Greene’s criterion for twist mappings asserts the existence of smooth invariant circles with preassigned rotation number if and only if the periodic trajectories with frequency approaching that of the quasi-periodic orbit are linearly stable. We formulate an extension of this criterion for conformally symplectic systems in any dimension and prove one direction of the implication, namely that if...

Usual inflation is realized with a slow rolling scalar field minimally coupled to gravity. In contrast, we consider dynamics of a scalar with a flat effective potential, conformally coupled to gravity. Surprisingly, it contains an attractor inflationary solution with the rapidly rolling inflaton field. We discuss models with the conformal inflaton with a flat potential (including hybrid inflati...

We show that complete conformally flat manifolds of dimension n > 3 with nonnegative Ricci curvature enjoy nice rigidity properties: they are either flat, or locally isometric to a product of a sphere and a line, or are globally conformally equivalent to R n or a spherical spaceform Sn/Γ. This extends previous results due to Q.-M. Cheng and B.-L. Chen and X.-P. Zhu. In this note, we study compl...