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Good morning everyone, I was wondering about the difference between manifolds carrying a Riemannian metric with negative sectional curvature and hyperbolic manifolds. I was told once "there are very f …

I was wondering if two compact oriented manifold carrying a Riemannian metric with negative sectional curvature, whose fundamental groups are isomorphic, are necessarily diffeomorphic (or homeomorphic …

Let $A = \pmatrix{1 & 0 \\ \alpha & 1} $ and $ B = \pmatrix{1 & 1 \\ 0 & 1}$, where $\alpha \in \mathbb{C}$ is a complex parameter. Now consider the family of representations $r_{\alpha}$ of the fre …

Consider hyperbolic metrics on $\Sigma_g$ a closed orientable surface of genus $g$. Let $[\gamma_1] , \cdots, [\gamma_n]$ be a finite collection of isotopy classes of simple closed curves on $\Sigma_g …

Good afternoon everyone, I have a very general question about hyperbolic manifolds and their fundamental groups in high dimension (at least $4$). If the theory of surfaces and $3$-manifolds provide …

Good afternoon everyone ! I have the following question of Riemannian geometry : Let $M$ be a smooth closed orientable manifold of dimension at least $3$, and let $\mathcal{T} = \{ $ smooth Riemanni …

Good morning, I'm trying to understand the following fact, that is stated in Gromov and Thurston's paper "Pinching constants for hyperbolic manifolds" : Let $M$ be a (at least) 3-dimensional compact …

1) Is there a (simple) way to characterise the convergence of complex curves toward a stable curve in term of hyperbolic metrics for the Deligne Mumford compactification of the moduli space of complex …

Good afternoon, Take a submanifold $V$ of codimension $1$ of the sphere at infinity of $\mathbb{H}^n$ which is not the sphere at infinity of a totally geodesic hyperplane $\mathbb{H}^{n-1} \subset \m …