Webb12 apr. 2024 · Find parametric equations for a simple closed curve of length 4π on the unit sphere which minimizes the mean spherical distance from the curve to the sphere; the solution must include proof of minimization. Can you solve this problem with arbitrary L > 2π instead of 4π? There seems to be little precedent for this problem. Webb12 mars 2013 · We investigate the relationship, in various contexts, between a closed geodesic with self-intersection number k (for brevity, called a k-geodesic) and its length. …
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WebbTheorem 1.1 The set of surfaces with simple simple length spectrum is dense and its complement is Baire meagre. If A is a path in Teichmüller space T then there is a surface on A which has at least two distinct simple closed geodesics of the same length. Let E denote the set of all surfaces with at least one pair of simple closed geodesics of In differential geometry and dynamical systems, a closed geodesic on a Riemannian manifold is a geodesic that returns to its starting point with the same tangent direction. It may be formalized as the projection of a closed orbit of the geodesic flow on the tangent space of the manifold. Visa mer On the unit sphere $${\displaystyle S^{n}\subset \mathbb {R} ^{n+1}}$$ with the standard round Riemannian metric, every great circle is an example of a closed geodesic. Thus, on the sphere, all geodesics are … Visa mer • Lyusternik–Fet theorem • Theorem of the three geodesics • Curve-shortening flow Visa mer lithocysts in ficus
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WebbThe study of closed geodesics on hyperbolic surfaces has multiple facets which links together topics as diverse as spectral theory, symbolic dynamics, geometric topology … Webbgeodesic current with length measure gives an invariant measure for the geodesic flow. Remark. The geodesic flow cannot be reconstructed from the topological action of Γ on S1, since its time parameterization determines the lengths of closed geodesics. Intersection number. Let I ⊂ G×G be the set of pairs of geodesics (α,β) that cross ... WebbShrinking all simple closed geodesics Consider a foliation E of the hyperbolic plane H2 by the set of curves that are equidistant from a given geodesic, and consider the foliation G of H2 by the curves that are orthogonal to the leaves of E … lithocysts in plants