Smooth Only Torsion-free G_3 Holonomy, Normal Holonomy and Isoparametric Hypersurfaces in Spheres

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Abstract of NSF Proposal #0103838 Dr. Chi proposes to find smooth torsion-free G_3 connections that are notanalytic by exploring a certain system of partial differential equations induced from such connections. The affirmative solution to this work willopen a new avenue of research about these connections, for which all the known examples by far have been analytic. He also proposes to investigatethe classification problem of isoparametric hypersurfaces with fourprincipal curvatures in spheres by investigating the normal holonomy Lie algebra of a focal submanifold of the hypersurface, which contains theinformation about the 2nd fundamental form of the focal submanifold, whichis, by his study, crucial for the classification. If one sets off from the North Pole and travels with a compass along a loop, one will discover that at the end of the trip back at the North Polethe compass needle has turned a degree. This says mathematically that paralleltranslation (induced by a linear connection) of vectors along loops candetect whether the space is flat, which is fundamental to Einstein's Theory of Relativity. In Hermann Weyl's terms, the existence of inertia systems in the universe warrants that its linear connection istorsion-free. The project on torsion-free G_3 connections is a study of the rather anomalous nature of these connections among all exotic holonomies whose development has been participated by the PI. The notion ofisoparametric hypersurfaces was first proposed in connection with wave fronts in Euclidean space and recently the more general Dupin hypersurfaces were tied with integrable hamiltonian systems of hydrodynamic type. Shortly after their introduction the isoparametric hypersurfaces were all classified in Euclidean and hyperbolic spaces of all dimensions. The complete classification of such hypersurfaces in spheres of arbitrary dimensions has defied people's efforts for nearly six decades, despite many outstandingresults achieved in the past three decades. The PI's second project is to utilize the normal holonomy and the new approach of him, T. Cecil and G. Jensen toward the classification in the case of four principal curvatures.