Foliations of codimension one and Milnor's conjecture
We prove that a fundamental group of leaves of a codimension one C²- foliation with nonnegative Ricci curvature on a closed Riemannian manifold is finitely generated and almost Abelian, i.e., it contains finitely generated Abelian subgroup of finite index. In particular, we confirm the Milnor conjec...
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irk-123456789-1458632019-02-02T01:23:21Z Foliations of codimension one and Milnor's conjecture Bolotov, D.V. We prove that a fundamental group of leaves of a codimension one C²- foliation with nonnegative Ricci curvature on a closed Riemannian manifold is finitely generated and almost Abelian, i.e., it contains finitely generated Abelian subgroup of finite index. In particular, we confirm the Milnor conjecture for manifolds which are leaves of a codimension one foliation with nonnegative Ricci curvature on a closed Riemannian manifold. Ми доводимо, що фундаментальна група шарiв C²-шарування ковимiрностi один невiд'ємно кривини Рiччi замкнутого рiманова многовиду є скiнченно породженою та майже абелевою, тобто мiстить скiнченно породжену абелеву пiдгрупу скiнченного iндексу. Зокрема, ми пiдтверджуємо гiпотезу Мiлнора щодо многовидiв, якi є шарами шарування ковимiрностi один невiд'ємно кривини Рiччi замкнутого рiманова многовиду. 2018 Article Foliations of codimension one and Milnor's conjecture / D.V. Bolotov // Журнал математической физики, анализа, геометрии. — 2018. — Т. 14, № 2. — С. 119-131. — Бібліогр.: 15 назв. — англ. 1812-9471 DOI: https://doi.org/10.15407/mag14.02.119 Mathematics Subject Classification 2010: 53A05 http://dspace.nbuv.gov.ua/handle/123456789/145863 en Журнал математической физики, анализа, геометрии Фізико-технічний інститут низьких температур ім. Б.І. Вєркіна НАН України |
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We prove that a fundamental group of leaves of a codimension one C²- foliation with nonnegative Ricci curvature on a closed Riemannian manifold is finitely generated and almost Abelian, i.e., it contains finitely generated Abelian subgroup of finite index. In particular, we confirm the Milnor conjecture for manifolds which are leaves of a codimension one foliation with nonnegative Ricci curvature on a closed Riemannian manifold. |
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Bolotov, D.V. |
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Bolotov, D.V. Foliations of codimension one and Milnor's conjecture Журнал математической физики, анализа, геометрии |
| author_facet |
Bolotov, D.V. |
| author_sort |
Bolotov, D.V. |
| title |
Foliations of codimension one and Milnor's conjecture |
| title_short |
Foliations of codimension one and Milnor's conjecture |
| title_full |
Foliations of codimension one and Milnor's conjecture |
| title_fullStr |
Foliations of codimension one and Milnor's conjecture |
| title_full_unstemmed |
Foliations of codimension one and Milnor's conjecture |
| title_sort |
foliations of codimension one and milnor's conjecture |
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Фізико-технічний інститут низьких температур ім. Б.І. Вєркіна НАН України |
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2018 |
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http://dspace.nbuv.gov.ua/handle/123456789/145863 |
| citation_txt |
Foliations of codimension one and Milnor's conjecture / D.V. Bolotov // Журнал математической физики, анализа, геометрии. — 2018. — Т. 14, № 2. — С. 119-131. — Бібліогр.: 15 назв. — англ. |
| series |
Журнал математической физики, анализа, геометрии |
| work_keys_str_mv |
AT bolotovdv foliationsofcodimensiononeandmilnorsconjecture |
| first_indexed |
2025-07-10T22:42:17Z |
| last_indexed |
2025-07-10T22:42:17Z |
| _version_ |
1837301606685605888 |
| fulltext |
Journal of Mathematical Physics, Analysis, Geometry
2018, Vol. 14, No. 2, pp. 119–131
doi: https://doi.org/10.15407/mag14.02.119
Foliations of Codimension One and the
Milnor Conjecture
Dmitry V. Bolotov
We prove that a fundamental group of leaves of a codimension one C2-
foliation with nonnegative Ricci curvature on a closed Riemannian manifold
is finitely generated and almost Abelian, i.e., it contains finitely generated
Abelian subgroup of finite index. In particular, we confirm the Milnor
conjecture for manifolds which are leaves of a codimension one foliation
with nonnegative Ricci curvature on a closed Riemannian manifold.
Key words: codimension one foliation, fundamental group, holonomy,
Ricci curvature.
Mathematical Subject Classification 2010: 53A05.
1. Introduction
In 1963, Bishop proved the following theorem [2].
