© Hindawi Publishing Corp. UNIVERSAL CONNECTIONS ON LIE GROUPOIDS

Given a Lie groupoid Ω , we construct a groupoid J 1 Ω equipped with a universal connection from which all the 
connections of Ω are obtained by certain pullbacks. We show 
that this general construction leads to universal connections on 
principal bundles (considered by Garcia (1972)) and 
universal linear connections on vector bundles (ultimately 
related with those of Cordero et al. (1989)).

Universal connections have been studied in, for example, [4, 8 , 12, 13], to name but a few o f the earlier attempts to the subject. Narasimhan and Ramanan [12], based on the Stiefel bundles and unitary groups, proved the existence o f universal bundles fo r principal bundles with a compact Lie group as a structure group. This result was extended to the case o f arbitrary connected Lie groups in [13]. However, [8] has a rather algebraic nature showing that the geometric construction o f [12]  Motivated by this approach, we study the existence o f universal connections on Lie groupoids. The geom etry o f Lie groupoids and algebroids is a topic o f current research which, beyond its significance per se, has found numerous applications in many areas (see, e.g., [2,9,10,18]). In this framework, universal connections might be an interesting complement o f the theory o f groupoid connections.
More explicitly, in Theorem 4.5, we prove that, roughly speaking, a given Lie groupoid Ω determines a Lie groupoid J iQ equipped with a universal connec tion from which all the connections o f Ω are derived (by appropriate pullbacks).
groupoid and Q iQ is the corresponding connection bundle, whose sections are in bijective correspondence with the connections o f Ω.
Before giving the p roo f o f the main result, we study in detail the bundle <2ιΩ, the differentiable structure o f J iQ , as well as the Lie algebroid ί(7 ι Ω ). We also give an explicit description o f certain mappings involved in the construction o f the universal connection.
The paper is completed with two applications.  [6 , 7]. For groupoids and algebroids, we mainly fo llow the notation and term inology o f [9]; however, fo r the reader's convenience, in this section, we briefly review some facts needed in the sequel.
A groupoid Ω is a small category all the morphisms o f which are invertible.
The manifold B is the base o f the algebroid and q is its anchor (fleche).
The Lie algebroid o f a differentiable groupoid Ω is defined to be be the induced trivializations o f TB and Ω, respectively. We also consider the if v g TgG and pg denotes the right translation o f G by g. Taking   (3.10) We obtain a chart o f <2ιΩ since Fu has a smooth inverse. In fact, the latter is given by (x , f ) -j\ s, the jet j * s being determined by the conditions The compatibility o f the previous charts is established by a direct verification; hence, (2 ιΩ becomes a differentiable manifold. The previous term inology is justified by the following proposition. It is easily checked that the assignments y -Sy and 5 -ys are mutually inverse. We recall that the total space / ιΩ e x Ω x ζ>ιΩ is the set We note that J\Cl here is not the first prolongation groupoid, often denoted also by the same symbol.  (Fu1(x,f),Zul (x,g,y), Fu1(y,h)). We w ill show that the desired universal connection is one defined on JιΩ.
Before its explicit description, we need a few preliminary facts. First, as an immediate consequence o f the relative definitions, we obtain that (see also [ On the other hand, since TyS(v) e 7s(y )Q iQ , for every v e TyB and y eB,

= E v ( S ( y ) , v )
= ys(v), for every v eTB\ thus, proving the universal property. Consequently, i f X * is the Killing vector field corresponding to X, On the other hand, for every iv,g) e ΓξΩ& x ω£,

E. VASSEJOU AND A. NIKOLOPOULOS
Pr o o f . We outline the main steps o f the procedure, omitting the computa tional details whose verification is a matter o f routine.
First, we observe that every h e Π (τγ *£ ) can be written, by the identification o f Corollary 5.5, in the form h = ( Y,h,X) with X = j*s, Y = jy S ', and h e (Π (Ε))χ. In these terms, the map ev is equivalently given by W ((Y,h,X),(X,e)) = (YMe)). we can show (see [14]) that the universal connections o f Section 5.2 are related with the universal connections on frame bundles considered in [3].