No Arabic abstract
We prove that if a closed, smooth, simply-connected 4-manifold with a circle action admits an almost non-negatively curved sequence of invariant Riemannian metrics, then it also admits a non-negatively curved Riemannian metric invariant with respect to the same action. The same is shown for torus actions of higher rank, giving a classification of closed, smooth, simply-connected 4-manifolds of almost non-negative curvature under the assumption of torus symmetry.
Motivated by a recent groundbreaking work of Ontaneda, we describe a sizable class of closed manifolds such that the product of each manifold in the class with the real line admits a complete metric of bounded negative sectional curvature which is an exponentially warped near one end and has finite volume near the other end.
Positively curved Alexandrov spaces of dimension 4 with an isometric circle action are classified up to equivariant homeomorphism, subject to a certain additional condition on the infinitesimal geometry near fixed points which we conjecture is always satisfied. As a corollary, positively curved Riemannian orbifolds of dimension 4 with an isometric circle action are also classified.
We describe the action of the fundamental group of a closed Finsler surface of negative curvature on the geodesics in the universal covering in terms of a flat symplectic connection and consider the first order deformation theory about a hyperbolic metric. A construction of O.Biquard yields a family of metrics which give nontrivial deformations of the holonomy, extending the representation of the fundamental group from SL(2,R) into the group of Hamiltonian diffeomorphisms of S^1 x R, and producing an infinite-dimensional version of Teichmuller space which contains the classical one.
Let $rho_0$ be an action of a Lie group on a manifold with boundary that is transitive on the interior. We study the set of actions that are topologically conjugate to $rho_0$, up to smooth or analytic change of coordinates. We show that in many cases, including the compactifications of negatively curved symmetric spaces, this set is infinite.