Quantitative Bi-Lipschitz embeddings of manifolds
Sylvester Eriksson-Bique
Tuesday 21st July, 2015
Contents
1 General description of collapsing theory
1
2 Examples
3
3 Background results on collapsing theory
6
Abstract
The goal of this write up is to provide a more detailed proof of
the collapsing result already contained in [2]. While reading this paper the author struggled to understand many details of the proofs,
particularly where the details of an argument were spread over several papers or simply assumed obvious. Nothing new is proven in this
write up, but results are explicated and an attempt is made to give
simple proofs to statements that bothered the author. Where the author finds the proof in an existing paper to be quite sufficient, it is left
without a proof.
1
General description of collapsing theory
Collapsing theory concerns the quantitative and qualitative description of
manifolds in the small scales. The structure of an arbitrary metrics on Rn
can already it’s self be very complicated, so we need to assume some control. Usually this is expressed in terms of the curvature tensor. We assume
one of the strongest controls: a bound on the sectional curvature |K| ≤ 1.
One could also do much of the collapsing theory with a lower bound on the
sectional curvature, or even just a Ricci-lower bound, but it is the authors
belief, that this theory is more limited.
1
The structure of manifolds in the small can be viewed as an outgrowth of
convergence theory of manifolds. One result of convergence theory is that if
M, N are manifolds with bounded curvature and the same dimension, and
they are sufficiently close in the sence of Gromov-Hausdorff, then in fact
they are bi-Lipschitz. Further, the class of manifolds with a lower bound on
injectivity radius and an upper bound on curvature and diameter, possesses
only finitely many diffeomorphism types. One could say that such a class of
manifolds has bounded geometry, as on a definite scale it can be modeled on
Euclidean space, and the size/complexity of the manifold in the large can be
effectively bounded.
One is then left with the study of manifolds with a small injectivity radius,
the so called collapsed manifolds. The term collapsing derives from convergence theory as well, since one can imagine a sequence of manifolds of given
dimension and with ever smaller injectivity radii. By a result of Gromov, a
subsequence of such manifolds will have a well defined limit, but the limit
may no longer be smooth and will have lower dimension. Hence, the dimension collapses. For conceptual reasons some examples will be provided in the
next section.
It is a somewhat unexpected result that the structure of collapsing can be
completely described. There are many ways to state this theorem, with the
focus on being on different aspects of collapsing. One can focus on giving a
description of the structure of possible limit spaces and their singularities.
An essential part of this is that a structure is proven to exist on a definite
scale δ(n). Whereas many results of differential geometry consider only the
structure in the arbitrarily small, in collapsing the power of the results derives from the ability to model the spaces at small, but bounded scales. This
is true for the limit spaces, as well as the spaces themselves. In fact, for every point there exists a local description of a vector bundle over a nilpotent
base. Also all the collapsing directions can be described, while simultaneously distinguishing between collapsing at different scales and locations. One
can differentiate between different types of singularities, and give a effective
group theoretic description of local fundamental groups.
A further benifitial aspect is that the local structure is quantitative with
estimates on the metric tensor it’s self, and even higher derivatives in the
case of higher order derivative bounds on the metric. The induced structures
2
are also differentiably dependent on the parameters. Not only does one obtain
a local description, but these “models” patch up to form a global structure
(singular fibration), which constrains the global geometry and topology of the
manifold. These attributes also distinguish collapsing theory with bounded
curvature from say Alexandrov space geometry, since in general one can not
prove structure theorems at a definite scales. In fact many global results in
Alexandrov space theory can only be stated in the topological category, and
Lipschitz category results are scarce or conjectural. A prominent example of
this is Perelman’s stability theorem, which generalizes the main theorem in
collapsing theorem stated above.
2
Examples
It will turn out that collapsed manifolds can be modeled on some very simple
classes of manifolds. As such, to understand the various technicalities of collapsing theory, we give examples of various phenomena relating to collapsing.
Example 1: S 1 × S 1 . This is a standard example, and illustrates how collapsing leads to a fibered structure, with fibers posessing particularly simple
structure. The fibration is in the most simple case a product.
Q
Example 2: i i S 1 . Here collapsing occurs at various scales, depending on
the relative sizes of i .
Example 3: Let H be the Heisenberg group defined as the upper triangular
matrices with diagonal entries equal to unity. Consider the lattices consisting
of matrices defined by


1 a/n c/n2
 0 1
b/n  ,
0 0
1
where n ∈ Z. If one quotients by the left action of this group one obtains
a manifold Hn , which collapses to a point. However, careful considerations
shows that the collapsing occurs at two scales 1/n and 1/n2 . As in if one
scales the manifold up by something comparable to n2 , then the space looks
like a circle-bundle over a flat base. On the other hand, at the scale of 1/n,
which is the diameter, it looks like a point.
