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Riemannian metrics on Lie groupoids

Matias del Hoyo EMAIL logo and Rui Loja Fernandes

Abstract

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.

Award Identifier / Grant number: Starting Grant No. 279729

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 GM be a Lie groupoid, and let E be a manifold. A quasi-actionθ:G~E with moment mapq:EM consists of a smooth map

θ:G×MEE,(g,e)θg(e)

satisfying q(θg(e))=t(g) for all (g,e)G×ME={(g,e)G×E:s(g)=q(e)}. In other words, a quasi-action associates to each arrow y𝑔x in G a smooth map θg:ExEy. The quasi-action is called

  1. unital if θ1x=idEx for all xM;

  2. flat if θg1θg2=θg1g2 for all g1,g2G(2);

  3. linear if q:EM is a vector bundle and θg:ExEy 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 θ:GE can be lifted to a quasi-action of the action groupoid GE over the tangent bundle TE with the help of a connection on the groupoid. By a connection σ on the Lie groupoid GM we mean a vector bundle map σ:s*TMTG such that dsσ=ids*TM and σ|M=du. 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 TGTM. Using a partition of the unity, one can show that every Lie groupoid admits a connection (see, e.g., [1]), however a groupoid may not have a multiplicative connection. For instance, a multiplicative connection for the pair groupoid M×MM is the same thing as a trivialization of the tangent bundle TMM which, of course, does not exist in general.

Definition A.1.

Given an action θ:GE and a connection σ on G, the tangent lift of θ is the quasi-action Tσθ:(GE)~TE which has as moment map the projection TEE and is defined by

Tσθ:G×MTETE,(Tσθ)(g,e)(v)=dθ(σg(deq(v)),v).

By transposition, we define the cotangent liftTσθ:(GE)~T*E:

Tσ*θ:G×MT*ET*E,(Tσ*θ)g,e(α),v=α,(Tσθ)(g-1,ge)(v).

We will often denote (Tσθ)(g,e)(v) just by gv and similar for the cotangent lift. With these notations we have gα,v=α,g-1v. The tangent lift Tσθ and the cotangent lift Tσ*θ 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.

Example A.2.

When GM is an étale groupoid, the map s* is an isomorphism and there exists a unique connection, namely σ=s*-1. 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:

Proposition A.3.

Let GM be a Lie groupoid, σ a connection, θ:GE an action, and OE an orbit. Then TOT*E is invariant for the cotangent quasi-action Tσ*θ, and the restriction (Tσ*θ)|TO agrees with the conormal representation of the action groupoid. Hence, it is an action which does not depend on σ.

Proof.

The connection σ consists in choosing for each (g,e)G×ME a retraction for the linear map

d(g,e)s*:Te*ET(g,e)*(G×ME)

in a smooth way. The value of such a retraction over the image d(g,e)s*(Te*E) is totally settled. Write O~=(G×ME)O and note that TO~d(g,e)s*(Te*E). The result follows. ∎

We end this subsection by stating the following naturality properties of the tangent lift, whose proofs are straightforward.

Proposition A.4.

Let GM be a Lie groupoid and fix a connection σ on it. Then:

  1. If θE:GE and θF:GF are two groupoid actions with moment maps qE,qF, respectively, then for any equivariant map p:EF the differential dp:TETF is also equivariant for the tangent lifts TσθE and TσθF.

  2. If θ1:GE and θ2:GE are two commuting actions with moment maps q1,q2:EM, then the tangent lifts Tσθ1 and Tσθ2 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 EM of rank r is a nowhere vanishing smooth section μ of the trivial line bundle (rE)(rE). For instance, when E is orientable, any volume form in E, i.e., a nowhere vanishing section ω of rE, determines a density ωω.

Let GM be a Lie groupoid with associated algebroid AM, and μ a smooth density on the underlying vector bundle. Denote by μx the pullback density on G(-,x)=s-1(x) through the target map

The family of densities {μx}xM satisfies the following two properties:

  1. (Smoothness) The function

    xG(-,x)f(g)μx(g)

    is smooth for all fC(G).

  2. (Right-invariance) For any arrow yx and fC(G(-,x)) one has

    G(-,y)f(gh)μy(g)=G(-,x)f(g)μx(g).

    In other words, we have μy=Rh*(μx), where Rh:G(-,y)G(-,x) denotes right multiplication.

Definition A.5.

We say that μ is a normalized Haar density if the family {μx}xM also satisfies the following property:

  1. (Normalization) The support supp(μx) is compact and for all xM:

    G(-,x)μx(g)=1.

Proposition A.6.

A proper groupoid GM 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 cμ~ of a nowhere vanishing density μ~ and a cut-off function c. Here by a cut-off function c:M we mean a function whose support intersects the saturation of any compact set in a compact set, or equivalently, such that s:supp(ct) is proper, plus the normalization condition

G(-,x)c(t(g))μx(g)=1.

Now let GM be a Lie groupoid and let θ:GE be an action with moment map q:EM. We say that a function fC(E) is θ-invariant if it is constant along the orbits, namely f(θge)=f(e) for all g,e for which the action is defined. A normalized Haar density allows us to construct for any fC(E)=Γ(E,E) a θ-invariant function Iθ(f) by averaging over the orbits.

In the same fashion it is possible to average sections of more general vector bundles Γ(E,V). More precisely, let VE be a vector bundle, let θE:GE be an action, and let θV:(GE)~V be a linear quasi-action. The main examples to keep in mind are the tangent and cotangent lifts of an action. Writing ge=θgE(e) and gv=θ(g,e)V(v), we say that a section fΓ(E,V) is θ-invariant if f(ge)=gf(e) for all g,e for which the action is defined.

Definition A.7.

Given a Lie groupoid GM with normalized density μ, an action θE:GE, and a linear quasi-action θV:(GE)V, the associated averaging operator is defined by

Iθ:Γ(E,V)Γ(E,V),Iθ(f)(e):=G(-,x)g-1f(g(e))μx(g),x=q(e).

Note that Iθ(f)(e) 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.

Proposition A.8.

With the above notations, the following hold:

  1. If θV is flat, then Iθ(f) is θ -invariant for any f.

  2. If f is already θ -invariant, then Iθ(f)=f.

  3. If θ1,θ2:GE are two commuting actions, then Iθ1Iθ2(f)=Iθ2Iθ1(f).

  4. For any equivariant map ϕ:V1V2 over vector bundles endowed with linear quasi-actions of GE, the averaging operators commute with ϕ.

Acknowledgements

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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Received: 2014-4-27
Revised: 2015-2-10
Published Online: 2015-6-16
Published in Print: 2018-2-1

© 2018 Walter de Gruyter GmbH, Berlin/Boston

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