Groupoïdes riemanniens. (Riemannian groupoids)
A Lie groupoid consists of submersive map between manifolds \(\alpha: \Gamma\to \Gamma_ 0\), and an inversion involution i: \(\Gamma\to \Gamma\), whose fixed point set is a copy of \(\Gamma_ 0\), the image of an embedding j: \(\Gamma\) \({}_ 0\to \Gamma\). The composite \(\alpha\circ i\) gives another submersion \(\alpha \circ i=\beta: \Gamma \to \Gamma_ 0.\) For example let \({\mathcal F}\) be a codimension p foliation of an n- manifold X, and define \(\Gamma\) to be the space of paths tangent to the foliation, modulo homotopies in the foliation which preserve end points. Then \(\Gamma\) is a 2n-p dimensional manifold. We let \(\Gamma_ 0\) be X, define i to be the map which reverses the parametrization of a path, and \(\alpha\),\(\beta\) to be the end point maps. (There is also a ``multiplication'' for suitable paths). This paper discusses the Riemannian geometry of Lie groupoids. The authors define the groupoid to be Riemannian if there are metrics g, \(g_ 0\) on \(\Gamma\), \(\Gamma_ 0\) such that i is an isometry and \(\alpha\) and \(\beta\) are Riemannian submersions. They consider the foliation \({\mathcal F}_ 0\) of \(\Gamma\) given by the images under \(\beta\) of the fibres of \(\alpha\), and show that this is Riemannian, in the sense that any geodesic in \(\Gamma_ 0\) which is initially orthogonal to the foliation remains so. They also show that the groupoid arising from a foliation F of X is Riemannian, for suitable choices of metrics, if \({\mathcal F}\) is a complete, regular Riemannian foliation of X.
scientific article; zbMATH DE number 4155359