Mathematical Sciences: Riemannian Geometry, Geometirc Complex Analysis, and Applications

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9626652 Greene The proposed research deals with Riemannian as well as Kahler geometry. Orbifolds, spaces locally isometric to Riemannian manifolds quotiented by finite groups of isometries, are to be investigated with a view towards understanding the isometry groups of nonsingular Riemannian manifolds. Estimation of numbers of topological types for Riemannian manifolds satisfying certain geometric conditions will also be pursued. In Kahler geometry, the investigator will study how curvature behavior at large distances controls the overall complex structure of the space. This is the higher dimensional analog of distinguishing between hyperbolic and parabolic Riemann surfaces. Much of this project has to do with how the local geometry, i.e., the geometric structure in small neighborhoods of each point, influences or even controls the global geometry and topology of a space, meaning the overall shape and nature of the space. This has a direct practical application in the construction of large-scale curved steel structure such as ship hulls. The deformation of steel plates by heat treatment into pre-specified curved shapes, for example, can be understood using this approach.