Positively and Non-Positively Curved Manifolds

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AbstractAward: DMS-0504534Principal Investigator: Xiaochun RongPositive curvature and non-positive curvature have been frequentsubjects in Riemannian geometry, where mathematics from severaldisciplines interact; such as differential geometry, analysis andpartial differential equations, transformation group theory andtopology. The PI is pursuing a research program concerning somebasic problems in these areas: 1. Interplay between positivecurvature (with Abelian symmetry) and topology. This project isamplified by the amazing fact that among the manifolds of thesame dimension whose sectional curvature is between two positiveconstants, all but finitely many admit (large) Abelian symmetry.2. The semi-rigidity of the moduli space of non-positively curvedmetrics on a closed manifold.Mathematics is the foundation of natural sciences, anddifferential geometry and Riemannian geometry is one of the mostimportant branches of mathematics. The PI is pursuing solvingsome basic problems in this field which would have a broadintellectual impact. PI will continue to actively pursuecollaborations with other mathematicians in US and abroad and tospeak at several national and international meetings a year.The PI organized Riemannian geometry seminars in summers of2001-2004, which were designed to provide a forum for thedissemination of new ideas, in particular for graduate students.The PI has had three papers with his graduate students andpostdoctoral fellow which were the products of the summerseminars. The PI will continue the Riemannian geometry summerseminars in the next three years. The PI has been writing twoadvance graduate textbooks for three years (titled: ``Theconvergence and collapsing theory in Riemannian geometry'' and``Positive curvature, symmetry and topology''). He wishes tofinish the two books in 2-3 years. This will not only benefitgraduate students and researchers from these areas (due to thelack of graduate textbooks in these topics) but also help toadvertise and disseminate Riemannian Geometry to students andmathematicians in other related fields, by giving them a quickpicture of some of the research directions of Riemanniangeometry, and by showing them how the subject vitally interactswith differential geometry, analysis and PDE, compacttransformation group theory and topology.