Riemannian metrics on Lie 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 G⇉M be a Lie groupoid, and let E be a manifold. A quasi-actionθ:G↷~E with moment mapq:E→M consists of a smooth map

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 y←𝑔x in G a smooth map θg:Ex→Ey. The quasi-action is called

1. unital if θ1x=idEx for all x∈M;

2. flat if θg1⁢θg2=θg1⁢g2 for all g1,g2∈G(2);

3. linear if q:E→M is a vector bundle and θg:Ex→Ey 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:G↷E can be lifted to a quasi-action of the action groupoid G⋉E over the tangent bundle TE with the help of a connection on the groupoid. By a connection σ on the Lie groupoid G⇉M we mean a vector bundle map σ:s*⁢T⁢M→T⁢G such that d⁢s⋅σ=ids*⁢T⁢M and σ|M=d⁢u. 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 T⁢G⇉T⁢M. 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×M⇉M is the same thing as a trivialization of the tangent bundle T⁢M→M which, of course, does not exist in general.

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

Although the tangent and cotangent lift depend on the choice of a connection, their action along the directions transversal to the orbits is intrinsic:

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

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 E→M 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 ω⊗ω.

Let G⇉M be a Lie groupoid with associated algebroid A→M, 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}x∈M satisfies the following two properties:

1. (Smoothness) The function

   x↦∫G⁢(-,x)f⁢(g)⁢μx⁢(g)

   is smooth for all f∈C∞⁢(G).

2. (Right-invariance) For any arrow y←ℎx and f∈C∞⁢(G⁢(-,x)) one has

   ∫G⁢(-,y)f⁢(g⁢h)⁢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.

For a proof we refer to [3, 18]. The basic idea is that one can construct such a density μ as the product c⁢μ~ of a nowhere vanishing density μ~ and a cut-off function c. Here by a cut-off function c:M→ℝ we mean a function whose support intersects the saturation of any compact set in a compact set, or equivalently, such that s:supp⁡(c∘t)→ℝ is proper, plus the normalization condition

Now let G⇉M be a Lie groupoid and let θ:G↷E be an action with moment map q:E→M. We say that a function f∈C∞⁢(E) is θ-invariant if it is constant along the orbits, namely f⁢(θg⁢e)=f⁢(e) for all g,e for which the action is defined. A normalized Haar density allows us to construct for any f∈C∞⁢(E)=Γ⁢(E,ℝE) 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 V→E be a vector bundle, let θE:G↷E be an action, and let θV:(G⋉E)↷~V be a linear quasi-action. The main examples to keep in mind are the tangent and cotangent lifts of an action. Writing g⁢e=θgE⁢(e) and g⁢v=θ(g,e)V⁢(v), we say that a section f∈Γ⁢(E,V) is θ-invariant if f⁢(g⁢e)=g⁢f⁢(e) for all g,e for which the action is defined.

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.