# Questions tagged [negative-curvature]

The negative-curvature tag has no usage guidance.

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### Examples of product negatively curved Riemannian manifolds

It is a well known theorem of Anderson that any vector bundle
over a negative sectional curvature Riemannian manifold
admits a metric of negative sectional curvature.
Q: Are there examples of complete ...

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### Deformed completion of negatively curved metric

Suppose X,g is a complete
negative sectional curvature Riemannian manifold. And $C \subset X$ is
compact submanifold.
What are the minimal conditions on $C$ so that
we can deform g on the restriction ...

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### Isometries of manifolds with non-positive sectional curvature

It is known since Bochner that for M compact with negative Ricci curvature, the group of isometries is discreet and hence finite. Are there any generalizations to compact non-positive sectional ...

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### Tzitzeica surface

A Tzitzeica surface has the property that the ratio of the surface’s Gaussian curvature and the fourth power of the distance from the origin to the tangent plane at any arbitrary point of the surface ...

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### Maximal geodesically convex function interpolating three points on the hyperbolic plane

Crossposted on MSE: https://math.stackexchange.com/questions/4282998/maximal-geodesically-convex-function-interpolating-three-points-on-the-hyperboli
Let $M$ be a two-dimensional Hadamard manifold. ...

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### Elliptic equations in asymptotically hyperbolic manifolds

I am interested in reading about existence and regularity theorems for elliptic equations on manifolds with negative (constant) curvature outside a compact subset. I am aware of some results in this ...

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### Birman-Series for variable negative curvature

A famous theorem of Birman and Series says that if $S$ is a compact hyperbolic surface, then the set of points that lie on simple geodesics is nowhere dense and has Hausdorff dimension one; in ...

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### Is this $a(p)=\lim_{r\to \infty} \frac{VolS(p,r)}{e^{h r}}$ exists and applied for manifolds with positive curvature?

In $1969$, Margulis proved, for suitable constant $h>0$and $r$ is a positive constant that :
$a(p)=\lim_{r\to \infty} \frac{VolS(p,r)}{e^{h r}}$ with ($(S(p,r)$ is geodesic spheres), exists at ...

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### 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 (...

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### 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 ...

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### 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 ...

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### 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

385
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### 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

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### 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 ...

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### 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 ...

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votes

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### 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 $f: US\rightarrow\mathbb{R}$...

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### 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 $r$...

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### 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, ...

11
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### 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)...

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### 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, ...

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votes

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### 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 ...

2
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1
answer

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### 2 questions about loops and negative curvature

$(M^n,g)$ is a compact $n$ dimensional manifold of negative curvature with n>2 . let $\alpha$ be a simple closed geodesic loop in $M$ based at a point $p$
1) will the geodesic in the free homotopy ...