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Nayatani's metric and conformal transformations of a Kleinian manifold

Author: Yasuhiro Yabuki
Journal: Proc. Amer. Math. Soc. 136 (2008), 301-310
MSC (2000): Primary 53A30; Secondary 22E40
Published electronically: October 5, 2007
MathSciNet review: 2350417
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Abstract: According to Schoen and Yau (1988), an extensive class of conformally flat manifolds is realized as Kleinian manifolds. Nayatani (1997) constructed a metric on a Kleinian manifold $ M$ which is compatible with the canonical flat conformal structure. He showed that this metric $ g_N$ has a large symmetry if $ g_N$ is a complete metric. Under certain assumptions including the completeness of $ g_N$, the isometry group of $ (M,g_N)$ coincides with the conformal transformation group of $ M$. In this paper, we show that $ g_N$ may have a large symmetry even if $ g_N$ is not complete. In particular, every conformal transformation is an isometry when $ (M,g_N)$ corresponds to a geometrically finite Kleinian group.

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Yasuhiro Yabuki
Affiliation: Mathematical Institute, Tohoku University, 980-8578 Sendai, Japan

Keywords: Nayatani's metric, geometrically finite, conformally flat.
Received by editor(s): June 15, 2006
Received by editor(s) in revised form: November 24, 2006
Published electronically: October 5, 2007
Communicated by: Richard A. Wentworth
Article copyright: © Copyright 2007 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.

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