Thesis (doctoral)--Rheinische Friedrich-Wilhelms-Universität Bonn, 2000. / Includes bibliographical references (p. 41-42).

Thesis (Dr. rer. nat.)--Rheinische Friedrich-Wilhelms-Universität Bonn, 2001. / Includes bibliographical references (p. 87-90).

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published_or_final_version / Mathematics / Doctoral / Doctor of Philosophy

by Lei Ka Keung. / Thesis (M.Phil.)--Chinese University of Hong Kong, 1999. / Includes bibliographical references (leaves 70-71). / Abstracts in English and Chinese. / Chapter 0 --- Introduction --- p.5 / Chapter 1 --- Dirichlet Problem at infinity --- p.9 / Chapter 1.1 --- The Geometric Boundary --- p.9 / Chapter 1.2 --- Dirichlet Problem --- p.15 / Chapter 2 --- The Martin Boundary --- p.29 / Chapter 2.1 --- The Martin Metric --- p.30 / Chapter 2.2 --- The Representation Formula --- p.31 / Chapter 2.3 --- Uniqueness of Representation --- p.36 / Chapter 3 --- The Geometric boundary and the Martin boundary --- p.42 / Chapter 3.1 --- Estimates for harmonic functions in cones --- p.42 / Chapter 3.2 --- A Harnack Inequality at Infinity --- p.49 / Chapter 3.3 --- The kernel function --- p.54 / Chapter 3.4 --- The Main Theorem --- p.55 / Chapter 4 --- Positive Harmonic Functions on Product of Manifolds --- p.61 / Chapter 4.1 --- Splitting Theorem --- p.61 / Chapter 4.2 --- Riemannian Halfspace and the parabolic Martin boundary --- p.62 / Chapter 4.3 --- Splitting of parabolic Martin kernels --- p.63 / Chapter 4.4 --- Proof of theorem 4.1 --- p.66 / Bibliography

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

Lau Chi Hin. / Thesis (M.Phil.)--Chinese University of Hong Kong, 1997. / Includes bibliographical references (leaves 88-91). / Chapter 1 --- Introduction --- p.4 / Chapter 1.1 --- Rough Isometries --- p.4 / Chapter 1.2 --- Discrete approximation of Riemannian manifolds --- p.8 / Chapter 2 --- Basic Properties of Rough Isometries --- p.19 / Chapter 2.1 --- Volume growth rate --- p.19 / Chapter 2.2 --- Sobolev Inequalities --- p.25 / Chapter 2.3 --- Poincare Inequality --- p.32 / Chapter 3 --- Parabolic Harnack Inequality --- p.39 / Chapter 3.1 --- Parabolic Harnack Inequality --- p.39 / Chapter 4 --- Parabolicity and Liouville Dp-property --- p.58 / Chapter 4.1 --- Parabolicity --- p.58 / Chapter 4.2 --- Liouville Dp-property --- p.67

We present an extension of Geodesics in Lorentzian Manifolds (Semi-Riemannian Manifolds or pseudo-Riemannian Manifolds ). A geodesic on a Riemannian manifold is, locally, a length minimizing curve. On the other hand, geodesics in Lorentzian manifolds can be viewed as a distance between ``events''. They are no longer distance minimizing (instead, some are distance maximizing) and our goal is to illustrate over what time parameter geodesics in Lorentzian manifolds are defined. If all geodesics in timelike or spacelike or lightlike are defined for infinite time, then the manifold is called ``geodesically complete'', or simply, ``complete''. It is easy to show that the magnitude of a geodesic is constant, so one can characterize geodesics in terms of their causal character: if this magnitude is negative, the geodesic is called timelike. If this magnitude is positive, then it is spacelike. If this magnitude is 0, then it is called lightlike or null. Geodesic completeness can be considered by only considering one causal character to produce the notions of spacelike complete, timelike complete, and null or lightlike complete. We illustrate that some of the notions are inequivalent.

In this thesis, we study two problems focusing on the interplay between geometric properties of differential equations and their invariants. / For the first project, we study the validity of the principle of symmetric criticality (PSC) in the context of invariant gauge fields over the four-dimensional non-reductive pseudo-Riemannian homogeneous spaces G/K recently classified by Fels & Renner (2006). Given H compact semi-simple, classification results are obtained for principal H-bundles over G/K admitting: (1) a G-action (by bundle automorphisms) projecting to left multiplication on the base, and (2) at least one G-invariant connection. There are two cases which admit nontrivial examples of such bundles and all G-invariant connections on these bundles are Yang--Mills. Using the invariant criteria obtained by Anderson--Fels--Torre, the validity of PSC is investigated for the bundle of connections and is shown to fail for all but one of the Fels--Renner cases. This failure arises from degeneracy of the scalar product on pseudo-tensorial forms restricted to the space of symmetric variations of an invariant connection. In the exceptional case where PSC is valid, there is a unique G-invariant connection which is moreover universal, i.e. it is a solution of the Euler--Lagrange equations associated to any G-invariant Lagrangian on the bundle of connections. This solution is a canonical connection associated with a weaker notion of reductivity which we introduce. / The second project is a study of the contact geometry of scalar second order hyperbolic equations in the plane of generic type. Following a derivation of parametrized contact-invariants to distinguish Monge--Ampere (class 6-6), Goursat (class 6-7) and generic (class 7-7) hyperbolic equations, we use Cartan's equivalence method to study the generic case. An intriguing feature of this class of equations is that every generic hyperbolic equation admits at most a nine-dimensional contact symmetry algebra. The nine-dimensional bound is sharp: normal forms for the contact-equivalence classes of these maximally symmetric equations are derived and explicit symmetry algebras are presented. Moreover, all such equations are Darboux integrable. An enumeration of several submaximally symmetric (eight and seven-dimensional) structures is also given.

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