Bishop’s Theorem. A complete manifold with nonnegative Ricci curvature
has a polynomial volume growth of balls.
In 1968, Milnor showed that a fundamental group of a complete manifold with
nonnegative Ricci curvature also has a polynomial growth (in the words metric)
and stated the following conjecture [8].
Milnor’s Conjecture. The fundamental group of any complete Riemannian
manifold with nonnegative Ricci curvature is finitely generated.
In this paper, we confirm the Milnor conjecture for leaves of codimension one
foliations with nonnegative Ricci curvature and prove the following theorem.
Main Theorem. Let L be a leaf of a codimension one foliation F with
nonnegative Ricci curvature on a closed manifold M . Then π1(L) is a finitely
generated almost abelian group. In particular, it satisfies the Milnor conjecture.
Remark 1.1. The Main theorem obviously holds for compact leaves (see The-
orem 1.5 bellow). For noncompact leaves, the structure of foliations with non-
negative Ricci curvature is successfully used.
c© Dmitry V. Bolotov, 2018
https://doi.org/10.15407/mag14.02.119
120 Dmitry V. Bolotov
Remark 1.2. The Main theorem can not be strengthened up to Abelian
groups. Indeed, consider the standard Reeb foliation FR = {Lα} on the round
sphere S3 (see [14]). It is well known that the leaves Lα in the induced metric
have a nonnegative curvature. The Riemannian product S3 with a closed non-
negative Ricci manifold Mn having non-Abelian fundamental group gives us a
foliation G = {Lα ×Mn} of codimension one with nonnegative Ricci curvature
on S3 ×Mn with leaves having non-Abelian fundamental group.
In the proof, we essentially use the properties of almost without holonomy
foliations and geometrical properties of the complete Riemannian manifolds with
nonnegative Ricci curvature well studied by many famous mathematicians. Thus,
in 1972, J. Cheeger and D. Gromoll generalized Toponogov’s splitting theorem
to the case of complete Riemannian manifolds with nonnegative Ricci curvature
and obtained the following results.
Theorem 1.3 (Splitting Theorem [5]). Assume that M is a complete Rie-
mannian manifold with Ricci curvature Ric(M) ≥ 0 which has a stright line, i.e.,
a geodesic γ such that d(γ(u), γ(v)) = |u−v| for all u, v ∈ R. Then M is isomet-
ric to a Riemannian product space N × R, where N is a complete Riemannian
manifold with Ric(N) ≥ 0.
Theorem 1.4 ([5]). Let Mn be a complete manifold with nonnegative Ricci
curvature. Then:
1. Mn has at most two ends;
2. Mn is isometric to the Riemannian product N × Ek of the manifold N and
Euclidian factor Ek, where N does not contain straight lines.
3. If Mn is closed, then its universal covering M̃n is isometric to the Riemannian
product P ×Ek, where P is compact and simply connected. Furthermore, the
following extension holds:
1→ E → π1(Mn)→ Γ→ 1, (1.1)
where E is a finite group and Γ is a crystallographic group.
Part 3 of Theorem 1.4 was generalized in 2000 by B. Wilking in the following
theorem.
Theorem 1.5 ([15]). Let Mn be a complete manifold with nonnegative Ricci
curvature and q : N × El → Mn be a regular isometric covering, where N has
a compact isometry group (it holds in particular when N is closed). Then there
exists a finitely sheeted covering N × T p × El−p →Mn. Moreover, this covering
can be isometric for some deformed Riemanniam metric on Mn. If Mn is closed,
then (1.1) is equivalent to the existence of the extension
0→ Zp → π1(Mn)→ F → 1, (1.2)
where F is a finite group.
Foliations of Codimension One and the Milnor Conjecture 121
2. Foliations
Let us recall the notion of a foliation defined on an n-dimensional manifold
M . We say that a family F = {Fα} of path-wises connected subsets (leaves) of
M defines a foliation of dimension p (or codimension q, where p+ q = n) on M if
• F is a partition of M , i.e., M =
∐
α
Fα.
• There is a foliated atlas U = {(Uλ, ϕλ)}λ∈Λ on M . This means that each
connective component of a leaf in the foliated chart with the coordinates
(x1, . . . , xp, y1, . . . , yq) has the form of a plane
y1 = const, . . . , yq = const,
and the transition maps
gij = ϕi ◦ ϕ−1
j : ϕj(Ui ∩ Uj)→ ϕi(Ui ∩ Uj)
have the form
gij(x, y) = (ĝij(x, y), gij(y)), (2.1)
where x ∈ Rp, y ∈ Rq.
The atlas U = {(Uλ, ϕλ)}λ∈Λ is supposed to be at least C2-smooth and good.
