Mathematical Sciences: Geometry of Compact Nonpositively Curved Manifolds of Rank 1

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The principal investigator will continue his study of the geometry of complete Riemannian manifolds of finite volume and nonpositive curvature. Particular emphasis will be given to manifolds of rank 1, a condition implied by but weaker than strictly negative sectional curvature. The research project will include the investigation of both geometric characterizations of symmetric spaces with strictly negative sectional curvature, and algebraic properties of the fundamental groups of the base manifold. Mathematicians have studied generalized surfaces, called manifolds, for more than a century. A so called "Riemannian manifold" may have a well-defined area or volume. For example, as we usually imagine them, spheres and tori have finite surface area. The principal investigator will study manifolds of nonpositive curvature. One example of such a manifold would be the two-holed torus. This line of research has been particularly active in the 1980's due in large part to earlier pioneering efforts of the principal investigator.