Rigidity theorems on Hermitian locally symmetric spaces.

透過使用調和映射的Bochner技巧, Siu[15, 16]證明了對於複維數≥2 時不可約對稱域緊致商空間的複結構的強剛定理. 其後在[9]中, Mok 證明了在任何秩≥2 的不可約對稱域緊致商空間上, 所有具備非正全純雙截曲率的Hermitian 度量必然和典範度量相差一個常數因子. 由這個定理和Siu 的定理可以得出Mostow 剛性定理[14]在特殊情形下的推廣.本論文會對Mok的結果作出研究.By using Bochner technique of harmonic maps, Siu[15, 16] proved a strong rigidity theorem concerning the complex structure of compact quotients of irreducible bounded symmetric domain of complex dimension≥ 2. Later in [9], Mok proved a metric rigidity theorem which asserts that any Hermitian metric of seminegative holomorphic bisectional curvature on a compact quotient of an irreducible bounded symmetric domain of rank≥ 2 is necessarily a constant multiple of the canonical metric. This theorem together with the theorem of Siu yields a generalization of a special case of Mostow's rigidity theorem[14]. This thesis is an exposition of Mok's results.Detailed summary in vernacular field only.Li, Ka Fai.Thesis (M.Phil.)--Chinese University of Hong Kong, 2012.Includes bibliographical references (leaves 102-104).Abstracts also in Chinese.Chapter 1 --- Introduction --- p.1Chapter 2 --- Symmetric Space --- p.5Chapter 2.1 --- Riemannian Symmetric Spaces --- p.5Chapter 2.2 --- Lie Groups and Lie Algebras --- p.10Chapter 2.3 --- Riemannian Symmetric Spaces of Compact and Non-compact type --- p.11Chapter 2.4 --- Hermitian Symmetric Spaces --- p.16Chapter 2.5 --- Duality --- p.19Chapter 3 --- Some Embedding Theorems --- p.22Chapter 3.1 --- The Borel Embedding Theorem --- p.22Chapter 3.2 --- Root Space Decomposition and Root System --- p.24Chapter 3.3 --- The Polydisc Theorem --- p.28Chapter 3.4 --- The Harish-Chandra Embedding Theorem --- p.36Chapter 4 --- Bounded Symmetric Domains --- p.42Chapter 4.1 --- Classical Bounded Symmetric Domains --- p.42Chapter 4.2 --- The Bergman metric --- p.57Chapter 5 --- Projective and Characteristic Bundle --- p.65Chapter 5.1 --- Projectivization of Hermitian Vector Bundle --- p.65Chapter 5.2 --- Characteristic bundle --- p.69Chapter 6 --- The Hermitian Metric Rigidity Theorem --- p.83Chapter 7 --- Appendix --- p.100Bibliography --- p.10