This article is cited in 6 scientific papers (total in 8 papers)
Certain homotopies in the space of closed curves
D. V. Anosov
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.
PDF file (3365 kB)
Mathematics of the USSR-Izvestiya, 1981, 17:3, 423–453
513.83 + 519.3
MSC: Primary 53C20, 53C22, 55P35; Secondary 55R10, 57R20, 55N99
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
\paper Certain homotopies in the space of closed curves
\jour Izv. Akad. Nauk SSSR Ser. Mat.
\jour Math. USSR-Izv.
Citing articles on Google Scholar:
Related articles on Google Scholar:
This publication is cited in the following articles:
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
I. A. Taimanov, “Closed extremals on two-dimensional manifolds”, Russian Math. Surveys, 47:2 (1992), 163–211
M Clapp, “Critical point theory of symmetric functions and closed geodesics”, Differential Geometry and its Applications, 6:4 (1996), 367
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
I. A. Taimanov, “The type numbers of closed geodesics”, Reg Chaot Dyn, 15:1 (2010), 84
Mark McLean, “The Growth Rate of Symplectic Homology and Affine Varieties”, Geom. Funct. Anal, 2012
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
I. A. Taimanov, “The spaces of non-contractible closed curves in compact space forms”, Sb. Math., 207:10 (2016), 1458–1470
|Number of views:|