1. Introduction

An -dimensional Riemannian orbifold, , is a metric space so that the following is true: for any , there exists and a Riemannian metric on , a finite group (the isotropy group) acting on by isometries, so that , and there is an isometry with (see [1]). We call a regular point if ; otherwise, is a singular point. We say the curvature of satisfies if the sectional curvature of every above satisfies . We say is collapsing, if admits a sequence of metrics, , with uniformly bounded curvature, so that, for any ,

As an example, consider the standard action on the sphere .

The quotient orbifold will be arbitrarily collapsed when (see Figure 1). However, for any fixed , can be collapsed only to a certain degree; it does not support a sequence of collapsing metrics. In fact, for each one of the two singularities on , there is a neighborhood that is isometric to , where is equipped with some invariant metric. Therefore if is a collapsing sequence of metrics on , we get a corresponding sequence of pullback metrics on ; every is smooth. Observe , where is fixed and , thus . If the diameter of stays bounded, we immediately get a contradiction to the Gauss-Bonnet theorem; in general, we can use the result in [2] to conclude that admits an F-structure, in particular the Euler characteristic vanishes—this is a contradiction since clearly .

Figure 1

On the other hand, consider the double of a 2-dimensional rectangle. Clearly it admits a flat metric, thus we obtain a sequence of collapsing metrics by rescale. Notice, in this example, for each of the four singularities, the isotropy group has order 2, a quite small number.

Intuitively, these examples suggest that when an orbifold is collapsing, a conelike singularity cannot be too “sharp,” that is, there should be some bound in . The main result of this paper is as follows.

Theorem 1.1. Assume is a compact, collapsing orbifold, is an isolated singularity. Then .

If has an isolated singularity , then the dimension of must be even, and . Theorem 1.1 fails if we drop the requirement that is an isolated singularity; for example, we can take any orbifold and let ; by shrinking the factor, we see is collapsing while there is no restriction on singularities of . The bound has its root in the Bieberbach theorem of crystallographic groups and Gromov's almost flat manifold theorem.

Clearly, Theorem 1.1 is a corollary of the following.

Theorem 1.2. For any , there is so that if is an orbifold with all singularities satisfying , , then for any isolated singularity .

Remark 1.3. The bound in Theorem 1.1 is not sharp. When , it is not hard to see that either or is a flat orbifold. Therefore by Polya and Niggli's classification of crystallographic groups on [3, page 105] or [4, page 228], we actually have for collapsing 2 orbifolds.

A nilmanifold, , is the quotient of the (left) action of a discrete, uniform subgroup , on a simply connected nilpotent Lie group . Left invariant vector fields (LIVFs) can be defined on . An affine diffeomorphism of is a diffeomorphism that maps any local LIVF to some local LIVF. In general, a right invariant vector field (RIVF) cannot be defined globally in , unless this vector field is in the center of the Lie algebra of . However, the right invariant vector fields, not the left invariant ones, are Killing fields of left invariant metrics on . An infranil orbifold is the quotient of a nilmanifold by the action of a finite group of affine diffeomorphisms. If the action is free, we get an infranil manifold.

In our previous work, [5], we generalized the Cheeger-Fukaya-Gromov nilpotent Killing structure [6] and the Cheeger-Gromov F-structure, [2, 7], to collapsing orbifolds. In particular, sufficiently collapsed can be decomposed into a union of orbits. Each orbit is the orbit of the action of a sheaf of nilpotent Lie algebras, which comes from local RIVFs on a nilmanifold fibration in the frame bundle . Therefore every is an infranil orbifold. The proof of Theorem 1.2 is based on the relation between singularities on and singularities within an orbit in , as well as the nilmanifold fibration on .

is called almost flat, if where is the diameter of , is a small constant that depends only on . In [8], Gromov proved that an almost flat manifold has a finite, normal covering space that is a nilmanifold. Subsequently, Ruh [9] proved that is diffeomorphic to , where is a discrete subgroup in the affine transformation group of . In [10], Ghanaat generalized this to an almost flat orbifold , under the assumption that is good in the sense of Thurston [1], that is, is the global quotient of a simply connected manifold . There are examples of orbifolds that are not good, see [1]. In fact, without much effort, one can remove the assumption that is good.

Proposition 1.4. If is an almost flat orbifold, then is an infranil orbifold.
Precisely, there is a nilmanifold , a finite group acting on by affine diffeomorphism, so that is diffeomorphic to . The order of is bounded by . Moreover, there is a sequence of metrics so that .

The proof is almost the same as [11, 12]; the only difference is one must replace the exponential map by the develop map (see [5, 13]) and modify the definition of Gromov product in [11] accordingly.

The proof of Theorem 1.2 does not depend on Proposition 1.4. On the other hand, Proposition 1.4 implies Theorem 1.2 for almost flat orbifolds immediately, even without the assumption that the singularities are isolated.

Remark 1.5. If is an isolated singularity, then, near , is homeomorphic to (and in the metric sense, close to) a metric cone over a space form of dimension . When , the -dimensional space forms were first classified by Threlfall and Seifert, they used the fact that is locally isomorphic to ; [3, chapter 7] or [4] for details.

Remark 1.6. By the work of Anderson, Gao, Nakajima, Tian, Yang, and others, orbifolds with discrete singularities appear naturally as Gromov-Hausdorff limits of noncollapsing Einstein metrics with a uniform curvature bound; see [14] for a recent survey. In particular, for Kähler-Einstein metrics, there is a complex structure on the limit .

2. Proof of Theorem 1.2

If is an infranil orbifold, then it is easy to obtain the bound in Theorem 1.1. Since the proof contains some ideas for the general case, we give full details.

