The Teichmüller space of pinched negatively curved metrics on a hyperbolic manifold is not contractible

Abstract

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