9
votes
Accepted
Hyperbolic groups and spaces of negative curvature
As already mentioned in the comments, it is still unknown whether hyperbolic groups are CAT(-1) or even CAT(0). A related question is:
Let $G$ be a hyperbolic group (endowed with a finite generated ...
- 8,401
7
votes
Accepted
Birman-Series for variable negative curvature
Here is another argument which reduces the general result to the constant curvature case:
For any negatively curved metric $g$ on $S$, the set of geodesics of the universal cover $\tilde{S}$ is ...
- 1,908
6
votes
Riemannian metric on the sphere with at least one negative sectional curvature at every point
Kazhymurat's answer is definitely correct -- every sphere has such a metric.
But, much more explicitly, if your sphere is odd-dimensional, then there are even homogeneous metrics (called Berger ...
- 5,891
4
votes
Birman-Series for variable negative curvature
Yes, Birman-Series is true in variable negative curvature. Here is a sketch.
Suppose that $S$ is a closed, connected, oriented surface of genus $g > 1$. Fix a metric $g$ of variable negative ...
- 28.1k
3
votes
Accepted
Examples of product negatively curved Riemannian manifolds
To expand on my second comment: Consider the warped product $T^n\!\times_{e^t}\!\mathbb R$, where $T^n$ is the Riemannian product of $n$ circles. The warped product is a complete Riemannian manifold ...
- 29.1k
3
votes
Accepted
Isometries of manifolds with non-positive sectional curvature
Bochner's theorem extends to nonpositive Ricci to give:
If $(M,g)$ is compact and has $\textrm{Ric}\leq 0$ then any Killing vector $X$ is parallel and $\textrm{Ric}(X,X) = 0$.
See Petersen (3rd ed) ...
- 7,197
3
votes
Accepted
Tzitzeica surface
The answer is 'no'.
Suppose that $M\subset\mathbb{E}^3$ is a smooth connected surface. If the ratio of the Gauss curvature $K$ and $p^4$ is constant (where $p(x)$ is the distance from $T_xM$ to the ...
- 108k
3
votes
Accepted
Elliptic equations in asymptotically hyperbolic manifolds
The definite reference for this is the monograph by John Lee "Fredholm operators and Einstein metrics on conformally compact manifolds", Mem. Am. Math. Soc. Series Profile 864, 83 p. (2006), ...
- 1,263
3
votes
Accepted
Convex hull of a connected subset on a complete surface of non-positive curvature
Yes, it is true.
First note that $S$ is CAT[0] space. This can be proved along the same lines as in "The intrinsic geometry of a Jordan domain" by Richard Bishop (you may also check the proof of ...
- 45k
3
votes
Ergodicity of geodesic flow in negative curvatutre as a possible obstruction for consideration of limit cycles as closed geodesics(4)
The geodesic flow on (possibly noncompact) hyperbolic surfaces of finite type is ergodic. The original reference for this is E. Hopf: „Fuchsian groups and ergodic theory“ Trans. AMS 39, 299-314 (1936)....
- 10.4k
1
vote
Accepted
Is this $a(p)=\lim_{r\to \infty} \frac{VolS(p,r)}{e^{h r}}$ exists and applied for manifolds with positive curvature?
When $M$ has positive curvature, the limit should always be 0. Indeed if $M$ has non-negative Ricci curvature, Bishop-Gromov tells you that $\frac{Vol S(p, r)}{n\omega_nr^{n-1}}\leq 1$, so the volumes ...
- 320
1
vote
Riemannian Manifolds of Bounded Curvature
Sorry, (since I got this theorem myself and found out after that it is Ambrose-Singer) I mistaken exercise 4 in doCarmo p.105 for it. The proof looks like this: let $D$ be the image of a unit square ...
- 763
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