Chidley, Jon Thomas Wood (1974) Lines of constant geodesic curvature and the k-flows. PhD thesis, University of Warwick.

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Abstract

Much work has been done on the geodesics of a Riemannian manifold and the flow it induces on the unit tangent bundle, particularly on manifolds of negative curvature. It is the purpose of this thesis to extend this work to a more general case, considering those curves of a manifold with constant geodesic curvature •

In the first chapter we define these k-lines and develop some ideas about their geometry, contrasting and comparing them with the geodesics. We show how they give flows on T1M, develop a variational theory for them and show how matrix methods may be used to solve the variational equations.

In the second chapter we investigate a particular property, that of Anosovity, well known and documented in the geodesic case. For a particular class of k-flows we solve the matrix variation equations, giving necessary and sufficient conditions for Anosovity in terms of the geodesic normals and curvatures. On manifolds of negative curvature we assign an ‘Anosov’ number to each flow such that if it is less than zero the flow is Anosov. We end by considering families of k-flows and for a class of flows indicate the topological similarities between members of the class and in particular the geodesic flow. We end with some conjectures and ideas for future work.

Also included are two appendices. The first catalogues the results we need on the metric of the unit tangent bundle, and we extend this to the Frame bundles defining k-flows on these spaces and investigating the volume preserving properties. The second derives k-lines and flow’ from a classical mechanics point at view and justifies their study as phenomena arising from physical situations.

Item Type:	Thesis or Dissertation (PhD)
Subjects:	Q Science > QA Mathematics
Library of Congress Subject Headings (LCSH):	Riemannian manifolds, K-theory, Curvature
Official Date:	1974
Dates:	Date Event 1974 Submitted
Institution:	University of Warwick
Theses Department:	Mathematics Institute
Thesis Type:	PhD
Publication Status:	Unpublished
Supervisor(s)/Advisor:	Walters, Peter, 1943-
Sponsors:	Science Research Council (Great Britain)
Extent:	175 leaves
Language:	eng

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