{"title":"Tubes in sub-Riemannian geometry and a Weyl's invariance result for curves in the Heisenberg groups","authors":"Tania Bossio, Luca Rizzi, Tommaso Rossi","doi":"arxiv-2408.16838","DOIUrl":null,"url":null,"abstract":"The purpose of the paper is threefold: first, we prove optimal regularity\nresults for the distance from $C^k$ submanifolds of general rank-varying\nsub-Riemannian structures. Then, we study the asymptotics of the volume of\ntubular neighbourhoods around such submanifolds. Finally, for the case of\ncurves in the Heisenberg groups, we prove a Weyl's invariance result: the\nvolume of small tubes around a curve does not depend on the way the curve is\nisometrically embedded, but only on its Reeb angle. The proof does not need the\ncomputation of the actual volume of the tube, and it is new even for the\nthree-dimensional Heisenberg group.","PeriodicalId":501444,"journal":{"name":"arXiv - MATH - Metric Geometry","volume":null,"pages":null},"PeriodicalIF":0.0000,"publicationDate":"2024-08-29","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":null,"platform":"Semanticscholar","paperid":null,"PeriodicalName":"arXiv - MATH - Metric Geometry","FirstCategoryId":"1085","ListUrlMain":"https://doi.org/arxiv-2408.16838","RegionNum":0,"RegionCategory":null,"ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"","JCRName":"","Score":null,"Total":0}
引用次数: 0
Abstract
The purpose of the paper is threefold: first, we prove optimal regularity
results for the distance from $C^k$ submanifolds of general rank-varying
sub-Riemannian structures. Then, we study the asymptotics of the volume of
tubular neighbourhoods around such submanifolds. Finally, for the case of
curves in the Heisenberg groups, we prove a Weyl's invariance result: the
volume of small tubes around a curve does not depend on the way the curve is
isometrically embedded, but only on its Reeb angle. The proof does not need the
computation of the actual volume of the tube, and it is new even for the
three-dimensional Heisenberg group.