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Izv. Akad. Nauk SSSR Ser. Mat., 1980, Volume 44, Issue 6, Pages 1219–1254 (Mi izv1960)  

This article is cited in 6 scientific papers (total in 8 papers)

Certain homotopies in the space of closed curves

D. V. Anosov


Abstract: It is shown that a smooth homotopy of a Riemannian manifold induces a smooth homotopy of the space of closed curves, and that it is possible to pass to a parametrization of the curves that is proportional to the arc length by means of a certain homotopy in this space. Applications are given to the homology of the space of nonoriented closed curves on a sphere, and errors in some previous articles on this topic are corrected. Despite these errors, it turns out to be possible to repair the proofs of theorems of Klingenberg and Al'ber on closed nonselfintersecting geodesics on a sphere with a Riemannian metric satisfying the $1/4$-pinching condition on the curvature (and, in the Al'ber theorem, also the Morse condition).
Bibliography: 10 titles.

Full text: PDF file (3365 kB)
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English version:
Mathematics of the USSR-Izvestiya, 1981, 17:3, 423–453

Bibliographic databases:

UDC: 513.83 + 519.3
MSC: Primary 53C20, 53C22, 55P35; Secondary 55R10, 57R20, 55N99
Received: 06.05.1980

Citation: D. V. Anosov, “Certain homotopies in the space of closed curves”, Izv. Akad. Nauk SSSR Ser. Mat., 44:6 (1980), 1219–1254; Math. USSR-Izv., 17:3 (1981), 423–453

Citation in format AMSBIB
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\by D.~V.~Anosov
\paper Certain homotopies in the space of closed curves
\jour Izv. Akad. Nauk SSSR Ser. Mat.
\yr 1980
\vol 44
\issue 6
\pages 1219--1254
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\mathscinet{http://www.ams.org/mathscinet-getitem?mr=603576}
\zmath{https://zbmath.org/?q=an:0489.58009|0456.58011}
\transl
\jour Math. USSR-Izv.
\yr 1981
\vol 17
\issue 3
\pages 423--453
\crossref{https://doi.org/10.1070/IM1981v017n03ABEH001367}
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    Citing articles on Google Scholar: Russian citations, English citations
    Related articles on Google Scholar: Russian articles, English articles

    This publication is cited in the following articles:
    1. D. V. Anosov, “Some homology classes in the space of closed curves in the $n$-dimensional sphere”, Math. USSR-Izv., 18:3 (1982), 403–422  mathnet  crossref  mathscinet  zmath
    2. I. A. Taimanov, “Closed extremals on two-dimensional manifolds”, Russian Math. Surveys, 47:2 (1992), 163–211  mathnet  crossref  mathscinet  zmath  adsnasa  isi
    3. M Clapp, “Critical point theory of symmetric functions and closed geodesics”, Differential Geometry and its Applications, 6:4 (1996), 367  crossref
    4. V. I. Arnol'd, A. A. Bolibrukh, R. V. Gamkrelidze, V. P. Maslov, E. F. Mishchenko, S. P. Novikov, Yu. S. Osipov, Ya. G. Sinai, A. M. Stepin, L. D. Faddeev, “Dmitrii Viktorovich Anosov (on his 60th birthday)”, Russian Math. Surveys, 52:2 (1997), 437–445  mathnet  crossref  crossref  mathscinet  zmath  adsnasa  isi
    5. I. A. Taimanov, “The type numbers of closed geodesics”, Reg Chaot Dyn, 15:1 (2010), 84  crossref  isi  elib
    6. Mark McLean, “The Growth Rate of Symplectic Homology and Affine Varieties”, Geom. Funct. Anal, 2012  crossref
    7. S. M. Aseev, V. M. Buchstaber, R. I. Grigorchuk, V. Z. Grines, B. M. Gurevich, A. A. Davydov, A. Yu. Zhirov, E. V. Zhuzhoma, M. I. Zelikin, A. B. Katok, A. V. Klimenko, V. V. Kozlov, V. P. Leksin, M. I. Monastyrskii, A. I. Neishtadt, S. P. Novikov, E. A. Sataev, Ya. G. Sinai, A. M. Stepin, “Dmitrii Viktorovich Anosov (obituary)”, Russian Math. Surveys, 70:2 (2015), 369–381  mathnet  crossref  crossref  mathscinet  adsnasa  isi  elib
    8. I. A. Taimanov, “The spaces of non-contractible closed curves in compact space forms”, Sb. Math., 207:10 (2016), 1458–1470  mathnet  crossref  crossref  mathscinet  adsnasa  isi  elib
  • Известия Академии наук СССР. Серия математическая Izvestiya: Mathematics
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