All Questions
Tagged with riemannian-geometry hyperbolic-geometry
101 questions
6
votes
0
answers
57
views
Connectedness of the space of negatively curved metrics of a compact 3-manifold
Is the space of metrics of negative sectional curvature over a closed 3-manifold connected? If so, in what paper is this result stated?
Note: as the Ricci flow hyperbolizes negatively curved metrics, ...
- 430
1
vote
1
answer
195
views
The length is bounded
Let $\Sigma$ be a surface of finite type. Let $\mathcal{S}$ be the set of non-trivial isotopy classes of simple closed curves on $\Sigma$. One denotes by $l_x(\alpha)$ the infimal length of curves in ...
- 1,043
3
votes
1
answer
135
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Geodesic convexity of Dirichlet Fundamental Domains
My question is motivated by this question, and this answer to it. Below, let's consider the setup in that answer:
Let $M$ be a Riemannian manifold. Let $G\times M\to M$ be a proper action of a ...
- 1,467
2
votes
0
answers
57
views
A sequence of conformal metrics with bounded negative curvatures on the disc
Let $\mathbb{D}$ denote the unit disk, and let $h_{-1}$ be the unique hyperbolic metric on $\mathbb{D}$ which is conformal to $dz^{2}$.
Take a sequence of smooth complete metrics $h_{n} = e^{\rho_{n}} ...
- 63
2
votes
1
answer
103
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Simple curves on hyperbolic tori
In the paper "Simple curves on hyperbolic tori" by McShane and Rivin, they show that if $T$ is a hyperbolic once punctured torus, one can define a norm on the homology $H_1(T,\mathbb{Z})$ by ...
- 7,527
10
votes
3
answers
2k
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Is there an absolute geometry that underlies spherical, Euclidean and hyperbolic geometry?
A space form is defined as a complete Riemannian manifold with constant sectional curvature. Fixing the curvature to +1, 0 & -1 and then taking the universal cover by the Killing–Hopf theorem ...
- 2,369
1
vote
0
answers
62
views
Gradient estimate of the eigenfunction of Laplacian on hyperbolic space
I am trying to understand the asymptotic behaviors of the gradient of the eigenfunction function of the Laplace-Beltrami operator on the hyperbolic plane $\mathbb{H}^2$. Specifically, my focus lies on ...
1
vote
2
answers
266
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Isometric embeddings of $\Bbb H^3$
Consider the upper-half space model of hyperbolic $3$-space $\Bbb H^{+}_{3}$, the unique, simply-connected, $3$-dimensional complete Riemannian manifold with a constant negative sectional curvature ...
user515818
1
vote
0
answers
85
views
Closed form ODE solutions for Jacobi field/eigenfunction of Laplacian on hyperbolic space
I'm trying to compute Jacobi fields of the hyperbolic disk $\mathbb{H}^m$ considered as a minimal hypersurface in $\mathbb{H}^{m+1}$ in the half model. References to literature or solutions to the ...
- 337
0
votes
1
answer
133
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Expansion of metric near boundary of 3 dimensional Poincaré-Einstein/hyperbolic manifolds
In Mazzeo-Alexakis, there is a brief discussion that if $(M^3,g) \sim \mathbb{H}^3/\Gamma$ (for $\Gamma$ convex cocompact), then the metric can be expanded near the topological boundary as
$$g = \frac{...
- 337
2
votes
0
answers
265
views
A Question about an article by Birman, Series
Birman and Series in their article GEODESICS WITH BOUNDED INTERSECTION NUMBER ON
SURFACES ARE SPARSELY DISTRIBUTED proved that the set of points on a hyperbolic surface (possibly with boundary) ...
- 231
2
votes
0
answers
71
views
Examples of elementary group of isometries of the ideal boundary of hyperbolic plane
A Riemannian manifold $(X,g)$ is called Hadamard manifold if it is complete and simply connected and has everywhere non-positive sectional curvature. An example of such a manifold would be the ...
- 41
5
votes
0
answers
164
views
Intersection of orbits of earthquake flow on Teichmüller space
Let $\Sigma$ be a closed oriented surface of genus $g\geq2$. We consider $\mu$ and $\nu$ two filling measured laminations on $\Sigma$. (We say that $\mu$ and $\nu$ fill $\Sigma$ if $\Sigma\setminus(\...
0
votes
1
answer
96
views
Hadamard submanifolds of $k$-fold product of hyperbolic plane
Let $\kappa>0$ and $d,k$ be a positive integers with $k\ge d$. For $k\in \mathbb{Z}^+$ large enough, can one find a geodesically complete and simply connected $d$-dimensional Riemannian ...
- 364
6
votes
0
answers
149
views
What is the minimum $n$ for which $\Bbb H^3$ can be isometrically embedded in $\Bbb R^n$ as a bounded set?
Consider the hyperbolic $3$-space $\Bbb H^3$ (i.e., the unique, simply-connected, $3$-dimensional complete Riemannian manifold with a constant negative sectional curvature equal to $-1$). The Nash ...
- 1,097
5
votes
1
answer
461
views
Ricci flow negative curvature
We denote by $\mathbb{H}^n$ the hyperbolic plane of dimension $n$.
