Metric Characteristics of Classes of Compact Sets on Carnot Groups with Sub-Lorentzian Structure

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Home Editorial board Publication ethics Peer review guidelines Latest issue All issues Rules for authors Online submission system’s guidelines Submit manuscript Contacts Address: Vatutina st. 53, Vladikavkaz, 362025, RNO-A, Russia Phone: (8672)23-00-54 E-mail: rio@smath.ru	Dear authors! Submission of all materials is carried out only electronically through Online Submission System in personal account. DOI: 10.46698/d9212-8277-5800-l Metric Characteristics of Classes of Compact Sets on Carnot Groups with Sub-Lorentzian Structure Karmanova, M. B. Vladikavkaz Mathematical Journal 2024. Vol. 26. Issue 3. Abstract: We consider classes of mappings of Carnot groups that are intrinsically Lipschitz and defined on compact subsets, and describe the metric characteristics of their images under the condition that a~sub-Lorentzian structure is introduced on the image. This structure is a sub-Riemannian generalization of Minkowski geometry. One of its features is the unlimitedness of the balls constructed with respect to the~intrinsic distance. In sub-Lorentzian geometry, the study of spacelike surfaces whose intersections with such balls are limited, is of independent interest. If the mapping is defined on an open set, then the formulation of space-likeness criterion reduces to considering the connectivity component of the intersection containing the center of the ball and analyzing the properties of the sub-Riemannian differential matrix. If the domain of definition of the mapping is not an open set, then the question arises what conditions can be set on the mapping that guarantee the boundedness of the intersection of the image of a compact set with a sub-Lorentzian ball. In this article, this problem is resolved: we consider that part of the intersection that can be parameterized by the connectivity component of the~intersection of the image of the sub-Riemannian differential and the ball. In addition, using such local parameterizations, a set function is introduced, which is constructed similarly to Hausdorff measure. We show that this set function is also a measure. As an application, the sub-Lorentzian area formula is established. Keywords: Carnot group, Lipschitz mapping, compact set, sub-Lorentzian structure, quasi-additive set function, area formula Language: English Download the full text For citation: Karmanova, M. B. Metric Characteristics of Classes of Compact Sets on Carnot Groups with Sub-Lorentzian Structure, Vladikavkaz Math. J., 2024, vol. 26, no. 3, pp. 56-64. DOI 10.46698/d9212-8277-5800-l + References 1. Miklyukov, V. M., Klyachin, A. A. and Klyachin, V. A. Maksimal'nye Poverkhnosti v Prostranstve-Vremeni Minkovskogo [Maximal Surfaces in Minkowski Space-Time], Volgograd, VolSU, 2011, 530 p. (in Russian). 2. 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, 2007, vol. 40, no. 1, pp. 52-60. DOI: 10.3103/S1063454107010062. 3. 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, 2008, vol. 41, no. 3, pp. 256-265. DOI: 10.3103/S1063454108030060. 4. Berestovskii, V. N. and Gichev, V. M. Metrized Left-Invariant Orders on Topological Groups, St. Petersburg Mathematical Journal, 2000, vol. 11, no. 4, pp. 543-565. 5. Karmanova, M. B. Lipschitz Images of Open Sets on Sub-Lorentzian Structures, Siberian Advances in Mathematics, 2024, vol. 34, no. 1, pp. 67-79. DOI: 10.1134/S1055134424010036. 6. Folland, G. B. and Stein, E. M. Hardy Spaces on Homogeneous Group, Princeton, Princeton University Press, 1982, 286 p. 7. 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, 2004, vol. 14, no. 4, pp. 78-125. 8. 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, 2005, vol. 15, no. 1, pp. 91-125. 9. Pansu, P. Metriques de Carnot-Caratheodory et Quasi-Isometries des Espaces Symetriques de Rang 1, The Annals of Mathematics, 1989, vol. 129, no. 1, pp. 1-60. DOI: 10.2307/1971484. 10. Vodopyanov, S. Geometry of Carnot-Caratheodory Spaces and Differentiability of Mappings, The Interaction of Analysis and Geometry. Contemporary Mathematics, vol. 424, Providence, RI, AMS, 2007, pp. 247-301. 11. Vodop'yanov, S. K. \(\mathcal P\)-Differentiability on Carnot Groups in Different Topologies and Related Topics, Trudy po Analizu i Geometrii [Proceedings in Analysis and Geometry], Novosibirsk, Institute of Mathematics of SB RAS, 2000, pp. 603-670. 12. Karmanova, M. B. An Area Formula for Lipschitz Mappings of Carnot-Caratheodory Spaces, Izvestiya: Mathematics, 2014, vol. 78, no. 3, pp. 475-499. DOI: 10.1070/im2014v078n03abeh002695. ← Contents of issue


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