The latter means that:
1) U is locally finite;
2) Uλ is relatively compact in M, and ϕλ(Uλ) = (−1, 1)n ⊂ Rn;
3) Ui ∪ Uj ⊂ Wij , where (Wij , ψij) is a foliated chart not necessarily belonging
to U .
Let π : (−1, 1)n → (−1, 1)q be a natural projection to the last q coordinates.
The preimage Pλ := ϕ−1
λ (π−1(x)) is called a local leaf. Denote the space of
local leaves by Qλ. Clearly, Qλ ' (−1, 1)q, and
Uλ =
⋃
x∈(−1,1)q
ϕ−1
λ (π−1(x)).
A foliation F is said to be oriented if the tangent to F p-dimensional dis-
tribution TFM ⊂ TM is oriented, and F is said to be transversely oriented if
some transversal to F distribution of dimension q = n − p is oriented. If the
manifold M is supposed to be Riemannian, then the transverse orientability of
F is equivalent to the transverse orientability of orthogonal distribution TF
⊥
M .
3. Holonomy
We recall the notion of holonomy.
Let l : [0, 1]→ L be a closed path in the leaf L ∈ F . It was shown in [14] that
there exists a chain of foliated charts C = {U0, . . . , Un−1, Un = U0}, which cover
l([0, 1]) such that:
122 Dmitry V. Bolotov
a) there exists a division of the segment [0, 1] : 0 = t0 < t1 < · · · < tn = 1 such
that l([ti, ti+1]) ⊂ Ui, i = 0, . . . , n− 1;
b) if the intersection Pi ∩ Ui+1 6= ∅, then it is connective, which means that the
local leaf Pi+1 is correctly defined.
It can be shown that a set of points z ∈ U0, which correctly define the
chain C from the initial condition z ∈ P0(z) is open in U0. Thus, there exists
some neighborhood O of the P0 consisting of local leaves for which the local
diffeomorphism Γl : V0 → (−1, 1)q of some neighborhood of zero V0 ⊂ (−1, 1)q to
(−1, 1)q is well defined as follows:
Γl(π ◦ ϕ0(P0(z)) = π ◦ ϕ0(Pn(z)).
In [14], it was shown that the local diffeomorphisms {Γl : V0 → (−1, 1)q}
define the homomorphism
Ψ : π1(L)→ Gq
of the fundamental group π1(L) to the group of diffeomorphism germs Gq in the
origin 0 ∈ Rq. The homomorphism Ψ is defined up to inner automorphisms and
it is called a holonomy of L. Its image is called a group of holonomy of L and
denoted by H(L).
Notice that if a foliation of codimension one is transversely oriented, then the
one-sided holonomy
Ψ+ : π1(L)→ G+
1
of the leaf L is also well defined, where G+
1 denotes the group of germs of one-sided
diffeomorphisms at 0 with a domain on the half-intervals [0, ε).
Recall the following important results obtained on the holonomy of leaves.
Theorem 3.1 ([12]). Let L be a leaf of a C2-foliation of codimension one.
If H(L) has a polynomial growth, then H(L) is a torsion free Abelian group.
Nishimori proved the next theorem which describes the behavior of a codimen-
sion one foliation in the neighborhood of a compact leaf with Abelian holonomy.
Theorem 3.2 ([9]). Let F be a transversely oriented Cr-foliation of codi-
mension one on the oriented n -dimensional manifold M and F0 be a compact
leaf of F . Suppose that 2 ≤ r ≤ ∞. Let T be a tubular neighborhood of F0 and
U+ be a union of F0 and a connected component T\F0. Suppose that H(F0) is
an Abelian group. Then only one of the three cases holds.
1. For any neighborhood V of F0, the restricted foliation F|V ⋂
U+
has a compact
leaf which is not F0.
2. There is a neighborhood V of F0 such that all leaves F|V ⋂
U+
except F0 are
dense in V
⋂
U+. In this case, H(F0) is a free Abelian group of rank ≥ 2.
3. There is a neighborhood V of F0 and a connected oriented submanifold N of
codimension one in F0 with the following properties. Denote by F∗ a compact
Foliations of Codimension One and the Milnor Conjecture 123
manifold with boundary obtained by attaching two copies N1 and N2 of N to
F0\N satisfying ∂F∗ = N1
⋃
N2. Let f : [0, ε) → [0, δ) be a contracting Cr-
diffeomorphism such that f(0) = 0. Denote by Xf a manifold obtained from
F∗ × [0, ε) by identifying (x, t) ∈ N1 × [0, ε) and (x, f(t)) ∈ N2 × [0, δ). After
factorization, we obtain the foliation Ff on Xf . It is claimed that for some
f as above, there is a Cr-diffeomorphism h : V
⋂
U+ → Xf which maps each
leaf of F|V ⋂
U+
onto some leaf of Ff . The foliation F|V ⋂
U+
uniquely defines
the homology class [N ] ∈ Hn−2(F0,Z), and the germ at zero of the map f is
unique up to conjugation. In this case, H(F0) is an infinite cyclic group.
