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I often hear mention of two theorems, Mostow's rigidity theorem and Liouville's theorem on conformal mappings, which superficially sound similar: they say that a set of geometric structures is, if nonempty, big in dimension 2, but small in dimension greater than 2.

(For Mostow's theorem, the set of structures in question is the set of hyperbolic metrics on a manifold; for Liouville's, it's the set of germs of flat metrics in a conformal equivalence class.)

I know that hyperbolic and conformal geometry are closely connected, at least in dimension 2. I'm curious as to whether this analogy is hinting at one such connection. Is there a "good reason" for this analogy?

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There is a connection in some way: if I remember right, you usually prove Mostow rigidity by looking at the hyperbolic space, which is the universal cover of your hyperbolic manifold, then you consider its boundary, which is the flat conformal sphere. Any isometry of the hyperbolic space induces a conformal transformation of its boundary, and vice-versa. But I don't think that you can derive Mostow rigidity from Liouville's conformal theorem.

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  • $\begingroup$ +1 Thanks for the thoughts. I'm aware of the "quasi-conformal" standard proof of Mostow rigidity. But, as you say, this doesn't seem to explain the connection in dim-2 floppiness/ higher-dim rigidity that intrigues me. If nothing else, the boundary of the universal cover, where the conformal phenomena occur, will have dimension one less than the universal cover itself, where the hyperbolic phenomena occur. $\endgroup$
    – macbeth
    Commented Mar 27, 2010 at 22:00
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    $\begingroup$ You're right, and in a sense the flexibility of lower dimensional cases seem different: in the conformal case, it comes from the analogy between conformality and holomorphy, while in the hyperbolic case the point is that the boundary is a circle, and has therefore very few structure (any diffemorphism is conformal!). There are many other situation where the lower dimensional cases lack rigidity: think of Poincaré's conjecture for instance, or of the fundamental theorem of affine geometry (a transformation that preserves alignment is affine as soon as the dimension is at least 2). $\endgroup$
    – Benoît Kloeckner
    Commented Mar 28, 2010 at 12:36

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