Lie groupoids of mappings taking values in a Lie groupoid

Abstract

Endowing differentiable functions from a compact manifold to a Lie group with the pointwise group operations one obtains the so-called current groups and, as a special case, loop groups. These are prime examples of infinite-dimensional Lie groups modelled on locally convex spaces. In the present paper, we generalise this construction and show that differentiable mappings on a compact manifold (possibly with boundary) with values in a Lie groupoid form infinite-dimensional Lie groupoids which we call current groupoids. We then study basic differential geometry and Lie theory for these Lie groupoids of mappings. In particular, we show that certain Lie groupoid properties, like being a proper étale Lie groupoid, are inherited by the current groupoid. Furthermore, we identify the Lie algebroid of a current groupoid as a current algebroid (analogous to the current Lie algebra associated to a current Lie group). To establish these results, we study superposition operators C (K, f) : C (K, M) → C (K, N) , γ → f • γ between manifolds of C-functions. Under natural hypotheses, C (K, f) turns out to be a submersion (an immersion, an embedding, proper, resp., a local diffeomorphism) if so is the underlying map f : M → N. These results are new in their generality and of independent interest.

Manifolds of mappings on Cartesian products

“…As known from classical work by Eells [9], the set C (M, N ) of all C -maps f : M → N can be given a smooth Banach manifold structure for each ∈ N 0 , compact smooth manifold M and σ -compact finite-dimensional smooth manifold N . More generally, C (M, N ) is a smooth manifold for each ∈ N 0 ∪{∞}, locally compact, paracompact smooth manifold M with rough boundary in the sense of [15] (this includes finite-dimensional manifolds with boundary, and manifolds with corners as in [7,8,21]) and each smooth manifold N modeled on locally convex spaces such that N admits a local addition (a concept recalled in Defini-H. Glöckner was supported by German Academic Exchange Service, DAAD Grant 57568548. tion 5.6); see [4,14,16,21,22,25] for discussions in different levels of generality, and [20] for manifolds of smooth maps in the convenient setting of analysis. For compact M, the modeling space of C (M, N ) around f ∈ C (M, N ) is the locally convex space C ( f * (T N)) of all C -sections in the pullback bundle f * (T N) → M, which can be identified with…”

“…In the case n = 1, for compact M and ∈ N 0 ∪ {∞}, canonical manifold structures on C (M, N ) as in Theorem 1.1 have already been considered in [4], in a weaker sense (fixing m = 1 in Definition 1.2). Parts of our discussion adapt arguments from [4] to the more difficult case of C α -maps.…”

“…product of smooth manifolds (possibly with rough boundary) modeled on locally convex spaces and f : 4 Note that due to Lemma 4.3 (a), the evaluation on a canonical manifold is a C ∞,α -map whence it is at least continuous. For a C k -manifold M which is C k -regular 1 and a locally convex space E = {0}, it is well known that for the compact-open C k -topology the evaluation ev : We now turn to smoothness properties of the composition map.…”

“…In particular, the proof of Lemma 4.3 (a) carries over and yields: Lemma 5. 4 In the situation of 5.1, the evaluation map…”

Direct limits of regular Lie groups

“…Let be a local addition for M , i.e., a smooth map on an open neighbourhood of such that for all and moreover the map has open image and is a ‐diffeomorphism onto its image 1 T(Σ|U∩TpM)false(0,·false)=prefixidTpM for all ; e.g., this holds for Riemannian exponential maps (an alternative argument can be found in [2, Lem. A.11]).…”