Abstract

The scalar curvature of a Riemannian metric is interesting not only in analysis, geometry, and topology, but also in physics. Enlargeable Length-structures will be introduced and showed that it is a new obstruction to the existence of a Riemannian metric with positive scalar curvature (PSC-metric). Thus, the connected sum of a closed manifold with some of locally CAT(0)-manifolds carry not PSC-metrics. We will also show that almost non-positive curved manifolds carry no PSC-metrics. On the other hand, harmonic maps with condition C will be used to show the rigidity theorem about the scalar curvature.