I don't know much about the Ricci flow, so my first question is probably naïve : can one define a normalized Ricci flow such that the ...
- 53
0
votes
0
answers
86
views
Deformation of hyperbolic structures
Let M be a hyperbolic 3-manifold, let $g_{t}$ be a smooth family of hyperbolic metrics on M. It is known that in this case we can find a family of devlopping maps $dev_{g_{t}}:\tilde{M} \to \mathbb{H}^...
- 63
2
votes
1
answer
118
views
Pair of laminations that fill on a closed surface
Let $S$ be a hyperbolic surface of genus $g \geq 2$.
A discrete geodesic lamination on $S$ is a set of disjoint, simple, closed geodesics.
Let $L_{1}$ and $L_{2}$ be two discrete geodesic laminations ...
- 63
2
votes
0
answers
83
views
What are the volume-preserving diffeomorphisms of hyperbolic space? [duplicate]
What are the volume-preserving diffeomorphisms of $d$-dimensional hyperbolic space (in say the hyperboloid model)?
In particular, I'm especially interested in: what are the volume-preserving ...
- 637
4
votes
0
answers
106
views
Geodesic foliations of open manifolds foliated by hyperbolic spaces
It is known that hyperbolic spaces admit geodesic foliations (that is, a smooth unit vector field all of whose integral curves are geodesics, see https://arxiv.org/abs/1411.6700).
Suppose a complete ...
- 4,729
5
votes
1
answer
278
views
For $f$ geodesically convex with $L$-Lipschitz-gradient on hyperbolic space, is $f(x)-f(x^*)\leq(\mathrm{const}) \cdot L r$ for all $x \in B(x^*, r)$?
$\DeclareMathOperator\dist{dist}$Setting: Let $M$ be a hyperbolic space of sectional curvature $-1$, and let $f \colon M\to \mathbb{R}$ be a $C^2$, geodesically convex function which has $L$-Lipschitz-...
- 637
10
votes
4
answers
667
views
Reference for shortest educational path to (Riemannian) hyperbolic plane
I am teaching an undergraduate class for math majors on axiomatic geometry, culminating in the proof that hyperbolic geometry is equiconsistent* with Euclidean geometry. I would like to make an end-of-...
- 2,218
4
votes
1
answer
245
views
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 ...
- 272
5
votes
1
answer
431
views
Isometric immersion of $\mathbb H^2$ into $\mathbb R^\infty$ built by Bieberbach
I'm analyzing the following isometric immersion of $(\mathbb H^2,g_D)$ in $(\ell^2,g_\infty)$ given by $f(x,y)=(x_1,x_2,\dots,x_{2m-1},x_{2m},\dots)$ with
\begin{align}\label{5.1}
x_{2m-1}=\color{...
- 143
1
vote
0
answers
112
views
The best lower bound for isometric immersions
I just read Azov's article in the considered two classes of Riemannian metrics,
\begin{align*}
ds^2&=du_1^2+f(u_1)\sum_{i=2}^ldu_i,&f>0\\
ds^2&=g^2(u_1)\sum_{i=2}^ldu_i^2
,&g>0\...
- 143
1
vote
0
answers
82
views
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. ...
- 637
2
votes
0
answers
177
views
Structure of hyperbolic manifolds of finite volume
Let $X$ be a hyperbolic manifold of finite volume.
I want to prove that $X$ has ends of the form $N\times \mathbb{R}$ where $N$ has a finite covering by a nilmanifold and $\pi_1N\to \pi_1 X$ is ...
- 563
4
votes
1
answer
321
views
A local isometric immersion from $\mathbb H^{n}$ into $\mathbb R^{2n-1}$
I found this local isometric immersion from $\mathbb H^{n}$ into $\mathbb R^{2n-1}$, given by Schur (1886) in Über die Deformation der Räume constanten Riemannschen Krümmungsmaasses as follows, $(1\...
- 143
5
votes
2
answers
207
views
Rozendorn's Article
I'm researching the isometric dips of the hyperbolic plane and in particular I'm interested in reading the results of Rozendorn who proved that the hyperbolic plane is isometrically immersed in $\...
- 143
6
votes
1
answer
375
views
Non-compact Dirichlet fundamental domains and free Fuchsian groups
Let $G$ be a finitely generated Fuchsian group, and let $\mathcal{F}$ denote the Dirichlet fundamental domain of $G$ with respect to $0$ in the Poincaré disc model.
Assume throughout that $\mathcal{F}$...
- 63
4
votes
1
answer
411
views
Explanation of perpendicularity of a Jacobian vector field
Here are some notes on hyperbolic manifolds. The aim is to prove that if $M_1$ and $M_2$ are simply connected, complete Riemannian manifolds having constant sectional curvature of $-1$, then $M_1$ and ...
- 1,755
11
votes
2
answers
314
views
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 ...
- 44.8k
3
votes
0
answers
250
views
Immersion of a part of the hyperbolic plane in $\mathbb{R}^3$
I know that the pseudosphere is a regular surface with Gaussian curvature $-1$ that is not complete, also this surface is not complete. Hilbert's theorem ensures that there is no isometric immersion ...
