Geodesic
On~the~existence of three nonselfintersecting closed geodesics on manifolds homeomorphic to the 2-sphere
Izvestiya. Mathematics , Tome 40 (1993) no. 3, pp. 565-590

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The author gives a complete proof of the Lyusternik–Shnirel'man theorem that on each smooth Riemannian manifold homeomorphic to the 2-sphere there exist at least three distinct nonselfintersecting closed geodesics (the proof by Lyusternik and Shnirel'man contains substantial gaps). 
@article{IM2_1993_40_3_a4,
     author = {I. A. Taimanov},
     title = {On~the~existence of three nonselfintersecting closed geodesics on manifolds homeomorphic to the 2-sphere},
     journal = {Izvestiya. Mathematics },
     pages = {565--590},
     publisher = {mathdoc},
     volume = {40},
     number = {3},
     year = {1993},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/IM2_1993_40_3_a4/}
}
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I. A. Taimanov. On~the~existence of three nonselfintersecting closed geodesics on manifolds homeomorphic to the 2-sphere. Izvestiya. Mathematics , Tome 40 (1993) no. 3, pp. 565-590. http://geodesic.mathdoc.fr/item/IM2_1993_40_3_a4/