3
Example 4: Let S 3 with the Hopf action T . Shrinking the metric along
the orbits by a factor of , and leaving it the same on the orthogonal complement, we obtain a family of collapsing manifolds S3 , where the fibration
corresponds to the Hopf-fibration, but it does not posess a simple product
structure.
Example 5: Consider a flat vector bundle of dimension n over S 1 with
holonomy generated by a rotation R. If n = 1, and R = −1, then we get the
Möbius bundle. If n = 2, and R is a rotation, we can get various structures,
that resemble cones. However, the bi-Lipschitz structure of a neighborhood
and the injectivity radius will depend on how far one is from the zero section
(the “Soul”). If n = 2 and R = eiθ , where θ >> and both are small, then
a neighborhood of the zero section will be Gromov-Hausdorff close to a ray,
indicating collapsing of two dimensions. The corresponding natural choice
of a fibration map will be the distance to the zero section r. This, however,
induces a singular fibration, with a collapse in dimension.
Other remarks of the last example are in order for small but positive values of r the second fundamental form of fibers becomes very large, and the
normal injectivity radius correspondingly decreases to 0. The “correct” fiber
corresponds to the soul r = 0, for which both of these quantities can be controlled. This is a frequent idea: the fiber of smallest dimension will generally
speaking have good control on both normal injectivity radius and second
fundamental form.
This family of examples, of vector bundles over a flat base indicate most of
the structural properties of collapsing, except with much simpler proofs. If
the base has larger dimension, one can get more complicated examples and
non-orbifold singularities. Also collapse at various scales, and with different
locations having different collapse, can be modeled by this family of examples. However, the examples don’t cover the non-abelian nature of example
3.
Example 6: Let K = [0, 1] × [0, ]/ ∼ by the action (x, y) ∼ (1 − x, y + ),
where operations are modulo 1 and , respectively. This can be called the
4
collapsed Klein bottle. As → 0, K → [0, 1/2] in the Gromov-Hausdorff
distance. A map can be constructed by (x, y) → x for x ∈ [0, 1/2] and
(x, y) → 1 − x otherwise. This is a continuous maps, with all fibers being
one-dimensional. The limit space, however develops singularities. Also, the
normal injectivity radius of the fibers is not bounded below, although the
fibration is non-singular and each fiber is a flat manifold embedded isometrically in locally Euclidean space. The issue can be described with observing
that the space has two special fibers 0 × [0, ] and 1/2 × [0, ], which do have
a lower bound on normal injectivity radius. They are invariants of the equivalence relation defining the space.
Example 7: Finally we illustrate the type of structure on a manifold,
which is not flat or infranil. Let S 3 be the 3-sphere and let Zn act on the
space by (z1 , z2 ) → (e2πik/n z1 , e2πikm/n z2 ), where m ⊥ n (coprime). This
defines the Lens space L(m, n). As n → ∞ these spaces become very
collapsed. One might ask what the local structure of these spaces are.
First let us consider what happens for various choices of parameters. If
n → ∞ and max(n/m, m/n) remains bounded, the limit spaces will be twodimensional spheres with conical singularities at the poles, and one obtains a
one-dimensional fibration. If on the other hand the ratio becomes unbounded,
the limit space will be the interval [0, 1], and the fibration will correspond
to the the orbits of the T 2 action (z1 , z2 ) → (z1 eθi , z2 eφi ). These orbits are
either singular (at the poles), or two-dimensional embedded torii. It turns
out there are three “regions” of the space L(m, n), corresponding to how
close one is to a singular orbit. Denote Ui the set of points were |zi | > 1 − δ
where one component dominates, and as U0 where both co-ordinates contribute (e.g. |z1 |, |z2 | > δ/2). The Ui contain special totally geodesic sets
Si corresponding to the projection of the polar circles (eiθ , 0) and (0, eiθ ). A
normal neighborhood of Si can be modeled as locally flat vector bundles over
Si with holonomy given by a rotation of e2πmi/n , and as such fall into the
category of exaples above. For U0 , a small ball can be modeled as a normal
neighborhood of a torus given by |z1 | = c in L(m, n), where the torus may
or may have one or two collapsed directions.
It is obvious, that our examples don’t exhaust all the combinations of phenomena that can occur, but hopefully they help in understanding some of
the complexities of collapsing theory.
5
3
Background results on collapsing theory
In this paper we will only consider compact Riemannian manifolds M . Let
injp (M ) denote the injectivity radius of a given manifold at the point p.
