Summary of Prof
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چکیده
of Prof. Yau's lecture, Monday, April 2 [with additional references and remarks] (for people who missed the lecture) A geometric structure on a manifold is a cover by coordinate systems [a " sub-atlas " ] in which the transition functions from one coordinate system to another are not arbitrary mappings but rather belong to some specific set of mappings from (open subsets of) Euclidean space to itself. [Precisely what is needed here is dealt with by the concept of " pseudo-group " of transformations]. Examples are C ∞ 1. projective structures—transformations are (restrictions of) projective transformations of RP n , namely linear fractional transformations 2. conformal structures: overlap maps are conformal 3. affine structures: overlap maps are affine transformations of Euclidean space 4. complex structures: overlap maps are holomorphic maps [considered as real C ∞ maps on an even dimensional Euclidean space] These situations are different from each other and higher dimensions are often different from dimension 2. Important differences: Holomorphic maps do not have to come from a map of the whole space whereas, e.g., projective maps extend by definition. In dimension 2, there are many [infinite dimensional, not described by a finite set of parameters] maps [holomorphic functions] whereas in higher dimensions, conformal maps are a more rigid[extend to whole space except for possible definitely determined singularities, finite number of parameters, e.g., Liouville's Theorem that conformal maps of R 3 are generated by inversions and isometries and dilations/contractions].
منابع مشابه
AFRILEX African Association for Lexicography Programme & Abstracts 17 th Annual International Conference University of Pretoria
reviewers: Dr M Alberts, Prof G-M de Schryver, Dr PA Louw, Prof. RH Gouws, Dr T Otlogetswe, Dr VM Mojela, Dr HS Ndinga-Koumba-Binza, Prof. DJ Prinsloo, Prof. E Taljard Abstract booklet editor: Ms J Wolvaardtbooklet editor: Ms J Wolvaardt © 2012 AFRILEX, African Association for Lexicography ISBN 978-0-620-53469-7
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