A foliation is said to be a foliation without holonomy if the holonomy of each
leaf is trivial, and it is said to be a foliation almost without holonomy if the
holonomy of noncompact leaves is trivial. For example, the Reeb foliation FR is
a foliation almost without holonomy on S3 since all leaves of FR, except a single
compact leaf homeomorphic to torus, are homeomorphic to R2 and thus have a
trivial fundamental group.
Let us call by block a compact saturated subset B of codimension one foliated
n-dimensional manifold which is an n-dimensional submanifold with a boundary.
Recall that a saturated set of the foliation F on a manifold M is called a subset
of M which is a union of leaves of F . Clearly that ∂B is a finite union of compact
leaves.
The following theorem is a reformulation of the results of Novikov [10] and
Imanishi [7] obtained for foliations without holonomy and for foliations almost
without holonomy, respectively.
Theorem 3.3. Let L be a noncompact leaf of a codimension one foliation F
almost without holonomy on a closed n-dimensional manifold M . Then one of
the following holds:
a) F is a foliation without holonomy whose all leaves are diffeomorphic to the
typical leaf L and dense in M . We have the group extension
1→ π1(L)→ π1(M)→ Zk → 0, (3.1)
where k > 0 and k = 1 iff the foliation F is a locally trivial fibration over the
circle.
The universal covering M̃ has the form
M̃ ∼= L̃× R.
b) L belongs to some block B whose all leaves in the interior B̊ are diffeomorphic
to the typical leaf L and are either dense in B̊ (B is called a dense block in
this case) or proper in B̊ (B is called a proper block in this case). We have
the group extension
1→ π1(L)→ π1(B)→ Zk → 0, (3.2)
where k > 0 and k = 1 iff B is proper and the foliation in B̊ is a locally trivial
fibration over the circle.
124 Dmitry V. Bolotov
The universal covering of B̊ has the form
˜̊
B ∼= L̃× R. (3.3)
Let us call the block from b) an elementary block.
4. Growth of leaves
A minimal set of the foliation F is a closed saturated set which has no other
closed saturated sets.
The following Plant’s theorem describes minimal sets of foliations of codimen-
sion one with leaves of subexponential growth. The growth means the volume
growth of balls Bx(R) ⊂ Lx ∈ F as a function of the radius R.
Theorem 4.1 ([11]). Assume that a C2-foliation of codimension one on a
compact manifold has the leaves of subexponential growth. Then each minimal
set of the foliation is either the whole manifold or a compact leaf.
We say that F is a foliation with nonnegative Ricci curvature on a Riemannian
manifold if each leaf of F has a nonnegative Ricci curvature in the induced metric.
We obtain the following corollary.
Corollary 4.2. One of the following holds:
1. All leaves of a C2-foliation of codimension one with nonnegative Ricci curva-
ture are dense.
2. The closure of each leaf contains a compact leaf.
A leaf L of a transversely oriented foliation F of codimension one is called
resilient if there is a transversal arc [x, y), x ∈ L, and a loop σ such that Γσ :
[x, y)→ [x, y) is the contraction to x, and L ∩ (x, y) 6= ∅.
It turns out that the theorem below holds.
Theorem 4.3 ([6]). Let M be a compact manifold and F be a C2-foliation of
codimension one on M . Then a resilient leaf of F must have exponential growth.
Corollary 4.4. A C2-foliation of codimension one with nonnegative Ricci
curvature on a compact manifold has no resilient leaves.
5. Mappings into foliations
Let us recall the definition of an exponential map along the leaf of a foliation
F on the Riemannian manifold M .
Each vector a tangent to a leaf Lx at the point x ∈ M is mapped to the
end of the geodesic of Lx which has the length |a| and is directed to a at the
initial point x. Since the foliation is supposed to be smooth, the constructed
exponential map expF : TFM → M must also be smooth. By analogy, exp⊥
denotes the orthogonal exponential map where each orthogonal vector p at x is
Foliations of Codimension One and the Milnor Conjecture 125
mapped to the end of the orthogonal to F trajectory of the length |p| which is
directed to p at the initial point x. Let us consider the composition of continuous
maps
F : I ×Dn−1(R)
i→ V (J,R)
expF
→ M, (5.1)
where I = [0, 1], Dn−1(R) is a Euclidean ball of radius R and i : I ×Dn−1(R)→
V (J,R) ⊂ TFM is a homeomorphism on the set V (J,R) = {(x, vx) : x ∈ J =
i(I×0), vx ∈ TFx M, |vx| ≤ R}. The interval J = i(I×0) is embedded and orthog-
onal to F . For each x ∈ J̊ , call the set F (I,Dn−1(R)) a V (J,R)-neighborhood
of x.
