by Chow Ha Tak. / Thesis (M.Phil.)--Chinese University of Hong Kong, 1998. / Includes bibliographical references (leaves 96-97). / Abstract also in Chinese. / Chapter 1 --- The Laplacian on a Riemannian Manifold --- p.5 / Chapter 1.1 --- Riemannian metrics --- p.5 / Chapter 1.2 --- L2 Spaces of Functions and Forms --- p.6 / Chapter 1.3 --- The Laplacian on Functions and Forms --- p.8 / Chapter 2 --- Hodge Theory for Functions and Forms --- p.14 / Chapter 2.1 --- Analytic Preliminaries --- p.14 / Chapter 2.2 --- The Hodge Theorem for Functions --- p.20 / Chapter 2.3 --- The Hodge Theorem for Forms --- p.27 / Chapter 2.4 --- Regularity Results --- p.29 / Chapter 2.5 --- The Kernel of the Laplacian on Forms --- p.33 / Chapter 3 --- Fermion Calculus and Weitzenbock Formula --- p.36 / Chapter 3.1 --- The Levi-Civita Connection --- p.36 / Chapter 3.2 --- Fermion calculus --- p.39 / Chapter 3.3 --- "Weitzenbock Formula, Bochner Formula and Garding's Inequality" --- p.53 / Chapter 3.4 --- The Laplacian in Exponential Coordinates --- p.59 / Chapter 4 --- The Construction of the Heat Kernel --- p.63 / Chapter 4.1 --- Preliminary Results for the Heat Kernel --- p.63 / Chapter 4.2 --- Construction of the Heat Kernel --- p.66 / Chapter 4.2.1 --- Construction of the Parametrix --- p.66 / Chapter 4.2.2 --- The Heat Kernel for Functions --- p.70 / Chapter 4.2.3 --- The Heat Kernel for Forms --- p.76 / Chapter 4.3 --- The Asymptotics of the Heat Kernel --- p.77 / Chapter 5 --- The Heat Equation Approach to the Chern-Gauss- Bonnet Theorem --- p.82 / Chapter 5.1 --- The Heat Equation Approach --- p.82 / Chapter 5.2 --- Proof of the Chern-Gauss-Bonnet Theorem --- p.85 / Chapter 5.3 --- Introduction to Atiyah-Singer Index Theorem --- p.87 / Chapter 5.3.1 --- A Survey of Characteristic Forms --- p.87 / Chapter 5.3.2 --- The Hirzenbruch Signature Theorem --- p.90 / Chapter 5.3.3 --- The Atiyah-Singer Index Theorem --- p.93 / Bibliography / Notation index

In the first part of this dissertation, we give a new definition of a Laplace operator for Finsler metric as an average, with regard to an angle measure, of the second directional derivatives. This operator is elliptic, symmetric with respect to the Holmes-Thompson volume, and coincides with the usual Laplace--Beltrami operator when the Finsler metric is Riemannian. We compute explicit spectral data for some Katok-Ziller metrics. When the Finsler metric is negatively curved, we show, thanks to a result of Ancona that the Martin boundary is H��lder-homeomorphic to the visual boundary. This allow us to deduce the existence of harmonic measures and some ergodic preoperties. In the second part of this dissertation, we study Anosov flows in 3-manifolds, with leaf-spaces homeomorphic to .... When the manifold is hyperbolic, Thurston showed that the (un)stable foliations induces an "orthogonal" flow. We use this second flow to study isotopy class of periodic orbits of the Anosov flow and existence of embedded cylinders.

In the first part of this dissertation, we give a new definition of a Laplace operator for Finsler metric as an average, with regard to an angle measure, of the second directional derivatives. This operator is elliptic, symmetric with respect to the Holmes-Thompson volume, and coincides with the usual Laplace--Beltrami operator when the Finsler metric is Riemannian. We compute explicit spectral data for some Katok-Ziller metrics. When the Finsler metric is negatively curved, we show, thanks to a result of Ancona that the Martin boundary is H��lder-homeomorphic to the visual boundary. This allow us to deduce the existence of harmonic measures and some ergodic preoperties. In the second part of this dissertation, we study Anosov flows in 3-manifolds, with leaf-spaces homeomorphic to .... When the manifold is hyperbolic, Thurston showed that the (un)stable foliations induces an "orthogonal" flow. We use this second flow to study isotopy class of periodic orbits of the Anosov flow and existence of embedded cylinders.

Thesis (Ph. D.)--University of Missouri-Columbia, 1999. / Typescript. Vita. Includes bibliographical references (leaves 162-169). Also available on the Internet.

Thesis (Ph. D.)--University of Missouri-Columbia, 1999. / Typescript. Vita. Includes bibliographical references (leaves 162-169). Also available on the Internet.

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Thesis (Ph. D.)--University of Missouri-Columbia, 1997. / Typescript. Vita. Includes bibliographical references (leaves 76-79). Also available on the Internet.

Thesis (Ph. D.)--University of Missouri-Columbia, 1997. / Typescript. Vita. Includes bibliographical references (leaves 76-79). Also available on the Internet.

Thesis (Ph. D.)--University of California, Riverside, 2009. / Includes abstract. Includes bibliographical references (leaves 92-93). Issued in print and online. Available via ProQuest Digital Dissertations.

Thesis (Ph.D)--Mathematics, Georgia Institute of Technology, 2010. / Committee Chair: Harrell, Evans; Committee Member: Chow, Shui-Nee; Committee Member: Geronimo, Jeffrey; Committee Member: Kennedy, Brian; Committee Member: Loss, Michael. Part of the SMARTech Electronic Thesis and Dissertation Collection.