Questions tagged [closed-geodesic]
The closed-geodesic tag has no usage guidance.
18
questions
2
votes
1
answer
156
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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 ...
- 399
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 ...
- 577
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 ...
- 23
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 ...
- 2,073
3
votes
1
answer
231
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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 ...
- 149k
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 ...
- 2,073
4
votes
1
answer
150
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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(\...
- 41
1
vote
0
answers
145
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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).
...
- 149k
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 $\gamma$.
...
- 91
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 ...
- 2,073
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 ...
- 42.3k
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 ...
- 2,236
24
votes
3
answers
1k
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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.
...
- 42.3k
1
vote
1
answer
181
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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 ...
- 270
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 ...
- 270
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 $\gamma_1, \gamma_2$ ...
- 270
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 $\, \...
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