Whitney's Extension Theorem and the finiteness principle for curves in\n the Heisenberg group

2021/07/09 by Scott Zimmerman, Zimmerman, Scott · 1 citation
Computer Science · Mathematics · #Topological and Geometric Data Analysis #Geometric Analysis and Curvature Flows #Morphological variations and asymmetry

paper · pdf · doi:10.48550/arxiv.2107.04554

Abstract

Consider the sub-Riemannian Heisenberg group \ℍ. In this paper, we\nanswer the following question: given a compact set K \⊆ \ℝ and\na continuous map f:K \→ \ℍ, when is there a horizontal C^m curve\nF:\ℝ \→ \ℍ such that F|_K = f? Whitney originally answered\nthis question for real valued mappings, and Fefferman provided a complete\nanswer for real valued functions defined on subsets of \ℝⁿ. We also\nprove a finiteness principle for C^m,\√(\ω) horizontal curves in the\nHeisenberg group in the sense of Brudnyi and Shvartsman.\n

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• Directional Pliability, Whitney Extension, and Lusin Approximation for Curves in Carnot Groups

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