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Metric Geometry of Nonregular Weighted Carnot–Carathéodory Spaces

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An Erratum to this article was published on 10 February 2015

Abstract

We investigate local and metric geometry of weighted Carnot–Carathéodory spaces which are a wide generalization of sub-Riemannian manifolds and arise in nonlinear control theory, subelliptic equations, etc. For such spaces, the intrinsic Carnot–Carathéodory metric might not exist, and some other new effects take place. We describe the local algebraic structure of such a space, endowed with a natural quasimetric (first introduced by A. Nagel, E. M. Stein, and S. Wainger) and compare local geometries of the initial Carnot–Carathéodory (CC) space and its tangent cone at some fixed (possibly nonregular) point. The main results of the present paper, in particular, the theorem on divergence of integral lines and other estimates obtained for the quasimetrics, are new even for the case of sub-Riemannian manifolds.

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Notes

  1. This research was partially supported by the Federal Target Grant “Scientific and educational personnel of innovative Russia” for 2009–2013, government contract no. 8206, by the Integration Project SB RAS–FEB RAS, no. 56, and by the Russian Foundation for Basic Research (project No. 12-01-31183).

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Acknowledgments

I thank Professor Sergei Vodopyanov for suggesting me this problematics, his interest to this work, and many fruitful discussions; Professor Andrei Agrachev for interesting discussions on the subject and useful references; Doctor Maria Karmanova for a consultation on her papers; and the anonymous referee for useful remarks.

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  1. Sobolev Institute of Mathematics, Russian Academy of Sciences, Acad. Koptyug av. 4, Novosibirsk, 630090, Russia

    Svetlana Selivanova

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  1. Svetlana Selivanova
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Correspondence to Svetlana Selivanova.

Additional information

This research was partially supported by the Federal Target Grant “Scientific and educational personnel of innovative Russia” for 2009–2013, government contract no. 8206, and by the Integration Project SB RAS–FEB RAS, no. 56.

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Selivanova, S. Metric Geometry of Nonregular Weighted Carnot–Carathéodory Spaces. J Dyn Control Syst 20, 123–148 (2014). https://doi.org/10.1007/s10883-013-9206-3

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  • DOI: https://doi.org/10.1007/s10883-013-9206-3

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