Mathematical Sciences: Differential Geometry and Differential Equations

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9505175 Croke The proposed research lies in the general area of Riemannian geometry. In particular, various dynamical properties of geodesic flow are to be investigated. Other topics include the study of periodic metrics: this portion of the project relates to the recent resolution of Hopf's conjecture on toral metrics. Hopf's conjecture states that a Riemannian metric on a torus without conjugate points must be flat. Riemannian geometry is currently a very active field of research in mathematics; length-minimizing paths, periodic behavior of the distance function, curvature and its relation to the global shape of the space are among the topics studied in this field. The familiar Euclidean geometry is a very special case of Riemannian geometry characterized by a total absence of curvature.