Geodesic
Area of images of measurable sets on depth 2 Carnot manifolds with sub-Lorentzian structure
Vladikavkazskij matematičeskij žurnal, Tome 26 (2024) no. 4, pp. 78-86
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The paper is devoted to analysis of metric properties of images of measurable sets in sub-Lorentzian geometry introduced on Carnot manifolds. The current research continues the results obtained earlier for classes of compact sets on Carnot groups. The main difference is that, firstly, the mapping is defined on a measurable set (not necessarily compact), and, secondly, the preimage and image of the mapping do not have a group structure. Also, the definition of sub-Lorentzian analog of Hausdorff measure (which is not a measure in general) is modified: in contrast to earlier research, it does not require “uniform” sub-Riemannian differentiability. One of results is the property of quasi-additivity of this sub-Lorentzian analog. The latter enables to derive its parameterization by sub-Riemannian Hausdorff measure. In turn, this property means that the sub-Lorentzian analog of Hausdorff measure has classical properties of measure on certain class of sets. The sub-Lorentzian area formula on Carnot manifold is the main result of the paper. We also demonstrate the main ideas of its proof and show their specificity. 
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M. B. Karmanova. Area of images of measurable sets on depth 2 Carnot manifolds with sub-Lorentzian structure. Vladikavkazskij matematičeskij žurnal, Tome 26 (2024) no. 4, pp. 78-86. http://geodesic.mathdoc.fr/item/VMJ_2024_26_4_a6/

[1] Karmanova, M. B., “Metric Characteristics of Classes of Compact Sets on Carnot Groups with Sub-Lorentzian Structure”, Vladikavkaz Mathematical Journal, 26:3 (2024), 47–55 | DOI

[2] Miklyukov, V. M., Klyachin, A. A. and Klyachin, V. A., Maximal Surfaces in Minkowski Space-Time, VolSU, Volgograd, 2011, 530 pp. (in Russian)

[3] Krym, V. R. and Petrov, N. N., “Equations of Motion of a Charged Particle in a Five-Dimensional Model of the General Theory of Relativity with a Nonholonomic Four-Dimensional Velocity Space”, Vestnik St. Petersburg University: Mathematics, 40:1 (2007), 52–60 | DOI | MR | Zbl

[4] Krym, V. R. and Petrov, N. N., “The Curvature Tensor and the Einstein Equations for a Four-Dimensional Nonholonomic Distribution”, Vestnik St. Petersburg University: Mathematics, 41:3 (2008), 256–265 | DOI | MR | Zbl

[5] Berestovskii, V. N. and Gichev, V. M., “Metrized Left-Invariant Orders on Topological Groups”, St. Petersburg Mathematical Journal, 11:4 (2000), 543–565 | MR

[6] Karmanova, M. B., “Lipschitz Images of Open Sets on Sub-Lorentzian Structures”, Siberian Advances in Mathematics, 34:1 (2024), 67–79 | DOI | MR

[7] Basalaev, S. G. and Vodopyanov, S. K., “Approximate Differentiability of Mappings of Carnot–Carathéodory Spaces”, Eurasian Mathematical Journal, 4:2 (2013), 10–48 | MR | Zbl

[8] Gromov, M., “Carnot–Carathéodory Spaces Seen from Within”, Sub-Riemannian Geometry, Birkhäuser Verlag, Basel, 1996, 79–318 | DOI | MR

[9] Karmanova, M. and Vodopyanov, S., “Geometry of Carnot–Carathéodory Spaces, Differentiability, Coarea and Area Formulas”, Analysis and Mathematical Physics, Birkhäuser, Basel, 2009, 233–335 | DOI | MR | Zbl

[10] Nagel, A., Stein, E. M. and Wainger, S., “Balls and Metrics Defined by Vector Fields I: Basic Properties”, Acta Mathematica, 155 (1985), 103–147 | DOI | MR | Zbl

[11] Postnikov, M. M., Lectures in Geometry. Semester V: Lie Groups and Lie Algebras, Mir, M., 1986, 446 pp. | MR

[12] Bonfiglioli, A., Lanconelli, E. and Uguzzoni, F., Stratified Lie Groups and Potential Theory for Their Sub-Laplacians, Springer Monographs in Mathematics, Springer, Berlin, 2007, 828 pp. | DOI | MR | Zbl

[13] Karmanova, M. B., “Fine Properties of Basis Vector Fields on Carnot-Carathéodory Spaces under Minimal Assumptions on Smoothness”, Siberian Mathematical Journal, 55:1 (2014), 87–99 | DOI | MR | Zbl

[14] Folland, G. B. and Stein, E. M., Hardy Spaces on Homogeneous Group, Princeton University Press, Princeton, 1982, 286 pp. | MR

[15] Vodopyanov, S. K. and Ukhlov, A. D., “Set Functions and Their Applications in the Theory of Lebesgue and Sobolev Spaces. I”, Siberian Advances in Mathematics, 14:4 (2004), 78–125 | MR | Zbl

[16] Vodopyanov, S. K. and Ukhlov, A. D., “Set Functions and Their Applications in the Theory of Lebesgue and Sobolev Spaces. II”, Siberian Advances in Mathematics, 15:1 (2005), 91–125 | MR | Zbl

[17] Vodopyanov, S., “Geometry of Carnot–Carathéodory Spaces and Differentiability of Mappings”, The Interaction of Analysis and Geometry, Contemporary Mathematics, 424, AMS, Providence, RI, 2007, 247–301 | DOI | MR | Zbl