I am looking for examples of compact manifold of dimension $>2$ with strictly negative sectional curvature (for dimension 2 it is well-known). Can anybody please help?
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http://en.wikipedia.org/wiki/Hyperbolic_3-manifold – Moishe Kohan Dec 19 '14 at 05:51
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What about any $n$ in general? – Bingo Dec 19 '14 at 08:45
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There are explicit examples in dimension 4, maybe 5. However, in general dimension I am not even sure what "explicit" means. – Moishe Kohan Dec 19 '14 at 15:50
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If you are looking for examples in relatively small dimensions $>3$ then consider reading this paper by M.Davis and this subsequent paper. Examples exist in all dimensions but they are much less constructive. The second linked paper in principle explains how to produce them, but getting explicit torsion-free finite-index subgroups becomes increasingly difficult as dimension grows. You may want to ask yourself: What are known "explicit" examples of compact manifolds of dimension, say, 9999, and you will find that they are all derived from a handful of constructions involving: (1) fiber bundles, (2) homogeneous spaces of some compact Lie groups, (3) some low-dimensional examples, (4) some quotients by finite groups. Here by "explicit" I mean that you have some known handle decomposition with a list of handles and attaching maps which fits into a couple of pages.
Moishe Kohan
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