Date of Award

Spring 1-1-2016

Document Type

Dissertation

Degree Name

Doctor of Philosophy (PhD)

Department

Mathematics

First Advisor

Markus J. Pflaum

Second Advisor

Jeanne N. Clelland

Third Advisor

Martin E. Walter

Fourth Advisor

Jonathan Wise

Fifth Advisor

Alexander Gorokhovsky

Abstract

The nascent theory of projective limits of manifolds in the category of locally R-ringed spaces is expanded and generalizations of differential geometric constructions, definitions, and theorems are developed. After a thorough introduction to limits of topological spaces, the study of limits of smooth projective systems, called promanifolds, commences with the definitions of the tangent bundle and the study of locally cylindrical maps. Smooth immersions, submersions, embeddings, and smooth maps of constant rank are defined, their theories developed, and counter examples showing that the inverse function theorem may fail for promanifolds are provided along with potential substitutes. Subsets of promanifolds of measure 0 are defined and a generalization of Sard's theorem for promanifolds is proven. A Whitney embedding theorem for promanifolds is given and a partial uniqueness result for integral curves of smooth vector fields on promanifolds is found. It is shown that a smooth manifold of dimension greater than one has the final topology with respect to its set of C1-arcs but not with respect to its C2-arcs and that a particular class of promanifolds, called monotone promanifolds, have the final topology with respect to a class of smooth topological embeddings of compact intervals termed smooth almost arcs.

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