1's) representing the points of the primary space fails to be complete; but it can be completed to give a 4- dimensional family, effecting a unique embedding of the original 3-fold in a 4-fold with conformal structure, of which the conformal curvature is selfdual, in such a way that the induced conformal structure is the original one and such that the conformal torsion is related to the second conformal fundamental form of the hypersurface in a canonical linear fashion. In any case, the small deformations of the complex structure of the space of null geodesies correspond precisely to the small deformations of the conformal connexion. It is shown that a space of torsion-free null geodesies admits a holomorphic contact structure, and that conversely, for n ≥ 4, the admission of a contact structure forces the conformal torsion to vanish; for n=3, the contact form constructs automatically a unique metric on the ambient 4-fold in the previously constructed self-dual conformal class which solves Einstein's equations with cosmological constant 1 and blov/s up on the 3-fold, which is a general umbilic hypersurface. These results are in turn used to show that a real-analytic 3-fold with real-analytic positive definite conformal structure and a real-analytic symmetric form of conformal weight 1 can be embedded (in a locally unique fashion) in a real-analytic 4-fold with positive-definite conformal structure for which the conformal curvature is self-dual in such a way as to realize the given structures as the first and second conformal fundamental forms of the hypersurface; and it is shown that a real analytic 3-fold with positivedefinite conformal bounds a locally unique positive-definite solution of Einstein's equations with cosmological constant -1 as its umbilic conformal infinity. By contrast, these results fail when "real-analytic" is replaced by "smooth".

Thesis (M. S.)--Electrical and Computer Engineering, Georgia Institute of Technology, 2009. / Committee Chair: Tannenbaum, Allen; Committee Member: Barnes, Christopher F.; Committee Member: Niethammer, Marc; Committee Member: Shamma, Jeff; Committee Member: Vela, Patricio.

Thesis (Ph. D.)--University of Texas at Austin, 2002. / Vita. Includes bibliographical references. Available also from UMI Company.

Thesis (M.Phil.)--Hong Kong University of Science and Technology, 2006. / Includes bibliographical references (leaves 64-68). Also available in electronic version.

Thesis (Ph. D. in Mathematics)--Vanderbilt University, May 2006. / Title from title screen. Includes bibliographical references.

Thesis (Ph.D.)--University of Waikato, 2006. / Includes bibliographical references (leaves 333-336)

No description available.

<P> In this thesis, we mainly study geodesics on various two dimensional surfaces. All the background material needed throughout the thesis is provided, including an explanation of the theory of geodesics. We will calculate geodesics using two numerical methods: Euler's method and Runge-Kutta method of fourth order. Using Maple, we will test the accuracy of the numerical methods on a test case surface, the Poincare half plane. Later, we proceed to investigate several interesting surfaces by numerically calculating geodesics. From the investigated surfaces, we will draw similarities between the human cerebral cortex and certain surfaces. The human cerebral cortex is the most intensely studied part of the brain and it is believe that their exists a relation between the function and structure of the cortex. Geodesic analysis can possibly be an essential tool in better understanding the cortical surface as it is in many disciplines of science to understand the nature of physical based problems. </P> / Thesis / Master of Science (MSc)

Given a hyperbolic surface, the set of all closed geodesics whose length is minimal form a graph on the surface, in fact a so called fat graph, which we call the systolic graph. The central question that we study in this thesis is: which fat graphs are systolic graphs for some surface -we call such graphs admissible. This is motivated in part by the observation that we can naturally decompose the moduli space of hyperbolic surfaces based on the associated systolic graphs. A systolic graph has a metric on it, so that all cycles on the graph that correspond to geodesics are of the same length and all other cycles have length greater than these. This can be formulated as a simple condition in terms of equations and inequations for sums of lengths of edges. We call this combinatorial admissibility. Our first main result is that admissibility is equivalent to combinatorial admissibility. This is proved using properties of negative curvature, specifically that polygonal curves with long enough sides, in terms of a lower bound on the angles, are close to geodesics. Using the above result, it is easy to see that a subgraph of an admissible graph is admissible. Hence it suffices to characterize minimal non-admissible fat graphs. Another major result of this thesis is that there are infinitely many minimal non-admissible fat graphs (in contrast, for instance, to the classical result that there are only two minimal non-planar graphs).

How many equivalence classes of geodesic rays does a graph contain? How many bounded automorphisms does a planar graph have? Neimayer and Watkins studied these two questions and answered them for a certain class of graphs. Using the concept of excess of a vertex, the class of graphs that Neimayer and Watkins studied are extended to include graphs with positive excess at each vertex. The results of this paper show that there are an uncountable number of geodesic fibers for graphs in this extended class and that for any graph in this extended class the only bounded automorphism is the identity automorphism.