Geodesic
Certain homotopies in the space of closed curves
Izvestiya. Mathematics, Tome 17 (1981) no. 3, pp. 423-453 Cet article a éte moissonné depuis la source Math-Net.Ru

Voir la notice de l'article

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. 
@article{IM2_1981_17_3_a0,
     author = {D. V. Anosov},
     title = {Certain homotopies in the space of closed curves},
     journal = {Izvestiya. Mathematics},
     pages = {423--453},
     year = {1981},
     volume = {17},
     number = {3},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/IM2_1981_17_3_a0/}
}
TY  - JOUR
AU  - D. V. Anosov
TI  - Certain homotopies in the space of closed curves
JO  - Izvestiya. Mathematics
PY  - 1981
SP  - 423
EP  - 453
VL  - 17
IS  - 3
UR  - http://geodesic.mathdoc.fr/item/IM2_1981_17_3_a0/
LA  - en
ID  - IM2_1981_17_3_a0
ER  - 
%0 Journal Article
%A D. V. Anosov
%T Certain homotopies in the space of closed curves
%J Izvestiya. Mathematics
%D 1981
%P 423-453
%V 17
%N 3
%U http://geodesic.mathdoc.fr/item/IM2_1981_17_3_a0/
%G en
%F IM2_1981_17_3_a0
D. V. Anosov. Certain homotopies in the space of closed curves. Izvestiya. Mathematics, Tome 17 (1981) no. 3, pp. 423-453. http://geodesic.mathdoc.fr/item/IM2_1981_17_3_a0/

[1] Klingenberg W., Lectures on closed geodesies, Grundlehren der mathematischen Wissenschaften, 230, Springer, Berlin, Heidelberg, New York, 1978 | MR | Zbl

[2] Klingenberg W., “Simple closed geodesies on pinched spheres”, J. of differential geometry, 2:3 (1968), 225–232 | MR | Zbl

[3] Flaschel P., Klingenberg W., Riemannsche Hilbertmannigfaltigkeiten. Periodische Geodätische, Lecture Notes in Math., 282, Springer, Berlin, Heidelberg, New York, 1972 | MR | Zbl

[4] Abraham R., Robbin J., Transversal mappings and flows, Benjamin, New York, Amsterdam, 1967 | MR

[5] Malgranzh B., Idealy differentsiruemykh funktsii, Mir, M., 1968

[6] Alber S. I., “O periodicheskoi zadache variatsionnogo ischisleniya v tselom”, Uspekhi matem. nauk, 12:4 (1957), 57–124 | MR

[7] Lyusternik L. A., Topologiya funktsionalnykh prostranstv i variatsionnoe ischislenie v tselom, Tr. Matem. in-ta im. V. A. Steklova AN SSSR, 19, M., L., 1947 | MR | Zbl

[8] Bredon G. E., “On the continuous image of a singular chain complex”, Pacific Journal of Mathematics, 15:4 (1965), 1115–1118 | MR | Zbl

[9] Alber S. I., “Topologiya funktsionalnykh mnogoobrazii i variatsionnoe ischislenie v tselom”, Uspekhi matem. nauk, 25:4 (1970), 57–122 | MR

[10] Kolmogorov A. N., Fomin S. V., Elementy teorii funktsii i funktsionalnogo analiza, Nauka, M., 1968 | MR | Zbl