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International youth conference "Geometry & Control"
April 18, 2014 12:35, Moscow, Steklov Mathematical Institute of RAS
 


Local and Metric Geometry of Nonregular Weighted Carnot–Carathéodory Spaces

Svetlana Selivanova

Sobolev Institute of Mathematics, Novosibirsk, Russia
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Flash Video 1,115.8 Mb
MP4 186.3 Mb
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Svetlana Selivanova


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Abstract: We investigate local and metric geometry of weighted Carnot–Carathéodory spaces in a neighbourhood of a nonregular point [8]. Such spaces are a wide generalization of classical sub-Riemannian spaces (which are smooth manifolds equipped by bracket-generating distributions of “horizontal” vector fields) and naturally arise in control theory (including cases when the dependence on control functions may be nonlinear), harmonic analysis, subelliptic equations etc.
For the spaces that we consider, there may be no analog of the intrinsic Carnot–Carathéodory metric (defined in sub-Riemannian geometry as the infimum of lengths of all “horizontal” curves joining the two given points) might not exist, and some other new effects, caused by the arbitrary weights of the vector fields, take place, which leads to necessity of introducing new methods of investigation of geometry of such spaces.
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 in [5]) induced by the given weighted structure. We compare local geometries of the initial CC space and its tangent cone (which is a homogeneous space of a nilpotent Lie group) at some fixed (maybe nonregular) point.
Our considerations heavily rely on similar results about equiregular Carnot–Carathéodory spaces [4,3] and adaptations of different “lifting” methods [6,2,1], which allow to reduce some questions about nonregular spaces to similar questions about the equiregular ones. Also, we use a generalisation to quasimetric spaces of the Gromov–Hausdorff spaces for metric spaces, which was constructed earlier in [7], and study new properties of the considered quasimetrics.

Materials: slides.pdf (111.8 Kb), abstract.pdf (60.9 Kb)

Language: English

References
  1. Jean F., “Uniform estimation of sub-Riemannian balls”, J. of Dynamical and Control Systems, 7:4 (2001), 473–500  crossref  mathscinet  zmath  scopus
  2. Hörmander L., Melin A., “Free systems of vector fields”, Ark. Mat., 16:1 (1978), 83–88  crossref  mathscinet  zmath  scopus
  3. Karmanova M., “Fine properties of basis vector fields of Carnot–Carathéodory spaces under minimal assumptions on smoothness”, Siberian Mathematical Journal, 55:1 (2014), 87–99  mathnet  crossref  mathscinet  zmath  isi  scopus
  4. Karmanova M., Vodopyanov S., “Geometry of Carnot–Carathéodory spaces, differentiability, coarea and area formulas”, Analysis and Mathematical Physics. Trends in Mathematics, Birkhauser, Basel, 2009, 233–335  mathscinet  zmath
  5. Nagel A., Stein E.M., Wainger S., “Balls and metrics defined by vector fields I: Basic properties”, Acta Math, 155 (1985), 103–147  crossref  mathscinet  zmath  isi  scopus
  6. Rotshild L. P., Stein E. M, “Hypoelliptic differential operators and nilpotent groups”, Acta Math, 137 (1976), 247–320  crossref  mathscinet  isi  scopus
  7. Selivanova S.V., “Tangent cone to a quasimetric space with dilations”, Siberian Mathematical Journal, 51:2 (2010), 388–403  mathnet  crossref  mathscinet  zmath  scopus
  8. Selivanova S., “Metric geometry of nonregular weighted Carnot–Carathéodory spaces”, Journal of Dynamical Control Systems, 20 (2014), 123–148  crossref  mathscinet  zmath  isi  scopus


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