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The homotopy type of spaces of locally convex curves in the sphere

Nicolau C Saldanha

Geometry & Topology 19 (2015) 1155–1203
Bibliography
1 S S Anisov, Convex curves in $\mathbb{R}\mathrm{P}^n$, Tr. Mat. Inst. Steklova 221 (1998) 9 MR1683685
2 V I Arnol’d, The geometry of spherical curves and quaternion algebra, Uspekhi Mat. Nauk 50 (1995) 3 MR1331356
3 D Burghelea, D Henderson, Smoothings and homeomorphisms for Hilbert manifolds, Bull. Amer. Math. Soc. 76 (1970) 1261 MR0283827
4 D Burghelea, N C Saldanha, C Tomei, Results on infinite-dimensional topology and applications to the structure of the critical set of nonlinear Sturm–Liouville operators, J. Differential Equations 188 (2003) 569 MR1955095
5 D Burghelea, N C Saldanha, C Tomei, The topology of the monodromy map of a second order ODE, J. Differential Equations 227 (2006) 581 MR2237680
6 D Burghelea, N C Saldanha, C Tomei, The geometry of the critical set of nonlinear periodic Sturm–Liouville operators, J. Differential Equations 246 (2009) 3380 MR2507961
7 L A Dikiĭ, I M Gel’fand, Fractional powers of operators, and Hamiltonian systems, Funkcional. Anal. i Priložen. 10 (1976) 13 MR0433508
8 L A Dikiĭ, I M Gel’fand, A family of Hamiltonian structures connected with integrable nonlinear differential equations, Akad. Nauk SSSR Inst. Prikl. Mat. Preprint (1978) 41 MR542603
9 Y Eliashberg, N Mishachev, Introduction to the h–principle, Graduate Studies in Mathematics 48, Amer. Math. Soc. (2002) MR1909245
10 W Fenchel, Über Krümmung und Windung geschlossener Raumkurven, Math. Ann. 101 (1929) 238 MR1512528
11 M Gromov, Partial differential relations, Ergeb. Math. Grenzgeb. 9, Springer, Berlin (1986) MR864505
12 M W Hirsch, Immersions of manifolds, Trans. Amer. Math. Soc. 93 (1959) 242 MR0119214
13 B A Khesin, V Y Ovsienko, Symplectic leaves of the Gel'fand–Dikiĭbrackets and homotopy classes of nonflattening curves, Funktsional. Anal. i Prilozhen. 24 (1990) 38 MR1052266
14 B A Khesin, B Z Shapiro, Nondegenerate curves on $\mathbb{S}^2$ and orbit classification of the Zamolodchikov algebra, Comm. Math. Phys. 145 (1992) 357 MR1162803
15 B A Khesin, B Z Shapiro, Homotopy classification of nondegenerate quasiperiodic curves on the $2$–sphere, from: "Geometric combinatorics" (editors R Živaljević, S Vrećica), Publ. Inst. Math. $($Beograd$)$ 66(80) (1999) 127 MR1765043
16 S Levy, D Maxwell, T Munzner (directors), Outside in, A K Peters, Ltd. (1995) MR1357901
17 J A Little, Nondegenerate homotopies of curves on the unit $2$–sphere, J. Differential Geometry 4 (1970) 339 MR0275333
18 M Morse, The calculus of variations in the large, AMS Colloq. Publ. 18, Amer. Math. Soc. (1996) MR1451874
19 N C Saldanha, The homotopy and cohomology of spaces of locally convex curves in the sphere, I, (2009) arXiv:0905.2111v1
20 N C Saldanha, The homotopy and cohomology of spaces of locally convex curves in the sphere, II, (2009) arXiv:0905.2116v1
21 N C Saldanha, B Shapiro, Spaces of locally convex curves in $\mathbb S^n$ and combinatorics of the group $B^+_{n+1}$, J. Singul. 4 (2012) 1 MR2872212
22 N C Saldanha, C Tomei, The topology of critical sets of some ordinary differential operators, from: "Contributions to nonlinear analysis" (editors T Cazenave, D Costa, O Lopes, R Manásevich, P Rabinowitz, B Ruf, C Tomei), Progr. Nonlinear Differential Equations Appl. 66, Birkhäuser, Basel (2006) 491 MR2187823
23 N C Saldanha, P Zühlke, On the components of spaces of curves on the $2$–sphere with geodesic curvature in a prescribed interval, Internat. J. Math. 24 (2013) 1350101, 78 MR3163615
24 B Z Shapiro, M Z Shapiro, On the number of connected components in the space of closed nondegenerate curves on $\mathbb{S}^n$, Bull. Amer. Math. Soc. 25 (1991) 75 MR1080005
25 M Z Shapiro, The topology of the space of nondegenerate curves, Izv. Ross. Akad. Nauk Ser. Mat. 57 (1993) 106 MR1252758
26 S Smale, The classification of immersions of spheres in Euclidean spaces, Ann. of Math. 69 (1959) 327 MR0105117
27 P Zühlke, Homotopies of curves on the $2$–sphere with geodesic curvature in a prescribed interval, arXiv:1304.3040