Remark 5.1. Since the length of J and the real number R can be taken
arbitrarily small, for any point x there exists J containing x and a V (J, ε)-
neighborhood W of x which is inside of a foliated chart (Uα, φα) of the foliation.
The restriction of φα on U ′α ⊂ W gives us a foliated chart (U ′α, φ
′
α) with diame-
ters of local leaves less than ε, where U ′α is a neighborhood of x homeomorphic
to (−1, 1)n.
Proposition 5.2. Let F : I ×Dn−1(R)→M as in (5.1). Then there exists
δ > 0 such that for each 0 ≤ t < δ the balls Bt(R) := expF ◦i(t×Dn−1(R)) belong
to an arbitrary ε-collar of the ball B0(R+ ε) ⊃ B0(R). By the ε-collar of B0(R+
ε), we understand the set ∆ = exp⊥(a), |a| < ε, where a are normal to B0(R +
ε) vectors directed toward a half-space defined by J = F (I × 0), and ε is small
enough for the deformation retraction pr : ∆ → B0(R + ε) to be well defined.
Here pr maps the point x ∈ ∆ to the initial point of the orthogonal trajectory.
Proof. The proof follows from the fact that I ×Dn−1(R) is compact for each
R ∈ R+ and F is uniformly continuous.
Recall also the definition by S. Adams and G. Stuck.
Definition 5.3 ([1]). Let F be a foliation of a Riemannian manifold M . Let
X be a connected locally compact Hausdorff space. Define CF (X,M) to be the
space of continuous maps of X into the leaves of F . Let us consider CF (X,M)
as a subspace of the space C(X,M) of continuous maps of X into M , where
C(X,M) is endowed with the topology of uniform convergence on compact sets.
The following important result was obtained.
Theorem 5.4 ([1]). Let M , F and X be as in Definition (5.3). Then
CF (X,M) is a closed subspace of C(X,M).
6. Foliations with nonnegative Ricci curvature
Proposition 6.1. Let F be a transversely oriented C2-foliation of codimen-
sion one with nonnegative Ricci curvature on a closed oriented manifold M . Then
F is a foliation almost without holonomy.
126 Dmitry V. Bolotov
Proof. As noted at the beginning of the article, Milnor proved that a fun-
damental group of a compact manifold with nonnegative Ricci curvature has a
polynomial growth in the words metric. Therefore, by Theorem 3.1, each compact
leaf has an Abelian holonomy and thus Theorem 3.2 can be applied. Suppose that
there exists a noncompact leaf L with nontrivial holonomy. Since F is supposed
to be transversely oriented, the holonomy mast be infinite. Let γ be a closed
path in L which represents a nontrivial holonomy element of π1(L). Consider
two cases of holonomy mappings.
Case 1. The one-sided holonomy Ψ+([γ]) is nontrivial and the holonomy
map Γγ : [0, ε)→ [0, ε′) does not have any fixed points in [0, ε) for some ε.
In this case, Ψ+(±[γ]) is represented by a contracting map. If L is locally
dense, then L must be a resilient leaf, which is impossible by Corollary 4.4.
Otherwise, another noncompact leaf P winds around L and, by Corollary 4.2,
there exists a compact leaf K ⊂ L
⋂
P . From part 3 of Theorem 3.2, it simply
follows that the leaf P must have infinitely many ends, which contradicts to
Theorem 1.4.
Case 2. The half-interval [0, ε) has a sequence of fixed points {Fi} of the
holonomy map Γγ : [0, ε)→ [0, ε′) which converges to zero.
Since {Fi}
⋂
[0, Fk] is closed in [0, ε), where Fk ∈ {Fi}
⋂
[0, ε), then either the
holonomy of L must be trivial or we have to find a half-interval [a, δ) ⊂ [0, ε) on
which Γγ′ is a contracting map. The leaf L′ corresponding to the point a ∈ [0, ε)
has a contracting holonomy on γ′ ⊂ L′, where γ′ is a closed path corresponding to
the fixed point a of the map Γγ . Since the set of compact leaves is closed (see [14])
and M is the normal topological space, we can choose ε small enough for the leaf
L′ to be noncompact. Thus, we have arrived to Case 1 and the proposition is
proven.
