Professor Ruh will investigate some problems in Riemannian geometry by using techniques of deformation theory. In particular, he will study deformation of metrics, connections, and curvature. The general goal is to obtain optimal structures on manifolds in order to draw conclusions about the geometry of the manifold. Professor Ruh will consider the deformation of almost Lie groups into Lie groups and the deformation of general Riemannian metrics into metrics with prescribed curvature conditions. The foundation of deformation theory lies in partial differential equations and the heat equation in particular. One underlying theme which has turned out to be extremely useful in mathematics is that of deformation to a desired structure. In order to obtain an optimal structure it is often useful to construct a structure which has some of the desired properties but not all of them. One then seeks to "deform" the initial structure to one with the desired properties. The mechanism used to make the deformation should preserve at least those aspects of the desired structure which one was able to attain as a first step.