by Yiu Chun Chit. / Thesis (M.Phil.)--Chinese University of Hong Kong, 1997. / Includes bibliographical references (leaves 81-82). / Chapter 1 --- Introduction --- p.7 / Chapter 1.1 --- Riemannian Manifolds --- p.7 / Chapter 1.1.1 --- Completeness --- p.8 / Chapter 1.1.2 --- Curvature tensor --- p.9 / Chapter 1.1.3 --- Holonomy --- p.11 / Chapter 1.2 --- Simply-connected Manifold of Nonpositive Sectional Curvature --- p.11 / Chapter 1.2.1 --- Topological structure --- p.12 / Chapter 1.2.2 --- Basic geometric properties --- p.13 / Chapter 1.2.3 --- Examples of nonpositively curved manifold --- p.20 / Chapter 1.2.4 --- Convexity properties --- p.23 / Chapter 1.2.5 --- Points at infinity for M --- p.27 / Chapter 2 --- Symmetric Spaces --- p.36 / Chapter 2.1 --- Symmetric Spaces of Noncompact Type --- p.36 / Chapter 2.1.1 --- Symmetric diffeomorphisms --- p.36 / Chapter 2.1.2 --- Transvections in I(M) --- p.38 / Chapter 2.1.3 --- Symmetric spaces as coset manifolds G/K --- p.39 / Chapter 2.1.4 --- Metric on TpM and the adjoint representation of Lie group --- p.41 / Chapter 2.1.5 --- Curvature tensor of M --- p.43 / Chapter 2.1.6 --- Killing form and classification of symmetric spaces --- p.44 / Chapter 2.1.7 --- Holonomy of M at p --- p.44 / Chapter 2.1.8 --- Rank of a symmetric space M --- p.45 / Chapter 2.1.9 --- Regular and singular points at infinity --- p.46 / Chapter 2.2 --- "The Symmetric Space Mn = SL(n,R)/SO(n,R)" --- p.46 / Chapter 2.2.1 --- Metric on TIMn --- p.47 / Chapter 2.2.2 --- Geodesic and symmetries of Mn --- p.48 / Chapter 2.2.3 --- Curvature of Mn --- p.48 / Chapter 2.2.4 --- Rank and flats in Mn --- p.49 / Chapter 2.2.5 --- Holonomy of Mn at I --- p.49 / Chapter 2.2.6 --- Eigenvalue-flag pair for a point in Mn(∞ ) --- p.50 / Chapter 2.2.7 --- Action of I0(Mn) on Mn(∞ ) --- p.52 / Chapter 2.2.8 --- Flags in opposition --- p.53 / Chapter 2.2.9 --- Joining points at infinity --- p.53 / Chapter 3 --- Group Action --- p.56 / Chapter 3.1 --- Action of Isometries on M(oo) --- p.56 / Chapter 3.1.1 --- Fundamental group as a group of isometries --- p.56 / Chapter 3.1.2 --- Lattices --- p.58 / Chapter 3.1.3 --- Duality condition --- p.59 / Chapter 3.1.4 --- Geodesic flows --- p.61 / Chapter 3.2 --- Action of Geodesic Symmetries on M(oo) --- p.62 / Chapter 3.3 --- Rank --- p.66 / Chapter 3.3.1 --- Rank of a manifold of nonpositive curvature --- p.66 / Chapter 3.3.2 --- Rank of the fundamental group --- p.68 / Chapter 3.4 --- Rigidity Theorems of Locally Symmetric Spaces --- p.69

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

Orientador: Thiago de Melo / Banca: Vanderlei Marcos do Nascimento / Banca: Nivaldo de Góes Grulha Junior / Resumo: A história da Geometria Diferencial começa com o estudo de curvas. Noções de retas tangentes à curvas podem ser encontradas em Euclides, Arquimedes e Apolônio. Também, o Cálculo está baseado em ideias geométricas e, portanto, é natural encontrar investigações sobre curvas entre os tópicos tratados pelos pioneiros da Análise, Newton, Leibniz e Euler. Neste trabalho, serão apresentados os conceitos que fundamentam a teoria de curvas, bem como exemplos envolvendo algumas curvas clássicas, como a cicloide / Abstract: The history of Differential Geometry begins with the study of curves. Notions of tangent lines to the curves can be found in Euclid, Archimedes and Apollonius. Also, the Calculus is based on geometrical ideas and therefore is natural to find researches on curves between topics treated by the pioneers of Analysis, Newton, Leibniz and Euler. In this work, the concepts that underlie the theory of curves and some examples involving classical curves are presented, as the cycloid / Mestre

