Positively and Non-Positively Curved Manifolds

Abstract

Award: DMS-0504534

Principal Investigator: Xiaochun Rong

Positive curvature and non-positive curvature have been frequent

subjects in Riemannian geometry, where mathematics from several

disciplines interact; such as differential geometry, analysis and

partial differential equations, transformation group theory and

topology. The PI is pursuing a research program concerning some

basic problems in these areas: 1. Interplay between positive

curvature (with Abelian symmetry) and topology. This project is

amplified by the amazing fact that among the manifolds of the

same dimension whose sectional curvature is between two positive

constants, all but finitely many admit (large) Abelian symmetry.

2. The semi-rigidity of the moduli space of non-positively curved

metrics on a closed manifold.

Mathematics is the foundation of natural sciences, and

differential geometry and Riemannian geometry is one of the most

important branches of mathematics. The PI is pursuing solving

some basic problems in this field which would have a broad

intellectual impact. PI will continue to actively pursue

collaborations with other mathematicians in US and abroad and to

speak at several national and international meetings a year.

The PI organized Riemannian geometry seminars in summers of

2001-2004, which were designed to provide a forum for the

dissemination of new ideas, in particular for graduate students.

The PI has had three papers with his graduate students and

postdoctoral fellow which were the products of the summer

seminars. The PI will continue the Riemannian geometry summer

seminars in the next three years. The PI has been writing two

advance graduate textbooks for three years (titled: ``The

convergence and collapsing theory in Riemannian geometry'' and

``Positive curvature, symmetry and topology''). He wishes to

finish the two books in 2-3 years. This will not only benefit

graduate students and researchers from these areas (due to the

lack of graduate textbooks in these topics) but also help to

advertise and disseminate Riemannian Geometry to students and

mathematicians in other related fields, by giving them a quick

picture of some of the research directions of Riemannian

geometry, and by showing them how the subject vitally interacts

with differential geometry, analysis and PDE, compact

transformation group theory and topology.