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559:– is 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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563:. 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.}
567:, 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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gives rise to an infinite sequence of critical points of the energy
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170:{\displaystyle E:\Lambda M\rightarrow \mathbb {R} }
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624:: "Manifolds all of whose geodesics are closed",
136:, closed geodesics of period 1 are precisely the
91:{\displaystyle \gamma :\mathbb {R} \rightarrow M}
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555:with the standard metric, every geodesic – a
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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628:, no. 93, Springer, Berlin, 1978.
450:{\displaystyle m\in \mathbb {N} }
511:. Thus every closed geodesic on
64:), a closed geodesic is a curve
332:is a closed geodesic of period
590:Theorem of the three geodesics
561:theorem of the three geodesics
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422:{\displaystyle \gamma ^{m}}
336:, the reparametrized curve
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626:Ergebisse Grenzgeb. Math.
125:{\displaystyle \Lambda M}
398:is a critical point of
391:{\displaystyle \gamma }
325:{\displaystyle \gamma }
140:of the energy function
652:Geodesic (mathematics)
585:Lyusternik–Fet theorem
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642:Differential geometry
605:Selberg zeta function
600:Selberg trace formula
595:Curve-shortening flow
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18:differential geometry
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571:of elements in the
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647:Dynamical systems
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542:{\displaystyle n}
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106:and is periodic.
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22:dynamical systems
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551:-dimensional
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42:tangent space
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38:geodesic flow
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610:Zoll surface
557:great circle
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553:unit sphere
429:, for each
636:Categories
616:References
98:that is a
48:Definition
622:Besse, A.
487:γ
466:γ
440:∈
411:γ
386:γ
350:γ
347:↦
320:γ
270:˙
267:γ
246:˙
243:γ
223:γ
204:∫
194:γ
160:→
154:Λ
117:Λ
83:→
72:γ
579:See also
523:Examples
100:geodesic
34:geodesic
565:surface
549:
529:
527:On the
40:on the
378:. If
52:In a
32:is a
28:on a
24:, a
20:and
312:If
16:In
638::
519:.
484::=
537:n
517:E
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490:(
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475:(
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444:N
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219:g
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164:R
157:M
151::
148:E
134:M
120:M
104:g
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79:R
75::
62:g
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58:M
56:(
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