We show that a certain class of manifolds admit metrics of positive Ricci curvature. This class includes many exotic spheres, including all homotopy spheres which represent elements of bP2n- §o.In this paper we investigate the Ricci curvature of a certain class of manifolds which includes many exotic spheres. In particular we will be concerned with constructing metrics of positive Ricci curvature. Our main result is as follows:Theorem 2.1. Homotopy spheres which bound parallelisable manifolds admit metrics of positive Ricci curvature.

We show that the moduli space of Ricci positive metrics on a certain family of homotopy spheres has infinitely many components. 53C20

No abstract

We consider performing surgery on Riemannian manifolds with positive Ricci curvature. We nd conditions under which the resulting manifold also admits a positive Ricci curvature metric. These conditions involve dimension and the form taken by the metric in a neighbourhood of the surgery.

Abstract. We consider Ptolemy's inequality in a metric space setting. It is not hard to see that CAT(0) spaces satisfy this inequality. Although the converse is not true in full generality, we show that if our Ptolemaic space is either a Riemannian or Finsler manifold, then it must also be CAT(0). Ptolemy's inequality is closely related to inversions of metric spaces. We exploit this link to establish a new characterization of Euclidean space amongst all Riemannian manifolds.2000 Mathematics Subject Classification. 53C20, 53C60, 51F99.