Calculus of Variations and Geometric Measure Theory

A. C. G. Mennucci

Neighborhoods and manifolds of immersed curves

created by mennucci on 22 Nov 2021
modified on 11 Feb 2022


Published Paper

Inserted: 22 nov 2021
Last Updated: 11 feb 2022

Journal: International Journal of Mathematics and Mathematical Sciences
Year: 2021
Doi: 10.1155/2021/6974292


We present some fine properties of immersions $i:M\to N$ between manifolds; with particular attention to the case of immersed curves $c:S^1\to {\mathbb R}^n$. 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, local unique representations of nearby such curves, possibly "up to reparameterization". We present examples and counter-examples 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, MascarĂ² and Michor: the quotient of the freely immersed curves by the action of reparameterization is a smooth (infinite dimensional) manifold.

Keywords: Immersed curves, tubular neighborhoods, diffeomorphisms, manifold of mappings