Nondegenerate homotopies of curves on the unit 2-sphere

Little, John A.

doi:10.4310/jdg/1214429507

Cited by 45 publications

(44 citation statements)

References 3 publications

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“…If ao is an embedding and ai is not, then there must exist t e (0, 1) such that the geodesic curvature of at changes sign. This somewhat strengthens a result of Little [4,Proposition 3].…”

Section: The Proof Of Theoremsupporting

confidence: 66%

Isoperimetric inequalities for immersed closed spherical curves

Weiner¹

1994

Proc. Amer. Math. Soc.

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“…If ao is an embedding and ai is not, then there must exist t e (0, 1) such that the geodesic curvature of at changes sign. This somewhat strengthens a result of Little [4,Proposition 3].…”

Section: The Proof Of Theoremsupporting

confidence: 66%

Isoperimetric inequalities for immersed closed spherical curves

Weiner¹

1994

Proc. Amer. Math. Soc.

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“…Some information about the topology of LS n (Q), mostly in the case Q = I or in the case n = 2, was earlier obtained in [1], [6], [8], [9], [11], [12] and [13]. In particular, it was shown that the number of connected components of LS n (I) equals 3 for even n and 2 for odd n > 1, which is related to the existence of closed globally convex curves on all even-dimensional spheres.…”

Section: Introduction and Main Resultsmentioning

confidence: 99%

Spaces of Locally Convex Curves in S^n and Combinatorics of the Group B_(n+1)^+

Shapiro¹,

Saldanha²

2012

jsing

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Abstract. In the 1920's Marston Morse developed what is now known asMorse theory trying to study the topology of the space of closed curves on S 2 ([7], [5]). We propose to attack a very similar problem, which 80 years later remains open, about the topology of the space of closed curves on S 2 which are locally convex (i.e., without inflection points). One of the main difficulties is the absence of the covering homotopy principle for the map sending a nonclosed locally convex curve to the Frenet frame at its endpoint.In the present paper we study the spaces of locally convex curves in S n with a given initial and final Frenet frames. Using combinatorics of B + n+1 = B n+1 ∩ SO n+1 , where B n+1 ⊂ O n+1 is the usual Coxeter-Weyl group, we show that for any n ≥ 2 these spaces fall in at most n 2 + 1 equivalence classes up to homeomorphism. We also study this classification in the double cover Spin(n + 1). For n = 2 our results complete the classification of the corresponding spaces into two topologically distinct classes, or three classes in the spin case.

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“…Such curves are also called nondegenerate or locally convex in the literature. Works on this problem include [21,22,24,33]; see also [3]. More generally, n-th order free curves on n-dimensional manifolds have been studied by several authors over the years; we mention [1,2,13,14,15,16,25,29,34,38,44].…”

Section: Introductionmentioning

confidence: 99%

Components of spaces of curves with constrained curvature on flat surfaces

Saldanha

Zühlke

2016

Pacific J. Math.

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Abstract. Let S be a complete flat surface, such as the Euclidean plane. We obtain direct characterizations of the connected components of the space of all curves on S which start and end at given points in given directions, and whose curvatures are constrained to lie in a given interval, in terms of all parameters involved. Many topological properties of these spaces are investigated. Some conjectures of L. E. Dubins are proved.

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Nondegenerate homotopies of curves on the unit 2-sphere

Cited by 45 publications

References 3 publications

Isoperimetric inequalities for immersed closed spherical curves

Isoperimetric inequalities for immersed closed spherical curves

Spaces of Locally Convex Curves in S^n and Combinatorics of the Group B_(n+1)^+

Components of spaces of curves with constrained curvature on flat surfaces

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