#### 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 C^{1}-arcs but not with respect to its C^{2}-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.

#### Recommended Citation

Krupa, Matthew Gregory, "Differential Geometry of Projective Limits of Manifolds" (2016). *Mathematics Graduate Theses & Dissertations*. 47.

https://scholar.colorado.edu/math_gradetds/47