We will frequently state that a statment is true, as long as the manifold
is sufficiently collapsed, meaning as long as the injectivity radius is small
enough. An almost flat manifold O is one such that diam(O)2 K ≤ c(n),
where c(n) is such that the structural theorems of Gromov-Ruh hold. Like
[2] we apply a result from Abresch to smooth out the metric to get universal
bounds on the covariant derivatives.
Theorem 1. (Abresch) Let (M, g) be a complete Riemannian manifold with
|K| ≤ 1, then for any > 0there exists a g such that
• ||g − g||g < • ||∇g − ∇ ||g < • ||(∇g )i Rg || < C(i, ).
A manifold (M, g) satisfying bounds |R| < A0 and ||∇i R|| < Ai will be
called Ai -regular. This allows us to do calculations in normal co-ordinates
with bounds on all derivatives of the metric tensor (as explicated by [6, 10]),
which can be used to justify the estimates on the various covariant derivatives in [2]. The authors interest is in constructing bi-Lipschitz mappings.
From the bounds it is obvious, that (M, g) is bi-Lipschitz to (M, g ) with
1+
, as stated in [10].
distortion 1−
There are two main ways to describe the structure of a collapsed manifold.
One of them is strictly local, but at a definite scale. It seems to be due to
Fukaya [8, 9], and its proof is also discussed in [2]. Usually the result is stated
only in the sence, that a local diffemorphism exists to a locally flat vector
bundle over an infranil base [12], but it is the authors understanding that
the metric result, while more useful, is only a little harder to prove.
Theorem 2. (Fukaya, [8]) Let (M, g) be a compact Riemannian manifold
of dimension n and sectional curvature |K| ≤ 1. For every > 0 there
exists a universal ρ(n, ) > 0 and δ(n) > 0 such that any point p ∈ M if
injp (M ) < δ(n), there is a point q such that d(p, q) < ρ(n) and on Bq (10ρ(n))
there exists a metric g 0 such that
||g − g 0 ||g < 6
and (Bq (5ρ(n)), g) is isometric to a subset of a locally flat vector bundle over
an infranil base O.
The local right action of N on O induces a local isometric action on the vector
bundle, and a sheaf of killing fields on a neighborhood of O. The orbits of
this structure, roughly, describe the most collapsed directions1 . The main
idea from Cheeger’s and Gromov’s work on F-structures is to patch togeather
these local structures, and provide the necessary quantitative estimates to
do so. This gives a global structure, which we do not discuss other than to
state the main aspects of that theory. In particular, we reffer to [9, 2, 4, ?]
for definitions of group sheafs and their local actions.
Theorem 3. (Cheeger-Gromov-Fukaya, [2]) Let (M, g) be a compact Riemannian manifold of dimension n and sectional curvature |K| ≤ 1 and diameter at most D. For every > 0 there exists a δ(n) > 0 such that if for
any p ∈ M injp (M ) < δ(n), then there exists a nilpotent group N and a group
sheaf g on M with stalk N and a local action thereof on M , and a sheaf k of
fields with stalk n on M , with compact orbits O with diam(O) < cδ(n) with
and a metric g invariant under k. The following properties hold.
• ||g − g ||g < • ||∇g − ∇g ||g < • ||∇g Rg ||g < Ci ()
• The orbits are almost flat manifolds.
• The manifold (M, g ) has (ρ, k)-bounded geometry from some k.
The last condition is of a technical nature and is discussed in [2]. This statement is useful for understanding the global structure of a collapsed manifold,
but for our work a local description is sufficient.
Theorem 4. (Gromov-Ruh, [16]) Let (M, g) be a complete Riemannian manifold with |K|diam(M )2 < (n), then (M, g) has a near-by metric g , with the
following.
• ||g − g ||g < • ||∇g − ∇g || < 1
The holonomy action might introduce further collapsing.
7
• (M, g ) is isometric to N Γ (i.e. it is infranil), where N is a nilpotent
Lie group and Γ ⊂ Aff(N ) is discrete.
• Further [Γ : Γ ∩ NL ] < c(n).
Gromov proved the statment about the manifold being infranil [13]. Ruh
showed that the covering transformations can be chosen to be affine, and
thus establishing the claim [16]. The metric result isn’t explicit in the statement, but can be derived using the bound on the connection and curvature.
The construction of Ruh depended on a choice of a basepoint p ∈ M , but by
the constructions in [2] this can be dispenced with.
The theorem of Gromov-Ruh gives a detailed description of the manifold in
the case when it is fully collapsed. The main acheavement of collapsing theory with bounded curvature is to provide a description for manifolds with
any number of collapsing scales, and where the description encompasses all
the collapsed directions. The statement comes in the form of a fibration,
with nilpotent fibers containing all the most collapsed directions.
