Questions tagged [closed-geodesic]

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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 ...
  • 549
2 votes
1 answer
73 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 ...
  • 23
6 votes
1 answer
100 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
168 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
84 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
138 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
135 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
668 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
191 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
406 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
85 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
157 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
100 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
351 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
436 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 $\, \...