimension one foliations, Uzbek. Mat. Zh. 4 (2000),
55–62. (Russian)
5. S. Helgason, Differential Geometry and Symmetric Spaces, Mir, Moscow, 1964. (Russian)
6. V. A. Rokhlin, D. B. Fuks, An Initial Course in Topology: Geometric Chapters, Nauka, Moscow,
1977. (Russian)
7. D. Gromoll, W. Klingenberg, W. Meyer, Riemannian Geometry in the Large, Mir, Moscow,
1971. (Russian)
8. Sh. Kobayashi, K. Nomizu, Foundations of Differential Geometry, Vol. I, Nauka, Moscow, 1981.
(Russian)
9. Sh. Kobayashi, K. Nomizu, Foundations of Differential Geometry, Vol. II, Nauka, Moscow,
1981. (Russian)
10. Ph. Tondeur, Foliations on Riemannian Manifolds, Springer-Verlag, New–York, 1988.
Department of Geometry and Applied Mathematics, National University of Uzbekistan,
Tashkent, 100174, Uzbekistan
E-mail address: narmanov@yandex.ru
Department of Geometry and Applied Mathematics, National University of Uzbekistan,
Tashkent, 100174, Uzbekistan
E-mail address: asharipov@inbox.ru
Received 29/05/2007; Revised 30/07/2008
|
| id | nasplib_isofts_kiev_ua-123456789-5705 |
| institution | Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| issn | 1029-3531 |
| language | English |
| last_indexed | 2025-11-24T16:13:09Z |
| publishDate | 2009 |
| publisher | Інститут математики НАН України |
| record_format | dspace |
| spelling | Narmanov, A.Yu. Sharipov, A.S. 2010-02-02T13:32:51Z 2010-02-02T13:32:51Z 2009 On the Group of Foliation Isometries / A.Ya. Narmanov, A.S. Sharipov // Methods of Functional Analysis and Topology. — 2009. — Т. 15, № 2. — С. 195-200. — Бібліогр.: 10 назв. — англ. 1029-3531 https://nasplib.isofts.kiev.ua/handle/123456789/5705 The purpose of our paper is to introduce some topology on the group GrF(M) of all Cr-isometries of foliated manifold (M, F), which depends on a foliation F and coincides with compact-open topology when F is an n-dimensional foliation. If the codimension of F is equal to n, convergence in our topology coincides with pointwise convergence, where n = dimM. It is proved that the group GrF(M) is a topological group with compact-open topology, where r ≥ 0. In addition it is showed some properties of F-compact-open topology. en Інститут математики НАН України On the Group of Foliation Isometries Article published earlier |
| spellingShingle | On the Group of Foliation Isometries Narmanov, A.Yu. Sharipov, A.S. |
| title | On the Group of Foliation Isometries |
| title_full | On the Group of Foliation Isometries |
| title_fullStr | On the Group of Foliation Isometries |
| title_full_unstemmed | On the Group of Foliation Isometries |
| title_short | On the Group of Foliation Isometries |
| title_sort | on the group of foliation isometries |
| url | https://nasplib.isofts.kiev.ua/handle/123456789/5705 |
| work_keys_str_mv | AT narmanovayu onthegroupoffoliationisometries AT sharipovas onthegroupoffoliationisometries |