Negatively curved conformally flat manifolds

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2 total messages Started by Deane Yang Thu, 23 Jul 1992 12:48

#15

Author: Deane Yang
Date: Thu, 23 Jul 1992 12:48

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Here's an easy sounding question for differential geometers
that I can't answer off the top of my head:

Is a negatively curved conformally flat manifold necessarily
isometric or diffeomorphic to a hyperbolic manifold?

Deane Yang
Polytechnic University
(visiting the Fields Institute until 8/31/92)

Re: Negatively curved conformally flat manifolds

#16

Author: John M. Lee
Date: Fri, 24 Jul 1992 00:07

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In article <BruE8L.AnD@watserv1.waterloo.edu> yang@fields.waterloo.edu (Deane Yang) writes:

   Here's an easy sounding question for differential geometers
   that I can't answer off the top of my head:

   Is a negatively curved conformally flat manifold necessarily
   isometric or diffeomorphic to a hyperbolic manifold?

Hi Deane--

Here are a couple of simple observations that might be useful.  

First, it certainly need not be isometric to a hyperbolic manifold: just
multiply a hyperbolic metric by a function very close to 1 in the C2 norm,
and the resulting metric is conformally flat and still negatively curved.

Second, it may not even be conformally diffeomorphic to a hyperbolic
manifold.  An example is the catenoid, which is conformally covered by the
plane but not by the disk.   

I don't know the answer in the diffeomorphism category, but I suspect it
might not be easy.

Cheers!

Jack Lee

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