The first theorem in this direction were given by Cheeger and Gromov in
[4, 5], but these provided a description of only the most collapsed directions
at a scale that could be arbitrarily small. They proved that there exists a
sheaf of local torus actions, with positive dimensional fibers of small diameter, and even constructed an invariant metric. They proceeded to provide
various collapsing structures along these so called T -structures. Around the
same time Fukaya constructeda global fibration for spaces, when they can
be approximated well by a lower dimensional manifold [7]. The estimates
were derived from [14], where Katsuda gave a detailed proof of Gromov’s
convergence theorem in the Lipschitz category. This initial paper works with
lower regularity, and thus the fibration maps produced have only C 1 -bounds.
Fukaya later improved his result in [10] by providing a smoothly varying,
parametrized, affine structure on the fibers, as long as the manifolds were
Ai -regular. One should state, that all constructions depend on the metric
and thus to bound higher order derivatives of the constructed maps, we need
corresponding bounds on the derivatives of the metric tensor. The proofs
were simplified and improved in [2], where in particular the affine structures
were constructed equivariantly, smoothly and canonically. This is the form
of the theorem that we state.
8
Theorem 5. (Fukaya-fibration, [2]) Let M n , N m be Ai -regular manifolds and
injp (N ) > δ for any p ∈ N . There exists an such that if dG H(M, N ) < ,
then there exists a fibration map f : M → N such that the following hold.
• f is an almost Riemannian submersion.
• ||∇i f || ≤ C(δ, Aj , i)
• diam(f −1 (p)) < cδ for every p ∈ N .
• II(f −1 (p)) ≤ C(δ, Aj , i). for every p ∈ N .
• If M n , N m admit an isometric action by G and h : M n → N m is
a continuous G − Gromov-Hausdorff approximation, then f can be
chosen to be G-equivariant.
• For any continuous Gromov-Hausdorff approximation h : M n → N m ,
we can choose d(f, h) < cδ.
A comment on the fifth condition. A map h : M → N is said to be a G − Gromov-Hausdorff approximation if
|d(h(p), h(q)) − d(p, q)| < for every p, q ∈ M , h(M ) is -dense in N and further for any g ∈ G and
p∈M
d(gh(p), h(gp)) < .
If is small enough a centre-of-mass argument produces a G-equivariant
Gromov-Hausdorff approximation.
Fukaya-fibration theorem resolves the case of collapsed manifolds that approximate lower dimensional manifolds. However, if (Mi , gi ) is a sequence
of collapsing manifolds then a limit space Mi → M might be more complicated. In general it is not even an orbifold, although it’s singularities can
be very well described. This problem disappears in the frame bundle, which
is partly the reason to consider equivariant convergence. This was observed
by Fukaya, but we collect here the elements of the proof from the various
papers.
Theorem 6. (Fibration on the frame bundle, [10, 2]) Let Mi be any sequence
of Ai -regular, then some subsequence converges F Mi → N , where F Mi and
N is Bi -regular.
9
Sketch of proof: It is easy to proove the existence of N . Consider pi ∈ F Mi
and Bpi i (10R) a definite sized ball in Tp (F Mi ) with normal co-ordinates. The
metric tensor is C k -smooth with definite bounds for all k. Therefore the
convergence arguments of [15] will allow us to find a limit Bp∞∞ (10R) with
a smooth metric. Further the local covering group defined in [10, 11] will
converge to a subgroup of isometries of Bp∞∞ (R) to the larger ball. This will
act smoothly on Bp∞∞ (2R), and we can quotient with respect to the local
action. The action of the limit group is fixed point free [10], and therefore
the quotient is smooth.
However to get Bi -regularity we need an argument from [10], where he shows
that r(g)/t(g) can be uniformly bounded. Here r(g) is the rotational part relative to a radially parallell field of frames, and t(g) is the translational part.
Using this bound [10] gives a sectional curvature bound. We need bounds on
all derivatives of the curvature tensor, so we provide some additional comments. First of all, by standard results from Montgomery-Zippin the local
action of G will be smooth2 . Further given a killing field X with |X|(p∞ ) = 1,
then |∇X| < C follows easily. Say take Y one of the co-ordinate vectors in
our reference frame, then
∇Y X = ∇X Y − [X, Y ].
The brackets are bounded by the rotational part and |∇Y | < is easy to
attain. Now since X is a killing field, it will be radially given by the Jacobiequation. This immediatly gives bounds on all derivatives of X. Let O be
the orbit of G at p and Tp O the embedded tangent space. Taking N to be
the normal space in the tangent space and give co-ordinates on the quotient
B ∞ /G by π(expp (n)), where n in a fixed ball in N and π is the quotient map.
The tangent spaces to the orbit O vary smoothly, so the projected metric
will have bounds on all derivatives.