Corollary 6.2. The structure of transversely oriented foliations of codimen-
sion one with nonnegative Ricci curvature on compact oriented manifolds is de-
scribed by Theorem 3.3.
Proposition 6.3. Let F be a transversely oriented C2-foliation of codimen-
sion one which is almost without holonomy (in particular, a foliation with nonneg-
ative Ricci curvature) on the oriented Riemannian manifold M . Let {[xi, yi]} ⊂
Li ∈ F be a sequence of the shortest (in Li) geodesic segments of the length li →
∞ and {zi ∈ [xi, yi]} be a sequence of points converging to some point z ∈ L,
where L is a noncompact leaf of F . Suppose that the lengths of the segments
{[xi, zi]} and {[zi, yi]} approach to ∞. Then there is a subsequence of {[xi, yi]}
converging to a straight line l ∈ L, which passes through z.
Proof. Starting from some i > i0, we can replace the sequence of the segments
[xi, yi] by the sequence of the segments [xli, y
l
i] ⊂ [xi, yi] of the length l such that
zi is the midpoint of [xli, y
l
i]. We have ρLi(xi, yi) ≥ ρM (xi, yi), where ρLi denotes
a Riemannian metric on the leaf Li induced by the Riemannian metric ρM on
M . By Arzela–Ascoli theorem (see [3, Theorem 2.5.14]) and Theorem 5.4, we
Foliations of Codimension One and the Milnor Conjecture 127
can see that the sequence {[xli, yli]} converges uniformly to the rectifiable curve
rl ⊂ L passing though z with the endpoints xl and yl in L. Show that rl is the
shortest geodesic segment [xl, yl]. Indeed, if it is not, then consider the loop rl ∪
[xl, yl] and a smooth transversal to F embedding of the square [xl, yl] × I ⊂ M
such that [xlt, y
l
t] := [xl, yl] × t ⊂ Lt ∈ F and [xl0, y
l
0] = [xl, yl]. The length lt
of [xlt, y
l
t] continuously depends on t (recall that F is C2-smooth and so is the
induced foliation on [xl, yl]× I) and there exists ε and δ such that for 0 ≤ t < δ
the inequality lt < l− 2ε holds. Since xli → xl, we can assume that for i > i0, the
points xlti and xli belong to the same local leaf of a foliated chart (Uα, φα) with
diameters of local leaves less than ε (see Remark 5.1). Therefore, ρLi(x
l
i, x
l
ti) <
ε. We also assume, without loss of generality, that if i > i0, then ti < δ and
limi→∞ ti = 0. But yli → yl, and starting from some i ≥ i1 > i0 the sequence
{yli} belongs to a foliated chart (Uβ, φβ) which contains yl and the diameters of
local leaves of (Uβ, φβ) are less than ε. If ylti belong to the same local leaf as yli,
then this contradicts our assumption that the segments [xli, y
l
i] are the shortest
segments:
l = ρLi(x
l
i, y
l
i) ≤ ρLi(x
l
i, x
l
ti) + ρLi(x
l
ti , y
l
ti) + ρLi(y
l
i, y
l
ti) = 2ε+ lti < l.
It means that for i > i1, the points yli and ylti belong to different local leaves of
(Uβ, φβ), and the loop rl ∪ [xl, yl] represents a nontrivial holonomy of L. This
is impossible since F is supposed to be a foliation almost without holonomy.
This implies that rl = [xl, yl]. So we have a sequence of the shortest geodesic
segments {[xlj , ylj ]}, which pass through the midpoint z ∈ L, that are limits of
sequences of the shortest segments {[xlji , y
lj
i ]} ⊂ Li, where lj →∞. The sequence
{[xlj , ylj ]} contains a subsequence converging to a straight line in L (see, for
example, [3]).
7. The main result
Definition 7.1. Let L be a noncompact leaf of the elementary block B and
K ∈ ∂B be a compact leaf. We call the curve γ(t) ⊂ L, t ∈ [0,∞) outgoing (to
K) if for each ε > 0 there is t0 such that γ(t), t ∈ [t0,∞) is inside of the ε-collar
of K. It is clear that γ ∩K 6= ∅.
Definition 7.2. Let L be a noncompact leaf of the elementary block B and
K ∈ ∂B be a compact leaf. We say that an element [α] ∈ π1(L, x0) is not
peripheral along the outgoing curve γ ⊂ L (to K) if there is some ε-collar UεK of
K and there is no loop αt with base point xt = γ(t) such that αt is free homotopic
to α and αt belongs to UεK. Otherwise, we say that α is peripheral along γ.
