Conformal automorphisms and conformally flat manifolds

Goldman, William M.; Kamishima, Yoshinobu

doi:10.1090/S0002-9947-1991-0987162-1

Conformal automorphisms and conformally flat manifolds
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by William M. Goldman and Yoshinobu Kamishima PDF

Trans. Amer. Math. Soc. 323 (1991), 797-810 Request permission

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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