Next we want a universal bound on the injectivity radius. Assume Ni is
a sequence of limit spaces with injectivity radius at injpi (Ni ) → 0. Using
a diagonal sequence of frame bundles we can find a limit space N of frame
bundles as above. This will be smooth and therefore must have a lowe bound
on the injectivity radius.
2
If the reader feels queezy about using such a powerful device, one could simply use
the more elementary result from Steenrod-Myer on continuous differentiability.
10
The previous statement is for sequences, although we want to work with a
fixed space. Another problem is that we do not have an effective bound
on the injectivity radius of the limit space. Let FM be the space of frame
bundles of Ai -regular spaces and of dimension m = n + (n − 1)n/2, and
CFM the closure. Define a stratification of CFM = ∪i CFMi of spaces with
dimension at most m − i.Actually, for i > n the space is empty, but for the
following argument we dont need it.
Theorem 7. (Compactness argument, [10, 2]) There exists δi such that if j
is the largest index such that d(F M, CFMj ) < δj , then there exists a space
N ∈ CFMj which is Bi -regular, and with injectivity radius > γj , such that
the inequalities in Fukaya’s fibration theorem are satisfied and in particular
there is a fibration f : F M → N satisfying the above conditions.
Proof: Let k be the largest integer such that CFMk is not empty. Let
Nj ∈ CFMk be a sequence such that injp (N ) → 0. One can use a diagonal
argument to find a sequence converging to a space of lower dimension 3 . But,
this leads to a contradiction, so there is a universal bound injp (N ) > γk for
all N ∈ CFMk . If now d(F M, CFMk ) < δk we can use the fibration theorem
for F M . If on the other hand d(F M, CFMk ) ≥ δk , any limit space will be
in CFMk−1 and at a distance δk from CFMk . Again the same argument
produces a lower bound on the injectivity radius, and another δk−1 such
that if d(F M, CFMk−1 ) < δk−1 but d(F M, CFMk ) ≥ δk , then there exists
a fibration over a m − k − 1 dimensional base with a lower bound on the
injectivity radius. One may now proceed inductively..
Since there is an injectivity radius bound on the base manifold N , and the
fibration is an almost riemannian submersion, the conclusion of Fukaya’s
theorem can be established.
Theorem 8. (Frame bundle fibration) Let (M, g) be a compact Riemannian
manifold of dimension n and sectional curvature |K| ≤ 1. For every > 0
there exists a universal ρ(n, ) > 0 and δ(n) > 0 such that any point p ∈ F M
3
The limit space is smooth, and has a lower bound on injectivity radius. If the dimension were the same, standard Alexandrov space arguments would give a lower bound on
the injectivity radius of the sequence.
11
if injp (F M ) < δ(n), there is a point q such that d(p, q) < ρ(n) and on
Bq (10ρ(n)) there exists a metric g 0 such that
||g − g 0 ||g < and (Bq (5ρ(n)), g) is isometric to a subset O × Rk .
Proof: Let q = p and consider the normal bundle of the orbit at Op . The
normal bundle can be trivialized by lifting a basis from N horicontally, and
performing Gram-schmidt to orthogonalize it. Taking the normal exponential
map and using the curvature bounds we immediately get the desired bounds
on the metric (since all the derivatives are bounded). Further the normal
injectivity radius can be bounded by that of N .
In this case the vector bundle is trivial. This is the case in general when
the limit space is a manifold. Frequently this will not be the case, and for
this reason we need to study an induced fibration on M . Note that since
the constructed fibration f : F M → N is O(n)-equivariant, the fibers of f
decend to those on M . One of these fibers will have normal injectivity radius
bounded from below, but to establish this we preffer to first work in the frame
bundle and modify the metric slightly in order to simplify the arguments.
Lemma 9. (invariant metric and affine structure, [2]) Let (M, g) be a compact Ai -regular Riemannian manifold of dimension n, then if M is sufficiently collapsed, the frame bundle admits an O(n)-equivariant fibration
f : F M → N , and there is a smooth family of affine connections on f −1 (p)
which is also O(n) invariant, with uniform bounds on all derivatives and further a sheaf of right invariant fields on the fibers k, and an O(n)-invariant
metric for which these are killing fields andg with bounds on derivatives and
in particular
||g − g || < .
