Curvature, geodesics, and rigidity

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AbstractAward: DMS 1207655, Principal Investigator: Benjamin SchmidtThe principal investigator will study the rigidity and deformation properties of curvature and geodesics in Riemannian manifolds. A specific goal is to classify the Riemannian manifolds having all sectional curvatures greater than or equal to one and with positive spherical rank. Conjecturally, these manifolds are isometric to compact rank one symmetric spaces. Another broad goal is to investigate the structure of constant vector curvature manifolds in dimensions four and higher.Objects in nature with a high degree of symmetry are quite often structurally rigid. When highly symmetric spaces are altered, numerical measurements of the way light propagates and the way space bends can be used to quantify changes. The principal investigator will study the influence of Riemannian curvature, a numerical measurement of bending, on the way that highly symmetric spaces can and cannot be perturbed.