The Teichmüller Space of Pinched Negatively Curved Metrics on a Hyperbolic Manifold is not Contractible

Details The Teichmüller Space of Pinched Negatively Curved Metrics on a Hyperbolic Manifold is not Contractible The Teichmüller Space of Pinched Negatively Curved Metrics on a Hyperbolic Manifold is not Contractible

2004-06-07

arxiv.org/abs/math/0406132v3

Mathematics

Differential Geometry

Scientific paper

For a smooth manifold $M$ we define the Teichm\"uller space $\cT(M)$ of all Riemannian metrics on $M$ and the Teichm\"uller space $\cT^\epsilon(M)$ of $\epsilon$-pinched negatively curved metrics on $M$, where $0\leq\epsilon\leq\infty$. We prove that if $M$ is hyperbolic the natural inclusion $\cT^\epsilon(M)\hookrightarrow\cT(M)$ is, in general, not homotopically trivial. In particular, $\cT^\epsilon(M)$ is, in general, not contractible.

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