Contributions to the General Geometry of Paths

Citation

Mewborn, Aladuke Boyd (1940) Contributions to the General Geometry of Paths. Dissertation (Ph.D.), California Institute of Technology. doi:10.7907/j19v-6k32. https://resolver.caltech.edu/CaltechTHESIS:01272025-200108353

Abstract

By general geometry of paths is here meant the study of the properties of a space of undefined elements having a Hausdorff topology and coordinates in a Banach space, and having as fundamental element of structure a class of distinguished subsets called paths. The paths are defined locally in terms of their mappings under certain coordinate systems (N.c.s.) analogous to normal coordinates. The topologic properties of the paths and their mappings in N.c.s. are studied in §2.1, and it is shown that the map of the aggregate of paths through any point is itself an open set. In §2.2 a class of allowable coordinate systems is introduced and it is shown that a linear connection (which was not postulated) can be defined and from it a curvature form. These are shown to have their usual properties. It is further shown that the space is locally flat. In §2.3 a modification of Fréchet differentiation is introduced and used to define another linear connection and curvature which do not necessarily imply a locally flat space. §2.4 is devoted to examples illustrating the above in particular instances.

The second part (Chapter III) of the thesis consists of certain results extending the properties of the normal coordinate component of linear connection and the curvature form as studied by Michal and Hyers {Annali di Pisa l938} and of a more extended study of the variety of uniform continuity of the Fréchet differential defined and introduced by Michal and the writer {Proceedings of N. A. S. 1939}.

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