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Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)

 

 

Conformal automorphisms and conformally flat manifolds


Authors: William M. Goldman and Yoshinobu Kamishima
Journal: Trans. Amer. Math. Soc. 323 (1991), 797-810
MSC: Primary 53C10; Secondary 53C20, 57R99, 57S99
DOI: https://doi.org/10.1090/S0002-9947-1991-0987162-1
MathSciNet review: 987162
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Abstract: A geometric structure on a smooth $ n$-manifold $ M$ is a maximal collection of distinguished charts modeled on a $ 1$-connected $ n$-dimensional homogeneous space $ X$ of a Lie group $ G$ where coordinate changes are restrictions of transformations from $ G$. There exists a developing map $ dev:wm \to X$ which is always locally a diffeomorphism. It is in general far from globally being a diffeomorphism. We study the rigid property of developing maps of $ (G,X)$-manifolds. As an application we shall classify closed conformally flat manifolds $ M$ when the universal covering space $ \tilde M$ supports a one parameter group of conformal transformations.


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DOI: https://doi.org/10.1090/S0002-9947-1991-0987162-1
Article copyright: © Copyright 1991 American Mathematical Society