Abstract. Let H be the n-dimensional hyperbolic space of constant sectional curvature −1 and let G be the identity component of the isometry group of H. We find all the G-invariant pseudo-Riemannian metrics on the space G n of oriented geodesics of H (modulo orientation preserving reparametrizations). We characterize the null, timeand space-like curves, providing a relationship between the geometries of G n and H. Moreover, we show that G 3 is Kähler and find an orthogonal almost complex structure on G 7 .2000 Mathematics Subject Classification. 53A55, 53C22, 53C35, 53C50, 53D25. Introduction. Let M be a Hadamard manifold (a complete simply connectedRiemannian manifold with nonpositive sectional curvature) of dimension n + 1. An oriented geodesic c of M is a complete connected totally geodesic oriented submanifold of M of dimension one. We may think of c as the equivalence class of unit speed geodesics γ : ‫ޒ‬ → M with image c such that {γ (t)} is a positive basis of T γ (t) c for all t. Let G = G(M) denote the space of all oriented geodesics of M. The space of geodesics of a manifold all of whose geodesics are periodic with the same length is studied with detail in [1]. The geometry of the space of oriented lines of Euclidean space is studied in [3, 9, 10].Let T 1 M be the unit tangent bundle of M and ξ the spray of M, that is, the vector field on

A smooth fibration of R 3 by oriented lines is given by a smooth unit vector field V on R 3 all of whose integral curves are straight lines. Such a fibration is said to be nondegenerate if dV vanishes only in the direction of V . Let L be the space of oriented lines of R 3 endowed with its canonical pseudo-Riemannian neutral metric. We characterize the nondegenerate smooth fibrations of R 3 by oriented lines as the closed (in the relative topology) definite connected surfaces in L. In particular, local conditions on L imply the existence of a global fibration. Besides, for any such fibration the base space is diffeomorphic to the open disc and the directions of the fibers form an open convex set of the two-sphere. We characterize as well, in a similar way, the smooth (possibly degenerate) fibrations.

For n = 3 or n = 7 let T n be the space of oriented lines in R n . In a previous article we characterized up to equivalence the metrics on T n which are invariant by the induced transitive action of a connected closed subgroup of the group of Euclidean motions (they exist only in such dimensions and are pseudo-Riemannian of split type) and described explicitly their geodesics. In this short note we present the geometric meaning of the latter being null, time-or space-like.On the other hand, it is well-known that T n is diffeomorphic to G (H n ), the space of all oriented geodesics of the n-dimensional hyperbolic space. For n = 3 and n = 7, we compute now a pseudo-Riemannian invariant of T n (involving its periodic geodesics) that will be useful to show that T n and G (H n ) are not isometrically equivalent, provided that the latter is endowed with any of the metrics which are invariant by the canonical action of the identity component of the isometry group of H.

In this article we apply a Bochner type formula to show that on a compact conformally flat riemannian manifold (or half-conformally flat in dimension 4) certain types of orthogonal almost-complex structures, if they exist, give the absolute minimum for the energy functional. We give a few examples when such minimizers exist, and in particular, we prove that the standard almost-complex structure on the round S 6 gives the absolute minimum for the energy. We also discuss the uniqueness of this minimum and the extension of these results to other orthogonal G-structures.

Let G be the Lie group of orientation preserving conformal diffeomorphisms of S n. Suppose that the sphere has initially a homogeneous distribution of mass and that the particles are allowed to move only in such a way that two configurations differ in an element of G. There is a Riemannian metric on G, which turns out to be not complete (in particular not invariant), satisfying that a smooth curve in G is a geodesic, if and only if (thought of as a conformal motion) it is force free, i.e., it is a critical point of the kinetic energy functional. We study the force free motions which can be described in terms of the Lie structure of the configuration space.