Abstract. We introduce a new integral invariant for isometric actions of compact Lie groups, the copolarity. Roughly speaking, it measures how far from being polar the action is. We generalize some results about polar actions in this context. In particular, we develop some of the structural theory of copolarity k representations, we classify the irreducible representations of copolarity one, and we relate the copolarity of an isometric action to the concept of variational completeness in the sense of Bott and Samelson.

We investigate orthogonal representations of compact Lie groups from the point of view of their quotient spaces, considered as metric spaces. We study metric spaces which are simultaneously quotients of different representations and investigate properties of the corresponding representations. We obtain some structural results and apply them to study irreducible representations of small copolarity. As an important tool, we classify all irreducible representations of connected groups with cohomogeneity four or five.Here, the boundary of the quotient space is understood in the sense of Alexandrov spaces, i.e., it is the closure of the union of all strata of codimension 1 in the quotient space (cf. subsection 2.2). Already this first observation shows that the non-triviality of quotient-equivalence classes is closely related to this intriguing property of representations: the presence or not of boundary in the quotient. This extremely restrictive property has been studied by Kollross and Wilking, who have announced a classification of all representations of simple connected groups with this property. In the case of semi-simple groups the problem seems to be much more involved.Before we proceed, we present a few important families of examples of quotient-equivalence. The simplest example of quotient-equivalence is given by orbit-equivalence. Recall that representations ρ 1 and ρ 2 are called orbit-equivalent if there exists an isometry from V 1 to V 2 that maps G 1 -orbits onto G 2 -orbits. In particular, any representation is orbit-equivalent to its effectivization. Thus we may and will always restrict ourselves to effective representations.The next very important family of examples is provided by the reduction of the principal isotropy groups, often used in geometry and algebra (cf. [LR79, Str94, GT00]). Assume that the principal isotropy group K of the effective representation ρ :A family that is at the origin of all ideas and results of this paper is given by polar representations (we refer the reader not acquainted with this species to subsection 2.3 below, or to [Dad85, PT88, BCO03]). For connected groups, these are representations orbit-equivalent to the isotropy representations of symmetric spaces and, already due to this fact, closely tied with many interesting geometric and algebraic objects. In our language, a representation is polar if and only if it is quotient-equivalent to a representation of a finite group. As a main example: the isotropy representation of a symmetric space is quotient-equivalent to the action of the corresponding Weyl group on a maximal infinitesimal flat.The preceding family of examples shows in an extreme way that a representation of a connected group may be quotient-equivalent to a representation of a disconnected group. This is a source of difficulties but gives rise to some nice geometric considerations. Recall that the Weyl group of a symmetric space is a finite Coxeter group generated by reflections. It turns out that in general the picture is not much different:Assume that G 1...

On contact sub-riemannian symmetric spaces Annales scientifiques de l'É.N.S. 4 e série, tome 28, n o 5 (1995), p. 571-589 © Gauthier-Villars (Éditions scientifiques et médicales Elsevier), 1995, tous droits réservés. L'accès aux archives de la revue « Annales scientifiques de l'É.N.S. » (http://www. elsevier.com/locate/ansens) implique l'accord avec les conditions générales d'utilisation (http://www.numdam.org/conditions). Toute utilisation commerciale ou impression systématique est constitutive d'une infraction pénale. Toute copie ou impression de ce fichier doit contenir la présente mention de copyright. Article numérisé dans le cadre du programme Numérisation de documents anciens mathématiques http://www.numdam.org/

A singular riemannian foliation F on a complete riemannian manifold M is said to admit sections if each regular point of M is contained in a complete totally geodesic immersed submanifold that meets every leaf of F orthogonally and whose dimension is the codimension of the regular leaves of F . We prove that the algebra of basic forms of M relative to F is isomorphic to the algebra of those differential forms on that are invariant under the generalized Weyl pseudogroup of . This extends a result of Michor for polar actions. It follows from this result that the algebra of basic function is finitely generated if the sections are compact. We also prove that the leaves of F coincide with the level sets of a transnormal map (generalization of isoparametric map) if M is simply connected, the sections are flat and the leaves of F are compact. This result extends previous results due to Carter and West, Terng, and Heintze, Liu and Olmos.

Let (M, 79, g) be a sub-Riemannian manifold (i.e. M is a smooth manifold, 79 is a smooth distribution on M and g is a smooth metric defined on 79) such that the dimension of M is either 3 or 4 and 79 is a contact or odd-contact distribution, respectively. We construct an adapted connection V on M and use it to study the equivalence pl:oblem. Furthermore, we classify the 3-dimensional sub-Riemannian manifolds which are sub-homogeneous and show the relation to Cartan's list of homogeneous CR manifolds. Finally, we classify the 4-dimensional sub-Riemannian manifolds which are sub-symmetric. (1991): Primary, 53C15, 53C30; Secondary, 32C16. Mathematics Subject Classifications O. IntroductionSub-Riemannian geometry is concerned with the study of a smooth manifold M equipped with a metric defined only on a sub-bundle D of the tangent bundle TM, henceforth a sub-Riemannian manifold, and of the related geometric structures in analogy with Riemannian geometry. When 7) = TM we recover Riemannian geometry. Despite the similarities between the two geometries, there are new interesting phenomena occurring in sub-Riemannian geometry; see [11] for a survey and references.It is worth noting that this subject is of more than only formal interest since the several applications and connections range from control theory and mechanics with non-holonomic constraints, sub-Laplacians and hypoelliptic differential equations, to contact geometry and Cauchy-Riemann structures. Now we come to the subject of this paper. A sub-Riemannian homogeneous space is a sub-Riemannian manifold which admits a transitive group of subRiemannian isometries. A sub-Riemannian symmetric space ([13]) is a homogeneous sub-Riemannian manifold for which there is an involutive isometry hich is a central symmetry when restricted to the distribution.This work is divided into two parts. In the first part, we study 3-dimensional sub-Riemannian homogeneous spaces. We use a connection adapted to the subRiemannian structure (the pseudo-Hermitian connection of Webster [ 15] which was