Remark 7.3. The term “peripheral” was used in [4]. But our definition of
peripheral elements of π1(L) along the outgoing curve γ is in some sense a foliated
analogue of elements having geodesic loops to infinity property along a ray γ used
in [13].
Let us call a dimension k of Ek in the splitting theorem, a splitting dimension
of the complete manifold M with nonnegative Ricci curvature.
128 Dmitry V. Bolotov
Now we turn to the proof of Main theorem.
Proof. Main theorem holds for compact leaves (see Remark 1.1). Suppose L
is noncompact. Suppose also that F is transversely oriented and M is oriented.
We have two cases.
Case 1. F does not contain compact leaves.
In this case, by Theorem 3.3 and Corollary 6.2, all leaves are dense and
diffeomorphic. The result follows from the main theorem of [1], which claims
that almost all leaves have a structure of a Riemannian product S × En, where
S is compact.
Case 2. F contains compact leaves and L is a typical leaf of the elementary
block B.
Recall that we have the monomorphism π1(L) → π1(B) (see (3.2)). Fix a
typical leaf L and consider the connective component of its universal covering
L̃ ⊂ ˜̊
B . Recall that the universal covering of B̊ has the form (3.3). According to
Theorem 1.4,
L̃ ∼= N × Ek, k ≥ 0, (7.1)
where N does not contain straight lines. Let us show that each leaf L̃x̃ passing
though x̃ of the pull back foliation F̃ of
˜̊
B , which is a preimage of Lx ∈ L ∈
B̊, splits with the same splitting dimension k as L̃. Indeed, let x̃ ∈ L̃x̃ be an
arbitrary point. We can find a sequence x̃i ∈ L̃x̃i such that limi→∞ x̃i = x̃, where
L̃x̃i are the preimages of L passing through x̃i ∈ ˜̊
B with respect to the covering
p :
˜̊
B → B̊. Notice that actually nontrivial is only the case of dense B. By (7.1),
we can find a sequence of k orthogonal straight lines passing through x̃i, which
converges to k orthogonal straight lines passing through x̃. This follows from
some modification of Proposition 6.3. Thus, we obtain that L̃x̃ ∼= Nx×El, l ≥ k.
By exchanging L and Lx, we obtain l = k. It is clear that the splitting directions
Rk form a subdistribution in TFM called a splitting distribution. Respectively, a
distribution orthogonal to a splitting distribution in TFM is called a horizontal
distribution.
Suppose that N is not compact in the splitting L̃ ∼= N ×Ek. Then there is a
horizontal ray γ̃ ∈ N×~0. Show that p(γ̃) =: γ is an outgoing curve. Suppose also
that γ is not an outgoing curve. Then we can find a sequence of points γ(ti), ti →
∞, which converges to the point x ∈ B̊. It means that we can find a sequence of
points x̃i := γ̃i(ti) which converges to the point x̃ such that p(x̃) = x. Here γ̃i
are the corresponding preimages of γ with respect to the covering map p :
˜̊
B →
B̊. Since ti → ∞, we can find a sequence of the shortest horizontal segments
[ai, bi] ∈ γ̃i passing through x̃i such that the lengths [ai, x̃i] and [bi, x̃i] tend to∞.
Thus, the sequence of the shortest horizontal segments contains a subsequence
converging to a horizontal straight line passing through x̃ by Proposition 6.3,
Foliations of Codimension One and the Milnor Conjecture 129
which is a contradiction as Nx̃ in the splitting L̃x̃ ∼= Nx̃ × Ek does not contain
straight lines. We can conclude that γ is outgoing.
Consider two possibilities.
a) Each element [α] ∈ π1(L) is peripheral along an outgoing to K ∈ ∂B curve γ.
Suppose γ(t) ⊂ UεK for t ∈ [t0,∞). In this case, in L, we can change each
loop α with the base point to the homotopic loop α′ ⊂ UεK with the same base
point. Since K is a deformation retract of UεK, we have that i∗(π1(L)) is isomor-
phic to a subgroup of i∗(π1(K)), where i∗ denotes in both cases homomorphisms
induced by the inclusions i : L → B and i : K → B. But a subgroup and an
image of a finitely generated almost Abelian group is finitely generated almost
Abelian, thus the result follows from (1.2) and Theorem 1.4.
b) There is a nonperipheral element [α] ∈ π1(L) along γ.
