One of the natural generalizations of conformal structure on a two dimensional surface is a conformally flat structure on an n-manifold. In higher dimensions, not every manifold admits such a structure and it is a difficult problem to give a good classification of conformally flat manifolds. Recall that conformally flat Riemannian manifolds are manifolds whose metrics are locally conformally equivalent to the Euclidean metric. Kuiper [Kul, Ku2] was the first to study the global properties of these manifolds. He classified those compact conformally flat manifolds with abelian fundamental group. For dimensions greater than two, the Liouville theorem tells us that the conformal transformations of S" are determined locally and are given by M6bius transformations. Hence by a standard monodromy argument, a simply connected conformally flat manifold (with dimension> 3) has a conformal immersion into S" which is unique up to

A Riemannian or Finsler metric on a compact manifoldM gives rise to a length function on the free loop space M, whose critical points are the closed geodesics in the given metric. If X is a homology class on M, the “minimax” critical level cr(X) is a critical value. Let M be a sphere of dimension > 2, and fix a metric g and a coefficient field G. We prove that the limit as deg(X) goes to infinity of cr(X)/ deg(X) exists. We call this limit = (M, g,G) the global mean frequency of M. As a consequence we derive resonance statements for closed geodesics on spheres; in particular either all homology on  of sufficiently high degreee lies hanging on closed geodesics of mean frequency (length/average index) , or there is a sequence of infinitely many closed geodesics whose mean frequencies converge to . The proof uses the ChasSullivan product and results of Goresky-Hingston [7].

We describe a general geometrical construction of representations of fundamental groups of 3-manifolds into PU(2, 1) and eventually of spherical CR structures defined on those 3-manifolds. We construct branched spherical CR structures on the complement of the figure eight knot and the Whitehead link. They have discrete holonomies contained in PU(2, 1,Z[ω]) and PU(2, 1,Z[i]) respectively.

No abstract uploaded!

No abstract uploaded!