Calculus of Variations and Geometric Measure Theory
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A. C. G. Mennucci

Neighborhoods and manifolds of immersed curves

created by mennucci on 22 Nov 2021


Accepted Paper

Inserted: 22 nov 2021
Last Updated: 22 nov 2021

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


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