Esrafillan, Ebrahim (1975) Normal structures on manifolds. University of Southampton, Doctoral Thesis.
Abstract
Many structures on a topological m-manifold M may be defined by means of an atlas of local co-ordinate systems for which the coordinate systems belong to some pseudogroup () of transformations in the model space Rm. To any symmetric affine connexion V on M there is associated family of normal coordinate systems in a canonical way, via the exponential map. However, the co-ordinate transformations that occur within this family do not form a pseudogroup of transformations in Rm . On the other hand, normal co-ordinate systems are abundant in the sense that there is at least one such system based at every point of M. The purpose of this thesis is to modify the pseudogroup notion of structure to obtain characterisations of symmetric affine convexions, Riemannian structures, and holomorphic affine connexion. A brief discussion of the relationship. of this theory to the general geometry of paths and to Finsler geometry is also given.
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