# Questions tagged [closed-geodesic]

The closed-geodesic tag has no usage guidance.

The closed-geodesic tag has no usage guidance.

16
questions

1
vote

0
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59
views

Consider the complex upper half plane $\mathbb{H}$ with the hyperbolic geometry. Fix a point $z \in \mathbb{H}$ and also a geodesic $c$. I want to find a hyperbolic translation $\gamma c$ passes that ...

2
votes

1
answer

73
views

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 ...

6
votes

1
answer

100
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Let $(\mathbb{S}^2,g)$ be a Besse sphere, that is, a Riemannian sphere all of whose geodesics are closed. By a result of Gromoll and Grove, all the geodesics are simple (no self-intersections) and ...

3
votes

1
answer

168
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I would appreciate a reference to support this statement that
appears under the Geodesic entry of the
CRC Encyclopedia of Mathematics:
"no matter how badly a sphere is distorted,
there exists an ...

5
votes

0
answers

84
views

Let $(S^2, g)$ be a Riemannian sphere and let $L := \Delta_{S^2} + q$ be a Schrödinger operator on $S^2$. Suppose that $L$ has index equal to one and that $u \in C^{\infty}(S^2)$ ($u \neq 0$) lies in ...

4
votes

1
answer

138
views

Given a Riemannian torus $(T,d)$ with fundamental group $\pi_1(T)=\langle a,b \mid ab=ba \rangle$. Denote for any $\gamma \in \pi_1(T)$ the infimum length of all representatives of $\gamma$ by $L(\...

1
vote

0
answers

135
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It is known that there are at least three simple, closed geodesics on
the surface of any smooth convex body $K$ in $\mathbb{R}^3$, the
Lusternik-Schnirelmann Theorem (see links below for references).
...

9
votes

3
answers

668
views

Given a closed 2-surface $M$ together with a Riemannian metric $g$.
We pick a free homotopy class $\gamma \in \pi_1(M)$ and consider the set $C(\gamma)$ of all closed geodesics homotopic to $\gamma$.
...

4
votes

0
answers

191
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My question is the following: if $(\Sigma, g)$ is any compact orientable Riemannian surface of genus $g \geq 1$, is it true that there are infinitely many closed, simple and geometrically distinct ...

8
votes

1
answer

406
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It is easy to see that any closed geodesic on a flat 2-torus is noncontractible.
Further the same holds true for a torus of revolution.
Indeed either a closed geodesic is a meridian and therefore ...

0
votes

2
answers

85
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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 ...

24
votes

3
answers

1k
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While teaching a course in differential geometry, I came up with the following problem, which I think is cool.
Assume $\gamma$ is a closed geodesic on a sphere $\Sigma$ with positive Gauss curvature.
...

1
vote

1
answer

157
views

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 ...

2
votes

1
answer

100
views

Is there a complete Riemannian metric on the cylinder such that the metric is not flat but the cylinder is foliated by closed geodesics with the same length?
A possibility non complete ...

3
votes

0
answers

351
views

First we explain our Motivation:
Motivation:
First note that there is no a Riemannian metric on an open set of the plane which possess two nested closed geodesics $\gamma_1, \gamma_2$ ...

8
votes

1
answer

436
views

Can we show that geodesics with a rational radius $ r_{mid-equator} =a q/p \,(p>q ) $ at mid-equator on a spindle type surface of revolution of constant Gauss curvature $ (K=1/a^2 \, $in $\, \...