On the structure of manifolds with positive scalar curvature

Abstract

Publisher Summary This chapter discusses some recent results by Richard Schoen and Shing-Tung Yau on the structure of manifolds with positive scalar curvature. The chapter presents theorems which are felt to provide a more complete picture of manifolds with positive scalar curvature: (1) let M be a compact four-dimensional manifold with positive scalar curvature. Then there exists no continuous map with non-zero degree onto a compact K(π,1). (2) Let M be n-dimensional complete manifold with non-negative scalar curvature. Then any conformed immersion of M into Sn is one to one. In particular, any complete conformally flat manifold with non-negative scalar curvature is the quotient of a domain in Sn by a discrete subgroup of the conformal group. (3.) Let M be a compact manifold whose fundamental group is not of exponential growth. Then unless M is covered by Sn, Sn–1 x S1 or the torus, M admits no conformally flat structure.