Discovering Geometric and Topological Properties of Ellipsoids by Curvatures

Abstract

Aims/ Objectives: We are interested in discovering the geometric, topological and physical properties of ellipsoids by analyzing curvature properties on ellipsoids. We begin with studying ellipsoids as a starting point. Our aim is to find a way to study geometric, topological and physical properties from the analytic curvature properties for convex hyper-surfaces in the general setting. Study Design: Multiple-discipline study between Differential Geometry, Topology and Mathematical Physics. Place and Duration of Study: Department of Mathematics (Borough of Manhattan Community College-The City University of New York), Department of Mathematics (University of Oklahoma), Department of Mathematics and Statistics (University of West Florida), and Department of Mathematics (Central Michigan University), between January 2014 and February 2015. Methodology: Calculating curvatures of a surface is now at the threshold of a better understanding regarding geometric, topological and physical properties on a surface. In order to calculate various curvatures, we demonstrate the way to compute the second fundamental form associated with curvatures by extending the calculation method from spheres to ellipsoids. Results: Just as curvatures of a sphere are determined by its radius, curvatures of an ellipsoid are determined by its longest axis and its shortest axis. On an ellipsoid, the value of the ratio of its longest axis to its shortest axis is also a critical index to characterize its geometric, topological and physical behaviors. Conclusion: Our results on ellipsoids are extensions or generalizations of results of Lawson-Simons, Wei, and Simons on spheres, and Kobayashi-Ohnita-Takeuchi on an ellipsoid with “one variable”. Methods and research findings in this paper can lead to future research on convex hyper-surfaces.