In the first part of the paper we introduce the theory of bundles with negatively curved fibers. For a space X there is a forgetful map FX between bundle theories over X, which assigns to a bundle with negatively curved fibers over X its subjacent smooth bundle. Our main result states that, for certain k-spheres \documentclass[12pt]{minimal}
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\begin{document}$${\mathbb{S}^k}$$\end{document}, the forgetful map \documentclass[12pt]{minimal}
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\begin{document}$${F_{\mathbb{S}^k}}$$\end{document} is not one-to-one. This result follows from Theorem A, which proves that the quotient map \documentclass[12pt]{minimal}
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\begin{document}$${\mathcal{MET}^{\,\,sec <0 }(M)\rightarrow\mathcal{T}^{\,\,sec <0 }(M)}$$\end{document} is not trivial at some homotopy levels, provided the hyperbolic manifold M satisfies certain conditions. Here \documentclass[12pt]{minimal}
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\begin{document}$${\mathcal{MET}^{\,\,sec <0 }(M)}$$\end{document} is the space of negatively curved metrics on M and \documentclass[12pt]{minimal}
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\begin{document}$${\mathcal{T}^{\,\,sec <0 }(M) = \mathcal{MET}^{\,\,sec <0 }(M)/ {\rm DIFF}_0(M)}$$\end{document} is, as defined in [FO2], the Teichmüller space of negatively curved metrics on M. In particular we conclude that \documentclass[12pt]{minimal}
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\begin{document}$${\mathcal{T}^{\,\,sec <0 }(M)}$$\end{document} is, in general, not connected. Two remarks: (1) the nontrivial elements in \documentclass[12pt]{minimal}
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\begin{document}$${\pi_{k}\mathcal{MET}^{\,\,sec <0 }(M)}$$\end{document} constructed in [FO3] have trivial image by the map induced by \documentclass[12pt]{minimal}
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\begin{document}$${\mathcal{MET}^{\,\,sec <0 }(M)\rightarrow\mathcal{T}^{\,\,sec <0 }(M)}$$\end{document} ; (2) the nonzero classes in \documentclass[12pt]{minimal}
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\begin{document}$${\pi_{k}\mathcal{T}^{\,\,sec <0 }(M)}$$\end{document} constructed in [FO2] are not in the image of the map induced by \documentclass[12pt]{minimal}
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\begin{document}$${\mathcal{MET}^{\,\,sec <0 }(M)\rightarrow\mathcal{T}^{\,\,sec <0 }(M)}$$\end{document} ; the nontrivial classes in \documentclass[12pt]{minimal}
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\begin{document}$${\pi_{k}\mathcal{T}^{\,\,sec <0 }(M)}$$\end{document} given here, besides coming from \documentclass[12pt]{minimal}
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\begin{document}$${\mathcal{MET}^{\,\,sec <0 }(M)}$$\end{document} and being harder to construct, have a different nature and genesis: the former classes – given in [FO2] – come from the existence of exotic spheres, while the latter classes – given here – arise from the non-triviality and structure of certain homotopy groups of the space of pseudo-isotopies of the circle \documentclass[12pt]{minimal}
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\begin{document}$${\mathbb{S}^1}$$\end{document}. The strength of the new techniques used here allowed us to prove also a homology version of Theorem A, which is given in Theorem B.
