# Questions tagged [closed-geodesic]

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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?
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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
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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 ...
• 577
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### 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
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### 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
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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
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### 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