Remote Access Transactions of the American Mathematical Society
Green Open Access

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
MathSciNet review: 987162
Full-text PDF

Abstract | References | Similar Articles | Additional Information

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.

References [Enhancements On Off] (What's this?)

  • [C-G] S. Chen and L. Greenberg, Hyperbolic spaces, Contribution to Analysis, Academic Press, New York, 1974, pp. 49-87. MR 0377765 (51:13934)
  • [G] W. Goldman, Projective structures with Fuchsian holonomy, J. Differential Geom. 25 (1987), 297-326. MR 882826 (88i:57006)
  • [G-K] W. Goldman and Y. Kamishima, Topological rigidity of developing maps with applications to conformally flat structures, Contemp. Math., vol. 74, Amer. Math. Soc., Providence, R.I., 1988, pp. 199-203. MR 957519 (90f:53060)
  • [H] G. Hochschild, The structure of Lie groups, Holden-Day, San Francisco, Calif., 1975. MR 0207883 (34:7696)
  • [Ka] Y. Kamishima, Conformally flat manifolds whose development maps are not surjective, Trans. Amer. Math. Soc. 294 (1986), 607-623. MR 825725 (87g:57060)
  • [Ki] B. Kimelfeld, Homogeneous domains on the conformal sphere, Math. Notes 8 (1970), 653-656 (Mat. Zametki 8 (1970), 321-328). MR 0277670 (43:3403)
  • [Ku] R. Kulkarni, Groups with domains of discontinuity, Math. Ann. 234 (1978), 253-272. MR 508756 (81m:30046)
  • [S-Y] R. Schoen and S. T. Yau, Conformally flat manifolds, Kleinian groups and scalar curvature, Invent. Math. 92 (1987), 47-91. MR 931204 (89c:58139)

Similar Articles

Retrieve articles in Transactions of the American Mathematical Society with MSC: 53C10, 53C20, 57R99, 57S99

Retrieve articles in all journals with MSC: 53C10, 53C20, 57R99, 57S99

Additional Information

Article copyright: © Copyright 1991 American Mathematical Society

American Mathematical Society