Nayatani’s metric and conformal transformations of a Kleinian manifold

Yabuki, Yasuhiro

doi:10.1090/S0002-9939-07-09022-3

Nayatani’s metric and conformal transformations of a Kleinian manifold
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by Yasuhiro Yabuki PDF

Proc. Amer. Math. Soc. 136 (2008), 301-310 Request permission

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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