# Questions tagged [negative-curvature]

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13
questions

**7**

votes

**1**answer

144 views

### Hyperbolic groups and spaces of negative curvature

Mikhail Gromov states that he "tried for about 10 years to prove that every hyperbolic group is realizable by a space of negative curvature" in his interview with Martin Raussen and Christian Skau (...

**1**

vote

**1**answer

120 views

### Ergodicity of geodesic flow in negative curvatutre as a possible obstruction for consideration of limit cycles as closed geodesics(4)

Does the ergodicity of geodesic flow of compact surfaces with negative curvature stile hold for non compact case?
Is not the ergocity theorems of geodesic flow an obstruction to have a ...

**8**

votes

**2**answers

644 views

### Riemannian metric on the sphere with at least one negative sectional curvature at every point

Does there exist a Riemannian metric on the $n$-sphere ($n > 2$) such that at each point some (but not every) sectional curvature is negative?
For $n=2$ it is easily seen that such a metric ...

**0**

votes

**0**answers

97 views

### Bounding Riemannian Distance

If $(M,g)$ is a geodesically complete Riemannian manifold of negative sectional curvature bounded below by $K<0$ then is it true that for any $x,y,x_0\in M$
$$
d_H(
Log(x_0,x),Log(y,x_0)
)
\leq
d_M^...

**1**

vote

**1**answer

288 views

### Riemannian Manifolds of Bounded Curvature

I am a complete newbie Riemannian Geometry with a particular application in mind so please excuse a lack of rigor in the question.
Suppose I have a manifold with sectional curvature everywhere ...

**4**

votes

**1**answer

119 views

### Convex hull of a connected subset on a complete surface of non-positive curvature

Let $S$ be a simply connected surface, possibly with boundary components, with a smooth complete metric of non-positive curvature. Let $X\subset S$ be a closed connected subset. I would like to know ...

**10**

votes

**0**answers

247 views

### Examples of quasi-negative but not negative holomorphic sectional curvature

Let $(X,\omega)$ be a compact Kähler manifold and call $\operatorname{HSC}_{\omega}(x,[v])$ the holomorphic sectional curvature of the Chern connection of $\omega$ at the point $x\in X$ in the ...

**3**

votes

**1**answer

511 views

### On the Birkhoff ergodic theorem for geodesic flows

Let $S$ be a closed surface endowed with a Riemannian metric of negative curvature and let $US$ be the unit tangent bundle. Let $\mu$ be the Liouville measure on $US$.
Let $f: US\rightarrow\mathbb{R}$...

**2**

votes

**0**answers

78 views

### Locus maximizing the holomorphic sectional curvature in a non-compact Hermitian symmetric space

Is there a quick way to prove the following statement, if possible without resorting to the classification of simple Lie groups?
Let $G$ be a simple Lie group of non-compact Hermitian type of rank $r$...

**4**

votes

**1**answer

359 views

### Geometry of ends of a finite volume negatively curved manifold

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, ...

**10**

votes

**1**answer

436 views

### Topological rigidity for negatively curved manifolds?

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)...

**5**

votes

**2**answers

431 views

### Are ramified covering of negatively curved manifolds negatively curved?

Gromov and Thurston proved in "Pinching constants for hyperbolic manifolds" that any finite ramified covering of a compact hyperbolic manifold, along a codimension $2$ totally geodesic submanifold, ...

**3**

votes

**1**answer

201 views

### Cusps as warped products

It is well-known that the ends of a finite-volume hyperbolic manifold are warped products $$(0,\infty)\times_f T$$
for some euclidean manifold $T$ and $f(t)=e^{-t}$.
Question: Is there a similar ...