ELSEVIER Journal of Geometry and Physics 17 (1995) 310-320 JOURNAL OF GEOMETRY,~o PHYSICS Ehresmann connections for lagrangian foliations Robert A. Wolak lnstytut Matematyki, Uniwersytet JagielloffskL WI. Reymonta 4, 30-059 Kraktw, Poland Received 12 November 1993; revised 15 September 1994 Abstract The notion of an Ehresmann connection was introduced by Ehresmann (1950). In recent years it has been extensively studied by some authors. The main aim of this paper is to demonstrate that under some relatively natural assumptions lagrangian foliations admit Ehresmann connections. Keywords: Lagrangian foliation; Ehresmann connection 1991 MSC: 53C05, 53C15, 57R30 1. Preliminaries In this section, for the convenience of the reader, we will recall some basic definitions and results. The only new result of this section is Proposition 2. 1.1. Totally geodesic foliations A foliation on a complete Riemannian manifold (M, g) is called totally geodesic if its leaves are totally geodesic submanifolds of (M, g). Let Q be the orthogonal complement of T.T'; it defines a natural splitting of the tangent bundle T M = T Jr ~ Q. Let us fix a point x ~ M. For any pair of curves t~ : [0, a] ~ M and/~ : [0, b] ~ M with the same starting point, i.e. a(0) = fl(0), such that the curve or is tangent to the leaf passing through the point x, i.e. it is a leaf curve, and fl is tangent to Q, i.e. it is orthogonal to thefoliation, there exists a smooth mapping tr : [0, a] x [0, b] ~ M such that: (1) the curves os : [0,a] ~ M, trs = o I [0,a] x {s}, s E [0,b], are contained in the corresponding leaves of ~'; (2) tr0 = a; 0393-0440/95/$09.50 © 1995 Elsevier Science B.V. All rights reserved SSDI 0393-0440(94)00049-2