Mathematics

The problem of whether or not a smooth closed manifold X admits a Riemannian metric of positive scalar curvature (psc-metric) has been extensively studied and is now largely understood. In the case where X admits such a metric we may ask about the topology of the space Riem+(X) consisting of all psc-metrics on X. In general, very little is known about the topology of this space, even at the level of path-connectedness. One interesting problem concerns the topological notions of isotopy and concordance (pseudo-isotopy) when applied to this space. It is known that isotopic psc-metrics are concordant. Whether or not the converse holds is an open question. In this talk we will discuss a theorem which states that for a certain type of concordance, constructed using the surgery techniques of Gromov and Lawson, this converse does indeed hold.