# Questions tagged [closed-geodesic]

The closed-geodesic tag has no usage guidance.

18
questions

2
votes

1
answer

156
views

### geodesics on a compact manifold

Let $M$ be a compact Riemann manifold without boundary. Please is this true that each homotopy class of closed curves contains a geodesic?

5
votes

2
answers

327
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### Closed geodesics on bumpy spheres

Main question:
Does every bumpy Riemannian metric on a sphere have at least three short and prime closed geodesics, for some reasonable definition of short?
E.g., a geodesic $\gamma$ could be called ...

1
vote

0
answers

73
views

### Translate of a geodesic that goes through a fixed point on $\mathbb{H}$

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

100
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 ...

6
votes

1
answer

116
views

### On properties of Besse spheres

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

231
views

### Infinite number of closed geodesics on distorted sphere

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

92
views

### When are nodal lines on a sphere geodesics?

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

150
views

### What is the minimal length of a “Diagonal” in a Torus?

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

145
views

### Angles between simple, closed geodesics on convex surface

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

706
views

### Is there the longest geodesic?

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

217
views

### Infinitely many simple closed geodesics in any compact orientable surface but the sphere

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

449
views

### Metrics on torus without closed contractible geodesics

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

100
views

### 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 ...

24
votes

3
answers

1k
views

### A problem on Gauss--Bonnet formula

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

181
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 ...

2
votes

1
answer

108
views

### A foliation of the cylinder by closed geodesics of the same length when the metric is complete but non flat

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

355
views

### (Some possible obstructions to ) Limit cycles as closed geodesics(3)

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

454
views

### Closed geodesics on constant positive Gauss curvature surfaces

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 $\, \...