Date of Award

Spring 1-1-2017

Document Type

Dissertation

Degree Name

Doctor of Philosophy (PhD)

Department

Mathematics

First Advisor

Stephen C. Preston

Second Advisor

Jeanne N. Clelland

Third Advisor

Magdalena Czubak

Fourth Advisor

Jem Corcoran

Fifth Advisor

Jordan Watts

Abstract

In this thesis we investigate some geometric properties of the contactomorphism group Dθ(M) of a compact, oriented, contact manifold of odd dimension. We endow this group with two different weak Riemannain metrics, the L2 and the H1, and investigate properties such as curvature, geodesic flows, conjugate points, and exponential maps. This group has applications to fluid mechanics as it is the natural generalization to D(S1), the diffeomorphism group of the circle, which is already a heavily studied object.

We end with providing numerical evidence that the sectional curvature of the group D(S1)/S1, the diffeomorphism group of the circle modulo its rotations, given the H1/2 metric is non-negative. We leave the proof, or search for counterexample, as open as well as discuss other possible topics of study which can follow as a result of this thesis.

Included in

Mathematics Commons

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