Abstract
In Kupka et al. 2006 appears the Focal Stability Conjecture: the focal decomposition of the generic Riemann structure on a manifold M is stable under perturbations of the Riemann structure. In this paper, we prove the conjecture when M has dimension two, and there are no conjugate points.
Highlights
Let M be a compact, smooth manifold of dimension m, and let R = Rr be the space of Cr Riemann structures on M, equipped with the natural Cr topology, 2 ≤ r ≤ ∞
In Kupka et al 2006 we investigated the concept of focal stability with an eye to proving the following
Focal Stability Conjecture: the generic Riemann structure is focally stable. (Since R is an open subset of a complete metric space, genericity makes sense.) The main result of this paper concerns Riemann structures that have no conjugate points
Summary
(Since R is an open subset of a complete metric space, genericity makes sense.) The main result of this paper concerns Riemann structures that have no conjugate points. It is most stated for the open set N ⊂ R of Riemann structures on T M whose Gauss curvature is everywhere negative. For a compact manifold of dimension two, the generic Riemann structure g ∈ N is focally stable. Fix a compact smooth surface of genus s ≥ 2, such as the bitorus, and let H denote the nonempty set of Riemann structures on M with curvature everywhere equal to −1. For the generic g ∈ H, the focal decomposition of Tp M is stable with respect to perturbations of g within H
Sign-in/Register to view highlights & summary Sign-in/Register to access full text options 
Free) 
Talk to us
Join us for a 30 min session where you can share your feedback and ask us any queries you have
Schedule a call
