Injectivity radius of manifolds with a Lie structure at infinity

Using Lie groupoids, we prove that the injectivity radius of a manifold with a Lie structure at infinity is positive.


Introduction
Manifolds with a Lie structure at infinity were introduced by Ammann, Lauter and Nistor in [1], forming a class of non-compact complete Riemannian manifolds of infinite volume. In the same article, they conjectured that the injectivity radius of a (connected) manifold with Lie structure at infinity is positive. In this paper, we give a proof of this conjecture using the associated groupoid given by [5] and [6]. Together with the results from [1], this implies that manifolds with a Lie structure at infinity are of bounded geometry. In particular, the hypothesis of injectivity radius in [2] is now automatically satisfied, as well as in [3], where positivity of the injectivity radius is used to obtain uniform parabolic Schauder estimates. Bounded geometry also yields uniform elliptic Schauder estimates, see [4] for a recent application in this direction. Acknowledgement. The author thanks his PhD advisor Frédéric Rochon for suggesting the problem and the approach, and also wishes to thank Bernd Ammann, Claire Debord and Victor Nistor for helpful discussions.

Preliminaries
Following [1] and [8], we recall some definitions and facts. Definition 1.1. A groupoid is a small category G in which every morphism is invertible.
The objects of the category are also called units, and the set of units is denoted by G 0 . The set of morphisms is denoted by G 1 . The range and domain maps are denoted respectively r, d : G 1 → G 0 . The multiplication operator µ is defined on the set of composable pairs of morphisms by: The inversion operation is a bijection ι : g → g −1 of G 1 . The identity morphisms give an inclusion u : x → id x of G 0 into G 1 .
is a groupoid such that G 0 and G 1 are manifolds with corners ( [7]), the structural maps d, r, µ, u, ι are differentiable, and the domain map d is a submersion.
Consequently, for an almost differentiable groupoid, ι is a diffeomorphism, r = d • ι is a submersion and each fiber G x = d −1 (x) ⊂ G 1 is a smooth manifold whose dimension n is constant on each connected component of G 0 . Following the convention in [5, p. 578], we require G 0 and d −1 (x) to be Hausdorff (for all x ∈ G 0 ), but not necessarily G 1 to avoid excluding important cases. From now on, Lie groupoid will stand for almost differentiable groupoid, manifold will stand for manifold with corners and smooth manifold will stand for manifold wihout corners. A Lie groupoid is called d-simply connected if its d-fibers G x = d −1 (x) are simply connected ( [5]).
for any smooth sections X and Y of A and any smooth function f on M .
There is a Lie algebroid A(G) associated to a Lie groupoid G, constructed as follows: let T vert G = kerd * = ∪ x∈G 1 T G x ⊂ T G 1 be the vertical bundle over G 1 . Then A(G) = T vert G| G 0 is the structural bundle of the Lie algebroid over G 0 . The anchor map is given by Remark 1.5. There might be more than one Lie groupoid integrating a Lie algebroid. However, by [5, Lie I], if a Lie algebroid over a smooth manifold is integrable, there is a unique d-simply connected Lie groupoid integrating it. 3. The space of continuous paths on a topological space modulo homotopy equivalence forms a groupoid which is called the fundamental groupoid.
We recall the definitions and basic properties of manifolds with Lie structures at infinity. For details and proofs, we refer to [1].

Injectivity radius of a manifold with Lie structure at infinity
The following theorem is due to Debord ([6, Theorem 2], see also [5,Corollary 5.9]).

Theorem 2.1 (Debord). Every almost injective Lie algebroid over a smooth manifold is integrable.
This has the following implication for Lie structures at infinity.
Theorem 2.2. Any Lie algebroid over a manifold with corners associated with a Lie structure at infinity is integrable.
Proof. This extension of Theorem 2.1 to manifolds with corners is well-known to experts. However, since no explicit proof seems to be available in the literature, we will provide one for the convenience of the readers. Let (M, V) be a Lie structure at infinity of M 0 and A = A V be the corresponding structural vector bundle. Taking two copies of M and gluing them along a maximal subset of disjoint boundary hypersurfaces, we obtain a compact manifold with corners M 1 with at least one hypersurface less. Repeating this operation finitely many times, we obtain a closed manifold M with a finite group Γ acting on M such that M /Γ ≃ M topologically. Now, by [7, Exercise 1.6.2], M is naturally an orbifold. Changing the smooth structure on M , one can in fact ensure that the quotient map q : M → M is such that In a suitable local chart, q can be written as (x 1 , . . . , x k , x k+1 , . . . , x n ) → (x 2 1 , . . . , x 2 k , x k+1 , . . . , x n ) where k is the depth of the point (0, 0, . . . , 0). To see this, it suffices to show that V is locally free of rank k for some k. Given p ∈ M , then since V is locally free of rank k for some k, there exist v 1 , . . . , v k ∈ V which locally and freely span V near q(p). This means V is locally and freely spanned by q * v 1 , . . . , q * v n ∈ V near p, showing that V is locally free of rank k as claimed. By the Serre-Swan theorem, we have a vector bundle A V over M with the space of smooth sections C ∞ ( M , A V ) = V. Clearly the inclusions V ⊂ V b ⊂ C ∞ ( M , T M ) induce an anchor map, so that A V is naturally an almost injective Lie algebroid. By the Theorem 2.1 and the Remark 1.5, there exists therefore a d-simply connected groupoid G integrating A V . Each element g ∈ Γ induces an automorphism ρ(g) : A V → A V , and by Lie II, an automorphism on G. Hence we have an action of the group Γ over G. The quotient G/Γ is then the desired d-simply connected Lie groupoid integrating (M, V).
Let M 0 be a connected smooth manifold with a Lie structure at infinity (M, V). By Theorem 2.2, there exists a d-simply connected groupoid G = (M, G 1 , d, r, µ, u, ι) with units M such that A(G) ≃ A as Lie algebroids over M . Therefore A(G) is equipped with an inner product also noted g. The anchor map is given by r * : A(G) → T M . We have an isomorphism r * A(G) ≃ T vert G where r * A(G) is the pull-back of A(G) via the range map r : G → M ([2, (19)]). Explicitly, for p ∈ G, (r * A(G)) p = A(G) r(p) = T r(p) G r(p) . The vector bundle r * A(G) is equipped with a metric induced by the metric g on A(G), hence so is T vert G. Therefore each G x becomes a Riemannian manifold for all x ∈ M . Proof. By [5, Proposition 1.1], for all x ∈ M 0 , r(G x ) ⊂ M 0 (which is the leaf of the singular foliation of A passing by x). On the other hand, G| M0 is the unique d-simply connected Lie groupoid which integrates T M 0 , and therefore it isomorphic to the homotopy groupoid ( M 0 × M 0 )/π 1 (M 0 ). Consequently, M 0 = r(G x ) for all x ∈ M 0 . Now, by definition of a Lie structure at infinity, r * : T y G x → T r(y) M 0 is an isomorphism. This means that r : G x → M 0 is a local diffeomorphism. Moreover, g 1 , g 2 ∈ G x with r(g 1 ) = r(g 2 ) if and only if there exists h = g −1 1 g 2 ∈ G x x such that g 2 = g 1 h. That is, r : G x → M 0 is a covering map with group G x x .