Highlights of the proof: The paper by Cheeger, Gromov and Fukaya
present this results quite well. One first proceeds to average the base-point
dependent Ruh-construction to find a canonical connection on each fiber that
varies smoothly. Using this one can produce local trivializations that respect
the nilpotent structure and therefore right invariant vector fields. One comment on this could be in order. Let Γ be a local fundamental group with
respect to a ball whose radius is much smaller than that of the injectivity
12
radius of N . This will map isomorphically to the virtually nilpotent fundamental group of the fiber f −1 (p). For near-by point these fundamental
groups are isomorphic using a translation argument. Futher the holonomies
are continuous (in fact: smooth), so that the translational normal subroup
Λ will be locally the same. Using this one can construct vector fields corresponding to generators of Λ, and further local trivializations. Averaging
the metric with respect to left multiplication gives a left invariant metric, for
which these right invariant metrics are killing fields. The averaging process
is permitted since nilpotent Lie groups are unimodular.
Because g , the fibration f and the sheaf of killing fields k are O(n)-invariant,
they decends to corresponding structures on M , which we denote by the
same symbol. The problem is that there may be singular fibers, and fibers
without approproate bounds on the second fundamental form. This occrus
for example, when the angle of a fiber O and the orbit of O(n) is ∠(O, O(n)) <
. The angle is defined as
inf
v∈O and v⊥O∩O(n)
∠(v, O(n)).
The angle may be small, but allways there will be a near-by point with a
definite lower bound on the angle. For this we need a technical result.
Lemma 10. Let > 0. For any one-parameter family etg ∈ O(n) for g ∈
o(n) s.t. |g| = 1, there exists a 10 < T < T (n, ) such that {etg , t ∈ [−T, T ]}
is -close in the Hausdorff distance to a closed compact abelian subgroup
T k ⊂ O(n) and an element g 0 in the tangent space to T k at the identy such
that ∠(g 0 , g) < .
Proof: For every g there exists a T (g) be the smallest value such that the
aforementioned holds, because the closure of the set {etg , t ∈ R} is a compact
abelian subgroup. The sets T (g) < N are open and exhaust the unit ball of
the space o(n), and as such there exists a common N which will work for all.
Lemma 11. Let > 0 and G a compact Lie group. Then for any closed
subgroups A, B of G there exists a N < N (, G) such that fort the set S
of finite producs of elements from A, B of length at most N there exists a
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closed subrgoup K such that dH (S, K) < . Further if the angle between the
subgroups A and B, which can be measured in terms of the tangent spaces,
∠(Te A, Te B) ≥ σ for some σ, we can choose dim(K) ≥ dim(Te A ⊕ Te B), but
N will also depend on σ.
Proof: For every A, B such a number exists, because the closure of the
group generated by A and B can be chosen as K. But then one can use
compactness, since the pairs (A, B) form a complete metric space with the
product metric of the Hausdorff metrics. The dimension bound follows also
by compactness, since if K is chosen as above, the condition follows.
Remark: We could just have one lemma, but the author finds these results
conceptually easier to work with.
Lemma 12. (angle lemma) For an Ai -regular n-dimensional Riemannian
manifold (M, g) and any p ∈ F M and any > 0 there is a δ(n, Ai ) > 0
such that either ∠(Oq , O(n)) > δ or d(p, q) < such that dim(Oq ∩ O(n)) >
dim(Op ∩ O(n)).
Proof: Consider the fibration f : F M → N with conjugate radius at least
γ. Let g ∈ o(n) such that |g| = 1, g ⊥ Op ∩ O(n) and ∠(Op , g) < δ
for a δ to be determined. Let be small and T (n, /2) and N (O(n), /2)
as above. Then for some T < T (n, ), N < N (O(n), /2) the partial orbit
{etg , t ∈ [−T, T ]} will be /2-close to a closed abelian subgroup T k and there
is a closed subgroup K to one containing the products of length N from
T k and Op ∩ O(n). If δ < (γ/10 − )/(10N T ), then each element of K
translates projected point f (p) by at most γ/10. Also we have dim(K) >
dim(Op , O(n)). Taking a center of mass of K of f (p) we obtain a point f (q)
for some q such that d(p, q) < γ and f (q) is invariant under K. Thus the
orbit dim(Oq ∩ O(n)) ≥ dim(K) > dim(Op , O(n)).
To abbreviate notation we simply speak of an angle of the orbit to denote
the angle discussed above and further define
∠(Op ) = ∠(Op , O(n)).
If we have an angle bound ∠(Oq ), then a normal neighborhood of the projected orbit Oq will have bounded geometry. This would seem like enough,
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but there is a slight issue relating to holonomy. Consider for example the
Klein bottle realized as quotient of [0, 1] × [0, 1]. The orbits of points near
the center line 1/2 × [0, 1] have small normal injectivity radius, although the
geometry is bounded. The center line it’s self is the orbit we seek, and thus a
further averaging is necessary. Recall the definition of the local fundamental
group of a Manifold with bounded covering geometry.
