We introduce a notion of metric on a Lie groupoid, compatible with multiplication, and we study its properties. We show that many families of Lie groupoids admit such metrics, including the important class of proper Lie groupoids. The exponential map of these metrics allows us to establish a linearization theorem for Riemannian groupoids, obtaining both a simpler proof and a stronger version of the Weinstein–Zung linearization theorem for proper Lie groupoids. This new notion of metric has a simplicial nature which will be explored in future papers of this series.
Funding source: H2020 European Research Council
Award Identifier / Grant number: Starting Grant No. 279729
Funding source: National Science Foundation
Award Identifier / Grant number: DMS 1308472
Award Identifier / Grant number: DMS 1405671
Funding statement: MdH was partially supported by the ERC Starting Grant No. 279729. RLF was partially supported by NSF grants DMS 1308472 and DMS 1405671, and FCT/Portugal. Both authors acknowledge the support of the Ciências Sem Fronteiras grant 401817/2013-0.
A Some technical background
In this appendix, we recall the concept of quasi-action, with focus on the tangent lift of an action. We then introduce averaging operators for quasi-actions. This is a crucial technique that we use in the paper to construct 2-metrics on proper groupoids.
A.1 The tangent lift of an action
Unlike the group case, an action of a Lie groupoid on a manifold does not induce a tangent action on the tangent bundle. We do have natural actions in the normal directions of the underline foliation, the so-called normal representations that we have already discussed. Still, sometimes it is necessary to put them all in a common framework. This can be done with the help of a connection on the groupoid, which allow us to define a quasi-action on the tangent bundle.
Let be a Lie groupoid, and let E be a manifold. A quasi-action with moment map consists of a smooth map
satisfying for all . In other words, a quasi-action associates to each arrow in G a smooth map . The quasi-action is called
unital if for all ;
flat if for all ;
linear if is a vector bundle and is linear for all g.
Thus, with these definitions, an action is the same as a unital flat quasi-action, and a representation is the same as a linear action.
An action can be lifted to a quasi-action of the action groupoid over the tangent bundle TE with the help of a connection on the groupoid. By a connection σ on the Lie groupoid we mean a vector bundle map such that and . Hence, a connection yields a splitting for the following sequence of vector bundles over G:
A connection σ is multiplicative if its image is a subgroupoid of . Using a partition of the unity, one can show that every Lie groupoid admits a connection (see, e.g., ), however a groupoid may not have a multiplicative connection. For instance, a multiplicative connection for the pair groupoid is the same thing as a trivialization of the tangent bundle which, of course, does not exist in general.
Given an action and a connection σ on G, the tangent lift of θ is the quasi-action which has as moment map the projection and is defined by
By transposition, we define the cotangent lift:
We will often denote just by gv and similar for the cotangent lift. With these notations we have . The tangent lift and the cotangent lift are both unital, but rarely flat. In fact, the tangent and cotangent lift are flat if and only if the connection is multiplicative. As we saw above in the example of the pair groupoid, this may be a quite restrictive condition, and that is why we need to consider quasi-actions to work with general groupoids.
When is an étale groupoid, the map is an isomorphism and there exists a unique connection, namely . Moreover, this connection is multiplicative. Therefore, when working with étale groupoids (and orbifolds), the tangent and cotangent lift are canonically defined, and they are actual actions, which greatly simplifies the whole theory.
Although the tangent and cotangent lift depend on the choice of a connection, their action along the directions transversal to the orbits is intrinsic:
Let be a Lie groupoid, σ a connection, an action, and an orbit. Then is invariant for the cotangent quasi-action , and the restriction agrees with the conormal representation of the action groupoid. Hence, it is an action which does not depend on σ.
The connection σ consists in choosing for each a retraction for the linear map
in a smooth way. The value of such a retraction over the image is totally settled. Write and note that . The result follows. ∎
We end this subsection by stating the following naturality properties of the tangent lift, whose proofs are straightforward.
Let be a Lie groupoid and fix a connection σ on it. Then:
If and are two groupoid actions with moment maps , respectively, then for any equivariant map the differential is also equivariant for the tangent lifts and .
If and are two commuting actions with moment maps , then the tangent lifts and also commute.
A.2 Haar systems and averaging methods
Haar systems on Lie groupoids generalize Haar systems on Lie groups, they always exist for proper groupoids, and allow some averaging arguments on functions and sections of equivariant vector bundles. We show that this can even be extended so as to include vector bundles endowed with quasi-actions, and apply in this way averaging arguments to metrics.
Recall that a smooth density on a vector bundle of rank r is a nowhere vanishing smooth section μ of the trivial line bundle . For instance, when E is orientable, any volume form in E, i.e., a nowhere vanishing section ω of , determines a density .
Let be a Lie groupoid with associated algebroid , and μ a smooth density on the underlying vector bundle. Denote by the pullback density on through the target map
The family of densities satisfies the following two properties:
(Smoothness) The function
is smooth for all .
(Right-invariance) For any arrow and one has
In other words, we have , where denotes right multiplication.
We say that μ is a normalized Haar density if the family also satisfies the following property:
(Normalization) The support is compact and for all :
A proper groupoid admits a normalized Haar density.
For a proof we refer to [3, 18]. The basic idea is that one can construct such a density μ as the product of a nowhere vanishing density and a cut-off function c. Here by a cut-off function we mean a function whose support intersects the saturation of any compact set in a compact set, or equivalently, such that is proper, plus the normalization condition
Now let be a Lie groupoid and let be an action with moment map . We say that a function is θ-invariant if it is constant along the orbits, namely for all for which the action is defined. A normalized Haar density allows us to construct for any a θ-invariant function by averaging over the orbits.
In the same fashion it is possible to average sections of more general vector bundles . More precisely, let be a vector bundle, let be an action, and let be a linear quasi-action. The main examples to keep in mind are the tangent and cotangent lifts of an action. Writing and , we say that a section is θ-invariant if for all for which the action is defined.
Given a Lie groupoid with normalized density μ, an action , and a linear quasi-action , the associated averaging operator is defined by
Note that only depends on the restriction of f to the orbit of e. The main properties of this averaging operator are summarized in the following proposition. The proof is straightforward.
With the above notations, the following hold:
If is flat, then is θ -invariant for any f.
If f is already θ -invariant, then .
If are two commuting actions, then .
For any equivariant map over vector bundles endowed with linear quasi-actions of , the averaging operators commute with ϕ.
We thank IST-Lisbon, IMPA and Universiteit Utrecht for hosting us at several stages of this project. We also thank H. Bursztyn, M. Crainic, I. Marcut, D. Martinez-Torres, H. Posthuma and I. Struchiner for many fruitful discussions.
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