From the definition, it follows that there is a countable family of minimal
length geodesic loops {αi} with the base points xi = γ(ti), ti →∞, which is free
homotopic to the loop α, and the sequence {zi ∈ αi} such that zi ∈ B \ UεK for
some ε-collar UεK of K ∈ ∂B. Let us suppose that the sequence {zi} converges
to z ∈ B \UεK. Since γ is an outgoing curve, we can choose a subsequence {xj}
in {xi} converging to x ∈ K to get ρ(xj , zj) → ∞. Indeed, by Proposition 5.2,
for an arbitrary fixed R > diamK, there exists δ > 0 such that for each 0 ≤ t <
δ, the balls Bt(R) := expF ◦i(t×Dn−1(R)) belong to the ω-collar UωK (ω < ε),
where i(0, 0) = x. Since xj → x, there exists i0 such that xj ∈ Bt(σ) ⊂ Bt(R) for
j > i0 and a small enough σ > 0 (see Remark 5.1) such that ρLi(xj , zj) ≥ R −
σ. As R is taken arbitrarily, we get that ρ(xj , zj) → ∞. It means that we have
a sequence of the shortest geodesic segments [x̃j , ỹj ] ⊂ L̃ ⊂ ˜̊
B of the length lj →
∞ with the ends x̃j ∈ γ̃ ⊂ L̃ and ỹj ∈ γ̃α := Θ([α])(γ̃) ⊂ L̃ such that p(x̃j) =
p(ỹj) = xj , where Θ defines an isometric action of π1(B) on
˜̊
B .
Let z̃j ∈ [x̃j , ỹj ] be such that p(z̃j) = zj . Since ρ(xj , zj) → ∞, we
have ρ(x̃j , z̃j) → ∞ and ρ(ỹj , z̃j) → ∞. Choose such gj ∈ π1(B) that
limj→∞Θ(gj)z̃j → z̃ ∈ ˜̊
B and p(z̃) = z. Pay attention to that [x̃j , ỹj ] do not
need to be horizontal since γ̃α does not need to belong to N×~0. However, [x̃j , ỹj ]
belong to N × ~a for some ~a ∈ Rk. Substitute the ray γ̃ with the ray γ̃′ := γ̃ + ~a
and the points x̃j with x̃′j := x̃j +~a. Clearly that the shortest geodesic segments
[x̃′j , ỹj ] are the projections of [x̃j , ỹj ] along ~a. Let z′j ∈ [x̃′j , ỹj ] be the images of
zj with respect to the projections. We can see that [xj , x
′
j , zj ] is a right triangle.
Since ρ(xj , x
′
j) = |a|, we have that limj→∞ ρ(x′j , yj) = ∞ and ρ(zj , z
′
j) < |a|.
Proposition 6.3 immediately implies that a sequence of the shortest horizontal
geodesic segments Θ(gj)[x̃
′
j , ỹj ] has a subsequence converging to the horizontal
straight line, which is impossible.
If N in the splitting L̃ ∼= N × Rk is compact, then the result follows from
Theorem 1.5.
If M is not oriented and (or) F is not transversely oriented, we can pass to
the finitely sheeted oriented isometric covering p : M → M such that the pull-
back foliation F is transversely oriented for which the result has been proven.
130 Dmitry V. Bolotov
Each leaf L ∈ F is finitely covered by some leaf L ∈ F . And the expected result
follows from the fact that p∗ : π1(L) → π1(L) is a monomorphism and π1(L) is
isomorphic to some subgroup of finite index in π1(L).
Corollary 7.4. The leaves of a foliation of codimension one with nonnegative
Ricci curvature on a closed Riemannian manifold satisfy the Milnor conjecture.
Acknowledgment. The author is grateful to the referee for useful com-
ments.
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Received May 30, 2017, revised July 31, 2017.
Dmitry V. Bolotov,
B. Verkin Institute for Low Temperature Physics and Engineering of the National
Academy of Sciences of Ukraine, 47 Nauky Ave., Kharkiv, 61103, Ukraine,
E-mail: bolotov@ilt.kharkov.ua
Шарування ковимiрностi один та гiпотеза Мiлнора
Dmitry V. Bolotov
Ми доводимо, що фундаментальна група шарiв C2-шарування кови-
мiрностi один невiд’ємної кривини Рiччi замкнутого рiманова многови-
ду є скiнченно породженою та майже абелевою, тобто мiстить скiнченно
породжену абелеву пiдгрупу скiнченного iндексу. Зокрема, ми пiдтвер-
джуємо гiпотезу Мiлнора щодо многовидiв, якi є шарами шарування
ковимiрностi один невiд’ємної кривини Рiччi замкнутого рiманова мно-
говиду.
Ключовi слова: шарування ковимiрностi один, фундаментальна гру-
па, голономiя, кривина Рiччi.
mailto:bolotov@ilt.kharkov.ua
Introduction
Foliations
Holonomy
Growth of leaves
Mappings into foliations
Foliations with nonnegative Ricci curvature
The main result
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