Marcos Salvai

ON THE GEOMETRY OF THE SPACE OF ORIENTED LINES OF THE HYPERBOLIC SPACE

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 of oriented geodesics of H (modulo orientation preserving reparametrizations). We characterize the null, time- and space-like curves, providing a relationship between the geometries of and H . Moreover, we show that is Kahler and find an orthogonal almost complex structure on .

On the dynamics of a rigid body in the hyperbolic space

Abstract Let H be the three-dimensional hyperbolic space and let G be the identity component of the isometry group of H . It is known that some aspects of the dynamics of a rigid body in H contrast strongly with the Euclidean case, due to the lack of a subgroup of translations in G . We present the subject in the context of homogeneous Riemannian geometry, finding the metrics on G naturally associated with extended rigid bodies in H . We concentrate on the concept of dynamical center, characterizing it in various ways.

On the Laplace and Complex Length Spectra of Locally Symmetric Spaces of Negative Curvature

We prove that if two compact oriented locally symmetric manifolds of negative curvature are locally isometric and have the same complex length spectrum, then they are strongly Laplace isospectral.

Force free conformal motions of the sphere

Abstract Let G be the Lie group of orientation preserving conformal diffeomorphisms ofxa0Sn. 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 ofxa0G. There is a Riemannian metric onxa0G, 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.

Solenoidal unit vector fields with minimum energy

In this article we give examples of compact manifolds P admitting homogeneous Riemannian metrics (depending on a real parameter) and unit vector fields V , which are critical for the total bending functional and have minimum energy among all solenoidal (that is, divergence free) unit vector fields. The family of manifolds P , introduced by Gary Jensen to provide new examples of Einstein metrics, consists of total spaces of principal bundles over symmetric spaces, and includes for instance Berger spheres. Those of Jensen’s examples involving classical groups (and one exceptional) are made explicit for instance as Grassmannor Stiefel-like manifolds.

AFFINE MAXIMAL TORUS FIBRATIONS OF A COMPACT LIE GROUP

By a generalization of the method developed by Gluck and Warner to characterize the oriented great circle fibrations of the three-sphere, we give, for any compact connected semisimple Lie group G, a general procedure to obtain the continuous fibrations of G by Weyl-oriented affine maximal tori, find conditions for smoothness and provide infinite dimensional spaces of concrete examples.

Generalized geometric structures on complex and symplectic manifolds

On a smooth manifold

Polar factorization of conformal and projective maps of the sphere in the sense of optimal mass transport

M