Definition 1. Let (M, g) be a Riemannian manifold, p ∈ M a point with a
lower bound R on the conjugate radius. Let expp : BR → M Then for any
δ < R (usually, δ is of the same order, but the ratio R/δ is large), define
SM (p, δ) = exp−1
p (M ) ∩ Bδ .
These points correspond to short geodesic loops γ. We define the Gromovproduct for loops γ, σ ∈ SM (p, R). Define τ = γ ∗ σ as the concatenation,
and assume |τ | < R. Then τ can be lifted to an element in exp−1
p (M ). This
way we have a partial product. Generate a group from SM (p, δ) ⊂ SM (p, R)
subject to the relations given by the partial product structure of SM (p, R) we
get a group ΓM (p, δ), which is called the local fundamental group.
Comment: There are other ways of defining local fundamental groups, such
as in [10]. One could consider partial isometries generated by γ, which map
BR/10 → BR . In this case the group constructed is the subgroup of local
isometries with composition as the multiplication. One may again only consider group elements near the identity and generate a group respecting the
product structure. This construction realizes a local action of the constructed
group, and thus shows that SM (p, δ) is injected into the set of generators for
ΓM (p, δ), i.e. no relations identify different generators. The above construction depends on R, which is usually chosen to be fixed, but small compared
to the conjugate radius, so that there is some slack. A priori this makes a
difference, but the structure can already be discerned by relations of a fixed
length, depending on δ. The author finds particularly helpful the discussion
of local fundamental groups in [1]. See also the beautiful proof of Klingenbergs lemma in [3].
Lemma 13. Let (M, g) be an Ai -regular manifold, that is sufficiently collapsed at p ∈ M (and R is chosen small enough above). Then for any q ∈ F M
such that π(q) = p, we can consider the local fundamental groups ΓF M (q, ρ)
and ΓM (p, ρ0 ). Assume that ρ0 , ρ sufficiently small and ρ0 /ρ sufficiently large,
and ρ0 up to a factor bigger than the diameter of a fiber in Oq . There is a
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group homeomorphism φ : ΓF M (q, ρ) → ΓM (p, ρ0 ), and Im(φ) is a normal
subgroup of ΓM (p, ρ0 ) of finite index. Further, both ΓF M (q, ρ), ΓM (p, ρ0 ) are
virtually nilpotent, and all indices can be bounded.
Comment: Much more can be said as to the relationship of the local fundamental groups of near by points. Further, the structure of the local groups
in the case of singular orbits can also be precicely described. These results
are implicitely contained in the various proofs and lemmas contained in the
literature. Also, the one doesn’t need ρ0 to be too small, as the fibration
theorem shows that it doesn’t depend on ρ0 as long as it is up to a factor
bigger than the diameter of the fiber and less than the normal injectivity
radius. Also note that the local fundamental groups do not fit into a long
exact sequence like the normal fundamental groups.
Proof: Let γ be a geodesic loop in F M , which is a non-trivial element
of SF M (p, ρ). Then π ◦ γ will be a loop in M , and can be homotoped to a
geodesic loop. If π◦γ were trivial, then one could homotope it along geodesics
to the identity, and this homotopy would lift to a homotopy of γ to a loop
contained in an O(n) fiber. Since the length of γ can not change by much,
this loop will be trivial, and this contradicts the non-triviality of γ. Further,
the relations are all satisfied by the projected elements, if they are satisfied
by elements in the covering. This is possible by taking R slightly smaller in
the above definition.
(One can also show, that π ◦ γ has small rotational part. This can be seen
by horizontally lifting π ◦ γ to σ and noting, that in a local trivialization of
the frame bundle, the vertical components |σv − γv | < 1. Integrating over the
short loop, will give, that d(σ(1), γ(1)) is small, and thus since γ(1) = γ(0)
the rotational part of π ◦ γ, which is σ(1)γ(0)−1
O(n) is small, say less than .
Thus, the local fundamental group in the Frame roughly corresponds to the
almost translational elements of the base, which is analogous to the situation
of flat manifolds.)
We next show that the image subgroup is normal and of finite index. First
we show, that if a geodesic loop τ of M has small rotational part, then it is
contianed in the image of φ. Take the horizontal lift σ, whose endpoint σ(1)
will be close to σ(0), due to the estimate on the rotational part. Say σ 0 is
the geodesic from σ(1) to σ(0), and let γ = σ 0 ∗ σ, be the concatenation of
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the curves. This curve will short-homotope to a curve γ 0 , and the projection
of this will be homotopic to τ , and thus τ ∈ Im(φ).
