Is there a survey of the geometry of manifolds with finite volume Riemannian metrics of negative sectional curvature? More specifically, I am interested in the geometry of cusp ends of such manifolds, which I think(?) are of the form $M \times \mathbb{R}_{+}$, where $M$ is a compact quotient of a nilpotent Lie group. I'm also interested in the geometry of the boundary of the universal cover. It's easy to find references on these subjects in the case of real or complex hyperbolic metrics, but I have yet to find a reference that summarizes what is known (and what techniques are available) in the case of non-locally symmetric negatively curved manifolds.

In the references, I am really looking for quantitative geometric, analytical, and dynamical information about such Riemannian manifolds, not coarse geometry related to its fundamental group. I also want information specific to pinched negative sectional curvature, as opposed to the more general case of nonpositive sectional curvature (about which I have been able to find references which summarize the basic techniques, e.g. the excellent text of Ballman, Gromov, and Schroeder on the subject).

Edit: I found the following recent survey on topological aspects of the subject http://arxiv.org/pdf/1306.1256.pdf which I think has enough references for me to follow up on to get some answers to my original question. Additional references would be greatly appreciated.

One of the reference questions I have which lies behind my original question is this: Is there a book or paper which collects techniques specific to strict negative sectional curvature, as opposed to nonpositive sectional curvature? Every general reference that I have found proves everything in the case $K \leq 0$, and then states additional improvements for $K < 0$ when applicable. I am wondering if there is a comprehensive text which prioritizes the viewpoint of negative sectional curvature and collects stronger results unique to this case.

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    $\begingroup$ I think the survey you found is the only comprehensive source, and in writing it I tried to include all known results. There was some work since, e.g. front.math.ucdavis.edu/1309.0043. My experience is that negativity of the curvature does not really give you much benefit, which is why most sources treat the nonpositively curved case. The condition $K<0$ is easy to check but hard to work with, as I explain in the survey. $\endgroup$ Commented Apr 9, 2014 at 1:01
  • $\begingroup$ @Igor: but in the non-positively curved case (even for symmetric spaces), the geometry at infinity is more complicated and certainly not a disjoint union of tubes along a compact nilmanifold. So the condition of negative curvature (or maybe rather the stronger condition that the curvature has a negative upper bound) sounds relevant to me. $\endgroup$
    – YCor
    Commented Nov 10, 2014 at 1:20
  • $\begingroup$ @YCor: again, my point is that the condition $K<0$ is hard to work with, and it is not natural geometrically. It sits between two better conditions on a Hadamard manifold: "visibility" and "no flat half-plane". Of course, I was not trying to suggest that all manifolds of $K\le 0$ look like those of $K<0$. On the other hand, in the finite volume case Rank Rigidity tells us that the manifold is either locally symmetric or has some negatively curved features, and the latter case is still poorly understood. $\endgroup$ Commented Nov 10, 2014 at 4:34

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Many papers of P. Eberlein on geodesic flows on negatively curved manifolds are good. Also, a paper of Heintze - Im Hof on the Geometry of horospheres could be useful ? There is also a book of Eberlein (very useful, as Ballmann-Gromov-Schroeder), which has maybe the same title (Geometry of nonpositively curved manifolds ?)

A book of Ballmann is also interesting and very useful http://people.mpim-bonn.mpg.de/hwbllmnn/archiv/NPC0606.pdf

Good lecture!



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