The space of non-degenerate closed curves in a Riemannian manifold

Abstract

Let LM be the semigroup of non-degenerate based loops with a fixed initial/final frame in a Riemannian manifold M of dimension at least three. We compare the topology of LM to that of the loop space ΩF T M on the bundle of frames in the tangent bundle of M . We show that ΩF T M is the group completion of LM , and prove that it is obtained by localizing LM with respect to adding a "small twist".

On the Components of Spaces of Curves on the 2-Sphere With Geodesic Curvature in a Prescribed Interval

“…The next result states that after we add enough loops to a curve, it becomes so flexible that any condition on the curvature may be safely forgotten. This is yet another instance of the "phone wire" construction already present in [8], [6] and [12]; we refer the reader to [14] for a thorough discussion of this kind of construction in terms of the h-principle.…”

“…14) Definition. Let B be a good band of width R. A track of B is a curve on S 2 ν of length R joining a point of ∂B + to a point of ∂B − .…”

Characterization of some convex curves on the 3-sphere

“…The study of the spaces of locally convex curves started in the seventies with the works of Litte on the 2-sphere. But the research on the topological aspects on these spaces of curves on the spheres of higher dimension as in related spaces is very productive area, here we mention some other relevant works: [9], [19], [26], [27], [28], [31], [32], [33] and [34]. A very hard and interesting question in this topic is to determine the homotopy type of the spaces of locally convex curves on the n-sphere, for n ≥ 3.…”

Components of spaces of curves with constrained curvature on flat surfaces

“…In the present notation, these correspond to CM 0 −∞ (u, u) CM +∞ 0 (u, u). Papers treating this problem for the simplest manifolds, such as R n , S n and RP n , include [1], [5], [6], [8], [9], [11], [12], [13], [16], [19] and [20]. In [18] the connected components of C(S 2 ) κ2 κ1 (u, u) are characterized for all κ 1 < κ 2 , and in [15] the homotopy type of spaces of (not necessarily closed) nondegenerate curves on S 2 is computed.…”

“…We denote by θ γ : [0, 1] → R the unique continuous function satisfying exp(iθ γ ) = t γ and θ γ (0) = 0. Also, (13) φγ := 1 2 max…”