Results on the homotopy type of the spaces of locally convex curves on 𝕊 3
Annales de l'Institut Fourier, Volume 69 (2019) no. 3, p. 1147-1185

A curve γ:[0,1]𝕊 n of class C k (kn) is locally convex if the vectors γ(t),γ ' (t),γ '' (t),,γ (n) (t) are a positive basis to n+1 for all t[0,1]. Given an integer n2 and QSO n+1 , let 𝕊 n (Q) be the set of all locally convex curves γ:[0,1]𝕊 n with fixed initial and final Frenet frame γ (0)=I and γ (1)=Q. Saldanha and Shapiro proved that there are just finitely many non-homeomorphic spaces among 𝕊 n (Q) when Q varies in SO n+1 (in particular, at most 3 for n=3). For any n2, one of these spaces is proved to be homeomorphic to the (well understood) space of generic curves (see below), but very little is known in general about the others. For n=2, Saldanha determined the homotopy type of the spaces 𝕊 2 (Q). The purpose of this work is to study the case n=3. We will obtain information on the homotopy type of one of these two other spaces, allowing us to conclude that none of the connected components of 𝕊 3 (-I) is homeomorphic to a connected component of the space of generic curves.

La courbe γ:[0,1]𝕊 n de classe C k (kn) est localement convexe si les vecteurs γ(t),γ ' (t),γ '' (t),,γ (n) (t) forment une base positive de n+1 pour chaque t[0,1]. Pour un entier n2 et QSO n+1 , soit 𝕊 n (Q) l’ensemble de toutes les courbes localement convexes γ:[0,1]𝕊 n avec repères de Frenet initial et final fixes γ (0)=I et γ (1)=Q. Saldanha et Shapiro ont demontré qu’il n’y a qu’un nombre fini d’espaces non-homéomorphes parmi les 𝕊 n (Q) avec QSO n+1 (en particulier, au plus 3 pour n=3). Pour n2, ils demontrent qu’un de ces espaces est homéomorphe à l’espace (bien compris) des courbes génériques (défini ci-dessous) mais on connaît très peu les autres espaces. Pour n=2, Saldanha a déterminé le type d’homotopie des espaces 𝕊 2 (Q). Le but de ce travail est d’étudier le cas n=3. On obtient des informations sur le type d’homotopie d’un de ces autres deux espaces, ce qui nous permet de déduire qu’aucune des composantes connexes de 𝕊 3 (-I) n’est homéomorphe à une composante connexe de l’espace des courbes génériques.

Received : 2017-04-12
Revised : 2017-10-16
Accepted : 2018-02-02
Published online : 2019-06-03
Classification:  57N12,  57N35,  57N65
Keywords: Locally convex curves, homotopy type, Bruhat decomposition
     author = {Alves, Em\'\i lia and Saldanha, Nicolau C.},
     title = {Results on the homotopy type of the spaces of locally convex curves on $\protect \mathbb{S}^3$},
     journal = {Annales de l'Institut Fourier},
     publisher = {Association des Annales de l'institut Fourier},
     volume = {69},
     number = {3},
     year = {2019},
     pages = {1147-1185},
     doi = {10.5802/aif.3267},
     zbl = {07067428},
     language = {en},
     url = {}
Results on the homotopy type of the spaces of locally convex curves on $\protect \mathbb{S}^3$. Annales de l'Institut Fourier, Volume 69 (2019) no. 3, pp. 1147-1185. doi : 10.5802/aif.3267.

[1] Alves, Emília Topology of the spaces of locally convex curves on the 3-sphere, PUC-Rio, Rio de Janeiro (Brazil) (2016) (Ph. D. Thesis)

[2] Anisov, Sergej Convex curves in ℝℙ n , Proc. Steklov Inst. Math., Tome 221 (1998) no. 2, pp. 3-39 | Zbl 1017.52001

[3] Arnol’D, Vladimir I. The geometry of spherical curves and the algebra of quaternions, Russ. Math. Surv., Tome 50 (1995) no. 1, pp. 1-68 | Zbl 0848.58005

[4] Burghelea, Dan; Saldanha, Nicolau C.; Tomei, Carlos Results on infinite dimensional topology and applications to the structure of the critical sets of nonlinear Sturm–Liouville operators, J. Differ. Equations, Tome 188 (2003) no. 2, pp. 569-590 | Article | MR 1955095 | Zbl 1025.34023

