Lo, Chi Yu. / On t.p. "n" is a superscript. / Thesis (M.Phil.)--Chinese University of Hong Kong, 2008. / Includes bibliographical references (leaves 92). / Abstracts in Chinese and English. / Chapter 0 --- Introduction --- p.5 / Chapter 1 --- "CR structures and the group SU(p, q)" --- p.7 / Chapter 1.1 --- Almost complex structure and CR manifolds --- p.7 / Chapter 1.2 --- Automorphism Groups of Ball and Polydisc --- p.17 / Chapter 1.3 --- "The group SU(p,q) and its Maurer Cartan form" --- p.24 / Chapter 2 --- Cartan´ةs construction on nondegenerate CR manifold --- p.33 / Chapter 2.1 --- A digression on the Frobenius Theorem and projective structure --- p.33 / Chapter 2.2 --- Cartan bundle and canonical forms --- p.45 / Chapter 2.3 --- Calculations of real hypersurface in C2 --- p.60 / Chapter 3 --- Geometric consequences and chain --- p.66 / Chapter 3.1 --- CR equivalence problem --- p.66 / Chapter 3.2 --- CR manifolds of dimension 3 --- p.71 / Chapter 3.3 --- Definition of chains --- p.78 / Chapter 3.4 --- Chains on a special kind of Reinhardt hyper surf ace in C2 --- p.87 / Bibliography --- p.92

In this work, we study how solutions of certain non-compact geometric flows of fast-diffusion type interact with their asymptotic geometries at infinity. In the first part, we show the long time existence theorem to the inverse mean curvature flow for complete convex non-compact initial hypersurfaces. The existence and behavior of a solution is tied with the evolution of its tangent cone at infinity. In particular, the maximal time of existence can be written in terms of the area ratio between the initial tangent cone at infinity and the flat hyperplane. In the second part, we study the formation of type II singularity for non-compact Yamabe flow. Assuming the initial metric is conformally flat and asymptotic to a cylinder, we show the higher order asymptotics of the metric determines the curvature blow-up rates at the tip in its first singular time. We also show the singularities of such solutions are modeled on rotationally symmetric steady gradient solitons.

Inaug.-Diss.-Bonn.

Thesis--Ph. D., University of Hong Kong.

The theory of affine connections is, roughly speaking, a generalization of certain concepts of parallelism and differentiation defined in plane differential geometry, to the differential geometry of surfaces, and, more generally, to the geometry of differentiable manifolds. It is the purpose of this essay to relate the various stages of this generalization, and to present the essentials of the classical theory of affine connections on a differentiable manifold. [...]

A quaternion-Hermitian manifold, of dimension at least 12, with closed fundamental 4-form is shown to be quaternionic Kähler. A similar result is proved for 8-manifolds. HyperKähler metrics are constructed on the fundamental quaternionic line bundle (with the zero-section removed) of a quaternionic Kähler manifold (indefinite if the scalar curvature is negative). This construction is compatible with the quaternionic Kähler and hyperKähier quotient constructions and allows quaternionic Kähler geometry to be subsumed into the theory of hyperKähler manifolds. It is shown that the hyperKähler metrics that arise admit a certain type of SU(2)- action, possess functions which are Kähler potentials for each of the complex structures simultaneously and determine quaternionic Kähler structures via a variant of the moment map construction. Quaternionic Kähler metrics are also constructed on the fundamental quaternionic line bundle and a twistor space analogy leads to a construction of hyperKähler metrics with circle actions on complex line bundles over Kähler-Einstein (complex) contact manifolds. Nilpotent orbits in a complex semi-simple Lie algebra, with the hyperKähler metrics defined by Kronheimer, are shown to give rise to quaternionic Kähler metrics and various examples of these metrics are identified. It is shown that any quaternionic Kähler manifold with positive scalar curvature and sufficiently large isometry group may be embedded in one of these manifolds. The twistor space structure of the projectivised nilpotent orbits is studied.

Thesis (M. Phil.)--University of Hong Kong, 2008. / Also available in print.

Inaug.-Diss.-Bonn.

In this thesis I will study noncommutative differential geometry, after the style of Connes and Woronowicz. In particular two examples of differential calculi on Hopf algebras are considered, and their associated covariant derivatives and Riemannian geometry. These are on the Heisenberg group, and on the finite group A4. I consider bimodule connections after the work of Madore. In the last chapter noncommutative fibrations are considerd, with an application to the Leray spectral sequence. NOTATION. In this thesis equations are numbered as round brackets (), where (a.b) denotes equation b in chapter a, and references are indicated by square brackets []. This thesis has been typeset using Latex, and some figures using the Visio program.

No description available.