Now if τi are representatives of left cosets of lenght at most Cρ0 . Then
d(r(τi ), r(τj )) > /2 > − C 0 ρ0 , for otherwise the product τi−1 ∗ τj will have
small rotational part, and would be contained in the image of φ, and the
separate τi would not belong to different co-sets. Thus among all loops less
than a sertain length there can be only a fixed number of left co-sets. As to
the selection of constants: δ(n) >> >> ρ0 >> ρ >> diam(Oq ). After is
fixed, we get that there are at most C() cosets, which can be used to give
an upper bound to the constant C such that all the cosets are enumerated
by those loops.
For the normality it is sufficient to check the loops that are generators in
SM (p, ρ0 ), which are short. Conjugation of an element with small rotational
part, will have also small rotational part, so for any τ ∈ SM (p, ρ0 ) and σ ∈
φ(SF M (q, ρ)) we have τ στ −1 ∈ im(φ), and since these generate the image we
have the desired normality. Combined with the finite index property of the
previous paragraph we obtain the result.
Consider now a point in the base p ∈ M with smallest dimensional orbit
within a ball or radius . Then by the angle lemma, we have that the normal
exponential has a lower bound in the conjugate radius. The manifold M has
bounded covering geometry, and we have a map expp : BR → M , and the
killing sheaf and fibration can be lifted to the ball BR . The lifted action
of the killing sheaf is trivial. For simplicity assume that the orbits all have
the same dimension, which is maximal. In this case the orbit space O of
the action of the killing sheaf on BR will be a bounded curvature manifold.
Define H = ΓM (p, ρ0 )/ΓF M (p, ρ). Observe, that ΓF M (p, ρ) leaves each orbit
in BR invariant, because they can be homotoped to loops contained in orbits
in F M . So H acts on the orbit space O by isometries. By the estimates from
the previous theorem we can take H to translate each element by a small
amount, and thus there is an element of O, which is invariant by the action
of H, and further the corresponding orbit in BR is invariant under the action
of the local fundamental group. The projected orbit Oq in M will then have
a lower bound on the normal injectivity radius, and one can trivialize M by
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that. To make the bundle locally trivial, we can choose a basis of vectors
invariant under the action of the killing sheaf.
In the case where the orbit at p is singular, we have a set ni of killing
vields leaving p invariant, and which correspond to directions in π −1 (p) ∩
π −1 (Op ). These vector fields define a totally geodesic submanifold of fixed
points, whose lift to BR is invariant under the local fundamental group.
Further, restricting to the submanifold, we have a non-singular fibration, and
can apply the previous argument. Thus we obtain the following theorem.
Lemma 14. For every p ∈ M . If Op is an orbit is the smallest in dimension within all orbits intersection Bp (), then there exists an orbit Oq within
distance δ << , with a lower bound of γ >> δ on the normal injectivy
radius.
Proof: The previous argument gives the construction of Oq . We will only
remark, that by choosing ρ0 and ρ small compared to the angle of π −1 (Op ),
which has a definite lower bound. Then the holonomy H will not translate
elements by too much, and the averaging will give us an effective bound on
d(Oq , Op ) and the angle of π −1 (Oq ), and thus the normal injectivity radius.
Finally we can conlude the proof of the main theorem.
Theorem 15. (Fukaya, [8]) Let (M, g) be a compact Riemannian manifold
of dimension n and sectional curvature |K| ≤ 1. For every > 0 there
exists a universal ρ(n, ) > 0 and δ(n) > 0 such that any point p ∈ M if
injp (M ) < δ(n), there is a point q such that d(p, q) < ρ(n) and on Bq (10ρ(n))
there exists a metric g 0 such that
||g − g 0 ||g < and (Bq (5ρ(n)), g) is isometric to a subset of a locally flat vector bundle over
an infranil base O.
Proof: If Op is the smallest dimensional orbit of M , then the statement is
obvious. Say the dimension of the smallest orbit is k. On the other hand if the
orbit Op is k + 1, and the angle of the orbit π −1 (Op ) is smaller than δk , then
Op must be within o(δk ) of an orbit of dimension k, and thus for δk sufficiently
small we can simply apply the previous lemma to the smaller dimensional
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orbit. Thus assume that ∠(π −1 (Op )) ≥ δk . Then one can use a smaller (depending on δk ) in the previous lemma and obtain the desired fibration.
This takes care of the k + 1 dimensional orbits. For k + 2 dimensional orbits
one defines δk−1 such that if a k + 2 dimensional orbit has angle smaller than
such, then it will be close to either a k or a k + 1 dimensional orbit. If on the
other hand the angle is larger than δk−1 , then one can choose an even smaller
in the previous theorem, and move on to those less singular points with
dimension of orbit k +3. This argument proceeds then in the same fashion as
in the case of the proof of the injectivity radius estimate for Frame bundles.
The moral of these proofs is that either one has a non-singular structure over
a large dimensional base, or one is too singular and therefor close to a lower
dimensional stratum.
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