Mathematical Sciences: Curvature and Topology

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The investigator will continue to study the relationship between the curvature and the topology of manifolds. In particular, he will continue to study the existence and concordance classification of positive scalar curvature metrics. He has recently shown that if a closed manifold M admits a positive scalar curvature metric, then the concordance classes of such metrics are in bijective correspondence to a certain bordism group which depends only on the fundamental group and the first two Stiefel- Whitney classes of M. Also, M admits a positive scalar curvature metric if and only if M represents zero in this bordism group. If the universal cover of M is spin, this bordism group maps to the real K-theory of the 'twisted' group C*-algebra of the fundamental group of M, the twist being determined by the first two Stiefel- Whitney classes of M. An optimistic hope is that this is in fact an isomorphism. The investigator intends to continue his study of the relations between positive Ricci curvature, elliptic genera and elliptic homology; in particular, to pursue his conjecture that the Witten genus of a positive Ricci curvature manifold M vanishes provided M is spin and its first Pontrjagin class is zero. In a different direction, he hopes to show that all the possible multiplicities of a Dupin hypersurface with four distinct eigenvalues are realized by the known examples, i.e. the homogeneous examples and the Clifford examples. Topology treats those properties of geometric objects which are not dependent upon distances or angles, which are so fundamental that they persist after stretching and bending an object short of tearing it. In a topological sense, a phonograph record and a wedding ring are the same, for no topological properties distinguish one from the other. In fact, each is topologically the same as a coffee cup. Their shapes are wildly different, and yet there are some geometric properties that all will have in common as a result of their common topology. The famous Gauss-Bonnet theorem, well-known in electromagnetic theory as well as differential geometry, requires that the total (Gaussian) curvature of any of their surfaces will be zero. Now curvature is a very non-topological property, and locally it can have any value whatsoever on one of these surfaces, but the values cannot be totally unrelated as one moves from point to point, for the theorem says they will all sum to zero if one weights them by elements of area. Prominent among the results this project seeks are modern variants of this wonderful theorem for manifolds of other dimensions and for other kinds of curvature.