Lemma 2.1. Assume is an infranil orbifold. Then .

Proof. Assume , where is a simply connected nilpotent Lie group, is a discrete group of affine diffeomorphisms on so that is compact. If is abelian, then is a flat orbifold, is a discrete group of isometries on that acts properly discontinuously. So the conclusion follows from (the proof of) Bieberbach's theorem on crystallographic groups. In fact, it is well known that the maximal rotational angle of any is either 0 or at least . Thus the bound comes from a standard packing argument; notice and the bi-invariant metric on has positive curvature.
We prove the general case by induction on dimension of . Remember that contains a normal subgroup of finite index, so that is a uniform, discrete subgroup of and is the quotient of the action on the nilmanifold . Clearly embeds in , that is, ; here we choose a point in that projects to .
Let be the center of , then is connected, of positive dimension. Since any is affine diffeomorphism, moves a -coset in to a -coset. Therefore acts on the nilmanifold , the quotient is an infranil orbifold of lower dimension. Let be the projection, and assume . Thus we have a homomorphism is a torus bundle over , the fiber is . Assume is in , the kernel of , then fixes every fiber in . If, in addition, fixes every point in the fiber passing through , we claim must be identity. In fact, on we have , where and is a Lie group automorphism of ; if fixes every point in one fiber, then is identity on the center . This implies that is a translation on every fiber. Since is of finite order and fixes every point in one -fiber, must be identity. Therefore any element is decided by its restriction on the fiber passing through ; so is isomorphic to a finite group of affine diffeomorphisms on that fixes , thus can be bounded by Bieberbach's theorem. Since the conclusion follows by induction.

In [5], the existence of nilpotent Killing structure of Cheeger-Fukaya-Gromov [6] is generalized to sufficiently collapsed orbifolds. We briefly review this construction.

As in the manifold case, one can define the frame bundle of an orbifold . If is isometric to , where is a finite group acting on , then locally is , where is the orthonormal frame bundle over , and acts on by differential, that is, moves a frame to . Therefore is a manifold; strictly speaking, is not a fiber bundle. Let be the projection.

Moreover, there is a natural action on ; on the frames over regular points, this action is the same one as in the manifold case; however, at (the frames over) singular points, this action is not free. As in the work of Fukaya [15], see also [5], any Gromov-Hausdorff limit of a collapsing sequence is a manifold. Following [6], for sufficiently collapsing orbifolds, locally we have an -equivariant fibration where the fiber is a nilmanifold, is a smooth manifold with controlled geometry.

As in [6], we can put a canonical affine structure on the fibers, that is, a canonical way to construct a diffeomorphism from a fiber to the nilmanifold . In particular, there is a sheaf , of a nilpotent Lie algebra of vector fields on . Sections of are local right invariant vector fields on the nilmanifold fibers . By integrating , we get a local action of a simply connected nilpotent Lie group, , on . Therefore we also call a fiber an orbit, and we can write .

The fibration is -equivariant, so any moves a fiber to a (perhaps another) fiber by affine diffeomorphism. Moreover, the action on is locally trivial, that is, if is sufficiently small, then moves a section, , of on any open set , to itself (over ) ( [6, Proposition 4.3]). In particular, the sheaf induces a sheaf, which we also denote by , on the orbifold away from the singular points. An orbit on projects down to an orbit on .

Assume is any frame over . Let be the the isotropy group of an orbit . We will simply write by . Let be the identity component of . It can be shown that, restricted on , the action of is identical to the action of a torus, and the Lie algebra of this torus, , is in the center of (see [5, 6] for more details). Consider the nilmanifold Therefore is a torus bundle over . Notice, on , moves fibers to fibers, thus the orbit is the quotient of by the action of the finite group . Therefore is an infranil orbifold. In particular, the singularities within satisfy the bound in Lemma 2.1.

It is important to remark that the above structure is not trivial.

Lemma 2.2. Let be any integer. Then there is , so that if is an orbifold with , for all , then every -orbit on is of positive dimension.

Proof (sketch). For any unit vector , the bound in implies that does not have fixed point in unless or . However, for sufficiently collapsed orbifolds, there is a vector in the center of so that generates a closed loop in that is shorter than , therefore cannot be in the Lie algebra of , which is in both and the center of . Thus the orbit is not contained in a single orbit in , so is of positive dimension in (see [5] for more details).

Proof of Theorem 1.2. Assume is an isolated singular point, is in . is the fiber that projects to . Let , , be as above. Let Thus is a subgroup of , and . Let Thus is a normal subgroup of .
Lemma 2.3. If fixes every point in , then is the identity in .
Proof. Potentially may fix every point in while moving some points of that are outside . We will rule out this possibility.
By assumption, is an isolated singularity. For any that is not identity, the connected component of the fixed point set of that passes through must project to under , because away from the action is free. Therefore is a single point in , and this contradicts the fact that the -orbits on are of positive dimension; see Lemma 2.2.
In particular, we have a faithful representation of in the affine group of , that is, we can identify with the restricted action of the group on .Take any that is not identity in . By definition fixes . If fixes every point in , as in Lemma 2.1, is a translation on every fiber; because is of finite order and moves every fiber to itself, necessarily fixes every point in ; thus is identity. So rotates the tangent plane of at . Therefore is isomorphic to a finite group of affine diffeomorphisms on the torus . By the Bieberbach theorem, Recall that Bieberbach's theorem implies that all finite subgroups of have a uniform upper bound in order.We have where is a finite group. Let be the subgroup of that fixes . Now we get an embedding By Lemma 2.1, Thus

Acknowledgment

The author is grateful to Professor Tian for very helpful suggestions.