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is an example of a closed geodesic. Thus, on the sphere, all geodesics are closed. On a smooth surface topologically equivalent to the sphere, this may not be true, but there are always at least three simple closed geodesics; this is the
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587:. Manifolds all of whose geodesics are closed have been thoroughly investigated in the mathematical literature. On a compact hyperbolic
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that returns to its starting point with the same tangent direction. It may be formalized as the projection of a closed orbit of the
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302:{\displaystyle E(\gamma )=\int _{0}^{1}g_{\gamma (t)}({\dot {\gamma }}(t),{\dot {\gamma }}(t))\,\mathrm {d} t.}
591:, whose fundamental group has no torsion, closed geodesics are in one-to-one correspondence with non-trivial
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Closed geodesics can be characterized by means of a variational principle. Denoting by
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is a closed geodesic of period 1, and therefore it is a critical point of
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170:{\displaystyle E:\Lambda M\rightarrow \mathbb {R} }
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648:: "Manifolds all of whose geodesics are closed",
578:with the standard round Riemannian metric, every
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571:{\displaystyle S^{n}\subset \mathbb {R} ^{n+1}}
504:{\displaystyle \gamma ^{m}(t):=\gamma (mt)}
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132:the space of smooth 1-periodic curves on
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367:{\displaystyle t\mapsto \gamma (pt)}
402:, so are the reparametrized curves
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652:, no. 93, Springer, Berlin, 1978.
450:{\displaystyle m\in \mathbb {N} }
511:. Thus every closed geodesic on
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332:is a closed geodesic of period
614:Theorem of the three geodesics
585:theorem of the three geodesics
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422:{\displaystyle \gamma ^{m}}
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650:Ergebisse Grenzgeb. Math.
125:{\displaystyle \Lambda M}
398:is a critical point of
391:{\displaystyle \gamma }
325:{\displaystyle \gamma }
140:of the energy function
676:Geodesic (mathematics)
609:Lyusternik–Fet theorem
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580:great circle
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529:unit sphere
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640:References
98:that is a
48:Definition
646:Besse, A.
548:⊂
487:γ
466:γ
440:∈
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350:γ
347:↦
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270:˙
267:γ
246:˙
243:γ
223:γ
204:∫
194:γ
160:→
154:Λ
117:Λ
83:→
72:γ
603:See also
523:Examples
100:geodesic
34:geodesic
589:surface
527:On the
40:on the
378:. If
52:In a
32:is a
28:on a
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312:If
16:In
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56:(
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