Neighborhoods and Manifolds of Immersed Curves

Abstract

We present some fine properties of immersions $\mathrm{ℐ}:M⟶N$ between manifolds, with particular attention to the case of immersed curves $c:S^1⟶{ℝ}^n$ . We present new results, as well as known results but with quantitative statements (that may be useful in numerical applications) regarding tubular coordinates, neighborhoods of immersed and freely immersed curve, and local unique representations of nearby such curves, possibly “up to reparameterization.” We present examples and counterexamples to support the significance of these results. Eventually, we provide a complete and detailed proof of a result first stated in a 1991-paper by Cervera, Mascaró, and Michor: the quotient of the freely immersed curves by the action of reparameterization is a smooth (infinite dimensional) manifold.