On the Structure of Complete Manifolds with Positive Scalar Curvature

One of the greatest contributions of Rauch in differential geometry is his famous work on manifolds with positive curvature. His comparison theorems, which are needed for his proof of the pinching theorem, are fundamental for later developments in Riemannian geometry. His work initiated a systematic research developed by Klingenberg, Berger, Gromoll, Meyer, Cheeger, Gromov, Ruh, Shio-hama, Karcher, etc. This work depends heavily on how a length-minimizing geodesic behaves under the influence of the curvature. Since geodesic is one-dimensional, the information we need from the curvature tensor is the curvature of the two planes which are tangential to the geodesic. This means that we need to know the behavior of the sectional curvature or the Ricci curvature of the manifold. Therefore, it seems very unlikely that arguments based only on length-minimizing geodesics can be used to deal with problems related to scalar curvature. The problem of scalar curvature, however, has drawn a lot of attention of the differential geometers in the late sixties and the seventies, partly because of its interest in general relativity.