- 143
9
votes
1
answer
180
views
When do the lengths of simple closed curves determine a hyperbolic surface?
Consider hyperbolic metrics on $\Sigma_g$ a closed orientable surface of genus $g$. Let $[\gamma_1] , \cdots, [\gamma_n]$ be a finite collection of isotopy classes of simple closed curves on $\Sigma_g$...
- 2,696
0
votes
0
answers
242
views
Exponential map and optimization
Apologies in advance for being somewhat vague. I'm trying to get pointers to establish a connection between a common trick used in practice in optimization, and the exponential map in differential ...
- 93
7
votes
1
answer
759
views
Complete geodesics on hyperbolic a pair of pants
I have asked this question on MSE. But I think Mo is a better place to ask my question. Here is the link to my question on MSE. I will rewrite it here:
I am trying to understand the article by Maryam ...
- 231
25
votes
1
answer
842
views
Alternate proofs that hyperbolic plane can’t be isometrically immersed in $\mathbb{R}^3$
A famous theorem of Hilbert says that there is no smooth immersion of the hyperbolic plane in 3-dimensional Euclidean space. The expositions of this that I know of (in eg do Carmo’s book on curves/...
- 251
3
votes
0
answers
109
views
An explicit (maybe algebraic) isometric embedding of the double torus with constant curvature K = -1
The following question is related to this previous question, Canonical immersion of the double torus:
Is there any known explicit (maybe algebraic) isometric embedding of a genus 2 surface endowed ...
- 31
10
votes
1
answer
320
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Systole of Riemann surfaces of genus $g$
In Buser and Sarnak's "On the period matrix of a Riemann surface
of large genus", we get
$$\frac4{3}\le\limsup_{g\rightarrow\infty}\frac{\max\{\operatorname{sys}(S)|S\in\mathcal{M}_g\}}{\log ...
- 181
11
votes
4
answers
1k
views
Surfaces with non-constant negative curvature
Are there any nice models of surfaces with non-constant negative curvature, analogous to the Poincare disk for constant negative curvature. I have found lots of general results and theory but no nice ...
- 680
1
vote
1
answer
271
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Does there exist a real-valued function on the hyperbolic plane which has bounded hessian norm and unbounded gradient norm?
Does there exist a real-valued function on the hyperbolic plane which has bounded hessian norm and unbounded gradient norm?
Specifically, consider the poincare half-plane model of the 2d hyperbolic ...
- 637
5
votes
1
answer
258
views
Smoothness of boundary of $r$-neighborhood of convex core
The boundary of an $r$-neighborhood of the convex core of hyperbolic $n$-manifold is smooth, by page 73 of Hyperbolic Manifolds and Kleinian Groups. The authors does not provide a proof for this fact. ...
- 113
4
votes
0
answers
127
views
Area lower bound given a mean curvature upper bound?
If $\Sigma$ is a smooth embedded closed hypersurface in $\mathbb R^n$ with (normalized) mean curvature $H\le 1$ (the mean curvature of the unit sphere), then its ($(n-1)$-dimensional) area is at least ...
- 41
11
votes
4
answers
649
views
Introductory textbook on geometry of hyperbolic space
I am looking for an introductory textbook to the geometry of the hyperbolic space $\mathbb{H}^n$. The book should include explicit description of geodesics and horospheres in various models (...
8
votes
0
answers
1k
views
Embed the hyperbolic plane into Euclidean spaces
Can the complete simply-connected surface with constant Gauss curvature -1 be embedded smoothly in the 5-dimensional Euclidean space?
Can the complete simply-connected surface with constant Gauss ...
- 81
2
votes
0
answers
41
views
Cylindrical coordinates in quotient of symmetric space
I am interested in the following situation. Suppose $G/K$ is a symmetric space of non-compact type and $\alpha$ is the axis of a hyperbolic isometry. I am interested in computing the Hessian of the ...
- 445
5
votes
1
answer
534
views
Isomorphism between the hyperbolic space and the manifold of SPD matrices with constant determinant
I'm studying properties of the manifold of symmetric positive-definite (SPD) matrices and I've learnt about the following connection to the hyperbolic space [1, Section 2.2], $$\mathcal{P}(n) = \...
- 281
7
votes
1
answer
546
views
Can a hyperbolic manifold be a product?
I was interested in whether a manifold which admits a metric of constant sectional curvature can be homotopy equivalent to a product of non-contractible manifolds. Of course, there are three cases: ...
- 19.3k
2
votes
0
answers
108
views
How to compute expansion factors for hyperbolic rational maps?
It is a commonly-referenced result about certain rational maps acting on $\mathbb{\hat{C}}$ that they are expanding on a neighborhood of their Julia sets. A sufficient condition to be expanding is ...
- 533
6
votes
1
answer
157
views
Configurations of $n$ points modulo isometries of the ambient space
Let $M$ be a Riemannian manifold and let $n$ a positive integer. Denote by $F_n(M) \subset M^n$ the space of all $n$-tuples of pairwise distinct points from $M$. The isometries of $M$ act co-ordinate ...
- 63