Tsoi, Hung Ming. / Thesis (M.Phil.)--Chinese University of Hong Kong, 2008. / Includes bibliographical references (leaves 88-90). / Abstracts in English and Chinese. / Preface --- p.7 / Chapter 1 --- Kahler Manifold --- p.10 / Chapter 1.1 --- Hermitian Manifold --- p.12 / Chapter 1.2 --- Kahler Manifold --- p.13 / Chapter 1.2.1 --- "Positive (l,l)-form" --- p.15 / Chapter 2 --- Vector Bundle --- p.16 / Chapter 2.1 --- Holomorphic Vector Bundle and Connection --- p.17 / Chapter 2.2 --- Hermitian Connection and Chern Connection --- p.18 / Chapter 2.2.1 --- Existence of Chern connection on a holomorphic vector bundle --- p.19 / Chapter 2.3 --- Curvature --- p.21 / Chapter 2.4 --- Positivity of Vector Bundles --- p.23 / Chapter 2.5 --- Chern Classes and Holomorphic Line Bundle --- p.24 / Chapter 2.5.1 --- Chern class in axiomatic approach --- p.25 / Chapter 2.5.2 --- Chern class in algebraic topology --- p.26 / Chapter 2.5.3 --- Chern class in terms of curvature --- p.27 / Chapter 2.5.4 --- In the case of hermitian line bundle --- p.28 / Chapter 3 --- Analytic Technique on Kahler Manifold --- p.30 / Chapter 3.1 --- Dolbeault Cohomology --- p.30 / Chapter 3.2 --- Commutator Relations on Kahler Manifold --- p.31 / Chapter 3.2.1 --- Commutator relation on a line bundle --- p.32 / Chapter 3.3 --- Hodge Theory --- p.33 / Chapter 3.4 --- Bochner Technique --- p.35 / Chapter 3.4.1 --- Bochner-Kodaira-Nakano identity --- p.36 / Chapter 4 --- Kodaira Vanishing Theorem and L2 estimate of d --- p.38 / Chapter 4.1 --- Kodaira Vanishing Theorem --- p.39 / Chapter 4.2 --- Extension of Kodaira Vanishing Theorem by L2 Method --- p.44 / Chapter 4.2.1 --- Plurisubharmonic functions and weakly pseudoconvex Kahler manifold --- p.47 / Chapter 5 --- Multiplier Ideal Sheaf --- p.55 / Chapter 5.1 --- Algebraic Properties of Multiplier Ideal Sheaf --- p.56 / Chapter 5.2 --- Some Calculations of Multiplier Ideal Sheaf --- p.59 / Chapter 6 --- Nadel Vanishing Theorem --- p.62 / Chapter 6.1 --- Nadel Vanishing Theorem by L2 Estimate of d --- p.62 / Chapter 6.2 --- The Original Setting of Nadel --- p.64 / Chapter 6.2.1 --- S-bounded and S-null sequence --- p.65 / Chapter 6.2.2 --- Multiplier ideal sheaf by Nadel --- p.67 / Chapter 6.3 --- Nadel Vanishing Theorem by Computation of Cech Cohomology --- p.69 / Chapter 6.3.1 --- L2 estimate of d --- p.69 / Chapter 6.3.2 --- Koszul cochain --- p.70 / Chapter 6.3.3 --- The cohomology vanishing theorem --- p.73 / Chapter 7 --- Kawamata-Viehweg Vanishing Theorem --- p.77 / Chapter 7.1 --- Numerically Effective Line Bundle --- p.77 / Chapter 7.2 --- Kawamata-Viehweg Vanishing Theorem --- p.85 / Bibliography --- p.88

Thesis (Ph.D.)--University of Missouri-Columbia, 2006. / The entire dissertation/thesis text is included in the research.pdf file; the official abstract appears in the short.pdf file (which also appears in the research.pdf); a non-technical general description, or public abstract, appears in the public.pdf file. Title from title screen of research.pdf file viewed on (February 26, 2007) Vita. Includes bibliographical references.

Thesis (Ph. D.)--University of Missouri-Columbia, 2000. / Typescript. Vita. Includes bibliographical references (leaves 106-111). Also available on the Internet.

Thesis (Ph. D.)--University of Missouri-Columbia, 2000. / Typescript. Vita. Includes bibliographical references (leaves 106-111). Also available on the Internet.

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Thesis (Ph. D.)--University of Hong Kong, 2009. / Includes bibliographical references (leaves 53-54) Also available in print.