[5] Casals, Roger; Pérez, Jose Luis; Pino, Álvaro Del; Presas, Francisco Existence of h-principle for Engel structures (2015) ( )

[6] Dubins, Lester E. On curves of minimal length with a constraint on average curvature, and with prescribed initial and terminal positions and tangents, Am. J. Math., Tome 79 (1957) no. 3, pp. 497-516 | Article | MR 89457 | Zbl 0098.35401

[7] Eliashberg, Yakov; Mishachëv, Nikolai Introduction to the h-principle, American Mathematical Society, Graduate Studies in Mathematics, Tome 48 (2002), xvii+206 pages | MR 1909245 | Zbl 1008.58001

[8] Fenchel, Werner Über Krümmung und Windung geschlossener Raumkurven, Math. Ann., Tome 101 (1929), pp. 238-252 | Article | MR 1512528 | Zbl 55.0394.06

[9] Gromov, Mikhael Partial Differential Relations, Springer, Ergebnisse der Mathematik und ihrer Grenzgebiete. 3. Folge., Tome 9 (1986), ix+363 pages | MR 864505 | Zbl 0651.53001

[10] Hirsch, Morris W. Immersions of manifolds, Trans. Am. Math. Soc., Tome 93 (1959), pp. 242-276 | Article | MR 119214 | Zbl 0113.17202

[11] Khesin, Boris A.; Shapiro, Boris Z. Nondegenerate curves on S 2 and orbit classification of the Zamolodchikov algebra, Commun. Math. Phys., Tome 145 (1992) no. 2, pp. 357-362 | MR 1162803 | Zbl 0849.17026

[12] Little, John A. Nondegenerate homotopies of curves on the unit 2-sphere, J. Differ. Geom., Tome 4 (1970) no. 3, pp. 339-348 | Article | MR 275333 | Zbl 0198.53603

[13] Alvaro Del Pino, Francisco Presas Flexibility for tangent and transverse immersions in Engel manifolds (2016) ( )

[14] Reeds, James A.; Shepp, Lawrence A. Optimal paths for a car that goes both forwards and backwards, Pac. J. Math., Tome 145 (1990) no. 2, pp. 367-393 | Article | MR 1069892

[15] Saldanha, Nicolau C. The homotopy and cohomology of spaces of locally convex curves in the sphere - I (2009) ( )

[16] Saldanha, Nicolau C. The homotopy and cohomology of spaces of locally convex curves in the sphere - II (2009) ( )

[17] Saldanha, Nicolau C. The homotopy type of spaces of locally convex curves in the sphere, Geom. Topol., Tome 19 (2015), pp. 1155-1203 | Article | MR 3352233 | Zbl 1318.53066

[18] Saldanha, Nicolau C.; Shapiro, Boris Z. Spaces of locally convex curves in 𝕊 n and combinatorics of the group B n+1 + , J. Singul., Tome 4 (2012), pp. 1-22 | MR 2872212 | Zbl 1292.58002

[19] Shapiro, Boris Z.; Khesin, Boris A. Homotopy classification of nondegenerate quasiperiodic curves on the 2-sphere, Publ. Inst. Math., Nouv. Sér., Tome 66 (1999) no. 80, pp. 127-156 | MR 1765043 | Zbl 1274.53055

[20] Shapiro, Boris Z.; Shapiro, Michael Z. On the number of connected components in the space of closed nondegenerate curves on 𝕊 n , Bull. Am. Math. Soc., Tome 25 (1991) no. 1, pp. 75-79 | Article | MR 1080005 | Zbl 0731.53003

[21] Shapiro, Michael Z. Topology of the space of nondegenerate curves, Izv. Ross. Akad. Nauk, Ser. Mat., Tome 57 (1993) no. 5, pp. 106-126 | Zbl 0821.57021

[22] Smale, Stephen The classification of immersions of spheres in Euclidean spaces, Ann. Math., Tome 69 (1959) no. 2, pp. 327-344 | MR 105117 | Zbl 0089.18201

[23] Zühlke, Pedro Homotopies of curves on the 2-sphere with geodesic curvature in a prescribed interval, PUC-Rio, Rio de Janeiro (Brazil) (2013) (Ph. D. Thesis)