Global Riemannian Geometry

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AbstractAward: DMS-9971045Principal Investigator: Peter PetersenThe PI proposes to investigate the relationship between geometricand topological invariants of Riemannian manifolds. The study canbe divided up into at least two parts. There are generalfiniteness results where one simply obtains some bounds ontopological invariants from certain bounds on the geometry. ThePI proposes to work on several finiteness phenomena for manifoldswith integral bounds for the curvature. The isospectralcompactness problem for 3-manifolds will also beinvestigated. There are more specific situations where one cancompute very specific topological invariants from certainspecific geometric information. Under this heading one alsostudies various pinching phenomena, where the goal is tounderstand the specific topological type of a Riemannian manifoldwith restrictions on some of its geometric invariants. The PIproposes to study Gauss-Bonnet type theorems for noncompact4-manifolds and their relationship with the study of Euclideanquantum gravity. This is well studied in the setting of compactmanifolds, but much less understood for noncompact manifolds. Itis also proposed to investigate various pinching phenomena bothfor curvature and eigenvalues. Finally the PI will undertake astudy of the Gromoll-Meyer sphere and try to determine whether itadmits metrics with positive sectional curvature.The main goal of the PI's research centers on understanding therelationship between on one hand geometric quantities such assize, volume and bending and on the other hand the complexity ofthe objects. Complexity is often measured by how many holes theobject has, a jungle gym is more complex than a doughnut which inturn is more complex than a ball. Depending on how good onesgeometric information is one can obtain very general bounds onthe complexity, but it is also possible to sometimes completelydetermine what the object is. For instance, only a ball has theproperty that it bends in the same manner everywhere. There arealso very interesting relationships between the geometricquantities themselves. For instance, any object that bends in thesame direction everywhere is bounded in size and volume. Suchresults also have counterparts in Einstein's theory of generalrelativity. There such results are related to the study of theBig Bang, black holes, and the ultimate fate on the universe. Onetechnical item the PI studies is the possibility that onesmeasurement of bending does not have to be very precise. In otherwords, slightly vision impaired people's ideas of bending are asgood as those of people with perfect eye sight.