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Double Lie algebroids and second-order geometry. I. (English) Zbl 0765.57025
From the author’s introduction: This is the first part of a two-part paper, the purpose of which is to show that the algebra of double groupoids — the algebra of squares, to paraphrase R. Brown and P. J. Higgins [J. Pure Appl. Algebra 21, 233-260 (1981; Zbl 0468.55007)] — underlies a possible second-order geometry in the same way that parallel translation or path lifting — the algebra of paths – - underlies first-order geometry. Here we are regarding the basic object of interest in geometry as, for example, a metric or, more generally, a $$G$$-structure or, more generally still, an abstract principal bundle. It was shown by us [Lie groupoids and Lie algebroids in differential geometry (Lond. Math. Soc. Lect. Note Ser. 124) (1987; Zbl 0683.53029)], following work of J. Pradines, that standard connection theory may be deduced from the Lie theory of (locally trivial) Lie groupoids and (transitive) Lie algebroids: Lie groupoids are a generalization of principal bundles which permit an analogy with Lie groups, and imitating the construction of the Lie algebra of a Lie group yields a first-order invariant, the Lie algebroid, which may be identified, for a Lie groupoid corresponding to a principal bundle, with the Atiyah sequence of the bundle. One can then show, for example, that the correspondence between Lie subgroupoids and Lie subalgebroids (with suitable connectivity and transitivity assumptions) includes the Ambrose-Singer and Reduction theorems of connection theory (see the author, op. cit. for a full account of these matters and further references).
In this paper we develop a corresponding Lie theory for double Lie groupoids. In Section 1 we give background material on double vector bundles and their cores; this is intended as an introduction to the core structure of double groupoids, which is recalled in the first part of Section 2. The second part of Section 2 describes the double groupoid structure of a double Lie group, and shows that all vacant double groupoids arise from situations of this type. Section 3 treats the case of vacant double Lie groupoids whose side groupoids are equivalence relations — the affinoids [A. Weinstein, Int. J. Math. 1, 343-360 (1990; Zbl 0725.58014)], or pregroupoids [A. Kock, Lect. Notes Math. 1348, 194-207 (1988; Zbl 0657.18007)] — and shows that they are equivalent to principal bundles with structure Lie groupoid, and to the butterfly diagrams of J. Pradines [“Lois d’action principales conjugées”, Manuscrit inéditée, 1977]. Section 4 introduces $${\mathcal{LA}}$$-groupoids, calculating the $${\mathcal{LA}}$$-groupoids of basic examples of double Lie groupoids, and showing that vacant $${\mathcal{LA}}$$- groupoids are determined by a pair of actions which satisfy differentiated forms of those required for a vacant double Lie groupoid. Section 5 defines the core diagram of suitable $${\mathcal{LA}}$$-groupoids and shows that in the presence of appropriate transitivity conditions, an $${\mathcal{LA}}$$-groupoid can be reconstructed from its core diagram. From this it follows that suitable connections in a transitive $${\mathcal{LA}}$$- groupoid are determined by their restriction to the core.

##### MSC:
 57R99 Differential topology 55R99 Fiber spaces and bundles in algebraic topology 53C99 Global differential geometry 58A99 General theory of differentiable manifolds
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