Metric Differential Geometry and Mathematical Gravity

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DMS-0104042Gregory G. Galloway Research under this award will be conducted in the areas ofLorentzian and Riemannian geometry, with applications to GeneralRelativity and String Theory. One of the specific aims of thisproject is to investigate the global structure of asymptoticallylocally anti-de Sitter spacetimes, especially in connection withthe AdS/CFT correspondence. We continue investigations into therelationship of the topology of an asymptotically locallyanti-de Sitter spacetime and that of its conformalboundary-at-infinity, and undertake a study of certaintopological and geometrical properties of higher dimensionalblack holes, and related objects arising in string theory,including certain solitions conjectured to be least energyconfigurations. Another aim of this project is to continueinvestigations into spacetime rigidity phenomena associated withthe occurence of timelike lines and null lines. Applications ofthe null splitting theorem will be considered, and a newapproach to well-known conjectures concerning the rigidity ofthe singularity theorems is proposed for study.Modern theories of gravity are geometrical in nature. Thegravitational field and other fields, black holes and relatedobjects, may be described and analyzed using geometric methods.In more general terms, this project is concerned with the studyof certain features of gravity of current scientific interestfrom this geometric point of view, utilizing the tools ofRiemannian geometry, a mathematical theory of space, andLorentzian geometry, a mathematical theory of spacetime. Thesetheories provide a method for studying the relationship amongthree fundamental aspects of the spacetime universe: curvature(i.e., the bending of space or spacetime), topology (i.e., theglobal shape and complexity of space or spacetime) and causalstructure (i.e., the large scale behavior of light rays andlight cones).