Jon Rosenberg
University of Maryland
Title
Positive Scalar Curvature and a Generalization
Abstract
The scalar curvature function is the simplest curvature invariant of a (closed) Riemannian manifold. In dimension 2, it is just twice the Gaussian curvature, and if it doesn't change sign, it must have the same sign as the Euler characteristic. But in dimensions 3 and up, there is no obstruction to negative scalar curvature, though there are many obstructions to positive scalar curvature, mostly related to the (generalized) index theory of the Dirac operator on spinors. We review this theory and discuss a "generalized scalar curvature" on spin^c manifolds, which is connected in a similar way to the (generalized) index theory of the spin^c Dirac operator. This is joint work with Boris Botvinnik and Paolo Piazza.