Geodesic
Topology of the space of nondegenerate curves
Izvestiya. Mathematics , Tome 43 (1994) no. 2, pp. 291-310.

Voir la notice de l'article provenant de la source Math-Net.Ru

A curve on a sphere or on a projective space is called nondegenerate if it has a nondegenerate moving frame at every point. The number of homotopy classes of closed nondegenerate curves immersed in the sphere or projective space is computed. In the case of the sphere $S^n$, this turns out to be 4 for odd $n\geqslant 3$ and 6 for even $n\geqslant 2$; in the case of the projective space $\mathbf P^n$, 10 for odd $n\geqslant 3$ and 3 for even $n\geqslant 2$. 
@article{IM2_1994_43_2_a4,
     author = {M. Z. Shapiro},
     title = {Topology of the space of nondegenerate curves},
     journal = {Izvestiya. Mathematics },
     pages = {291--310},
     publisher = {mathdoc},
     volume = {43},
     number = {2},
     year = {1994},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/IM2_1994_43_2_a4/}
}
TY  - JOUR
AU  - M. Z. Shapiro
TI  - Topology of the space of nondegenerate curves
JO  - Izvestiya. Mathematics 
PY  - 1994
SP  - 291
EP  - 310
VL  - 43
IS  - 2
PB  - mathdoc
UR  - http://geodesic.mathdoc.fr/item/IM2_1994_43_2_a4/
LA  - en
ID  - IM2_1994_43_2_a4
ER  - 
%0 Journal Article
%A M. Z. Shapiro
%T Topology of the space of nondegenerate curves
%J Izvestiya. Mathematics 
%D 1994
%P 291-310
%V 43
%N 2
%I mathdoc
%U http://geodesic.mathdoc.fr/item/IM2_1994_43_2_a4/
%G en
%F IM2_1994_43_2_a4
M. Z. Shapiro. Topology of the space of nondegenerate curves. Izvestiya. Mathematics , Tome 43 (1994) no. 2, pp. 291-310. http://geodesic.mathdoc.fr/item/IM2_1994_43_2_a4/

[1] Gelfand I. M., Dikii L. A., Semeistvo gamiltonovykh struktur, svyazannykh s integriruemymi nelineinymi differentsialnymi uravneniyami, Preprint IPM AN SSSR, 1978

[2] Lazutkin V. P., Pankratova T. F., “Normalnye formy i versalnye deformatsii uravnenii Khilla”, Funk. analiz i ego prilozh., 9:4 (1975), 41–48 | MR | Zbl

[3] Ovsienko V. Yu., “Klassifikatsiya lineinykh differentsialnykh uravnenii tretego poryadka i simplekticheskikh listov skobki Gelfanda–Dikogo”, Matem. zametki, 47:5 (1990), 62–69 | MR

[4] Ovsienko V. Yu., Khesin B. A., “Simplekticheskie listy skobok Gelfanda–Dikogo i gomotopicheskie klassy neuploschayuschikhsya krivykh”, Funk. analiz i ego prilozh., 24:1 (1990), 38–47 | MR | Zbl

[5] Fuks D. B., Fomenko A. T., Kurs gomotopicheskoi topologii, Nauka, M., 1989 | MR | Zbl

[6] Khartman F., Obyknovennye differentsialnye uravneniya, Mir, M., 1970 | MR | Zbl

[7] Shapiro B. Z., “Prostranstvo neostsillyatsionnykh differentsialnykh uravnenii i mnogoobraziya flagov”, Izv. AN SSSR. Ser. matem., 54:1 (1990), 173–187 | MR | Zbl

[8] Hamenstadt U., “Zur Theorie von Carno–Caratheodory Metriken und ihren Anwendungen”, Bonner Math. Schr., 180 (1987), 1–64 | MR

[9] Kirillov A. A., “Infinite-dimensional Lie groups: their orbits, invariants and representations Geometry of moments”, Lect. Notes in Math., 970, 1982, 101–123 | MR | Zbl

[10] Kuiper N. H., “Locally proective spaces of dimension one”, Michigan Math. J., 2:2 (1953–1954), 95–97 | DOI | MR

[11] Khesin B. A., Shapiro B. Z., Nondegenerate curves on $S^2$ and orbit classification of the Zamolodchicov algebra, Preprint, Center for pure and applied Math. Univ. of California, Berkeley, PAM-521 | Zbl

[12] Little J., “Nondegenerate homotopies of curves on the unit 2-sphere”, J. Diff. Geom., 4 (1970), 339–348 | MR | Zbl

[13] Little J., “Third order nondegenerate homotopies of the space curves”, J. Diff. Geom., 5 (1971), 503–515 | MR | Zbl

[14] Segal G., “Unitary representations of some infinite dimensional groups”, Coram. Math. Phys., 80:3 (1981), 301–342 | DOI | MR | Zbl

[15] Shapiro B. Z., Shapiro M. Z., “On the number of connected components of the space of closed nondegenerate curves”, Bull. of Amer. Math. Soc.

[16] Sherman T. L., “Conjugate points and simple zeros for ordinary differential equations”, Trans. of the AMS, 146 (1969), 397–411 | DOI | MR