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Lipschitz Images of Open Sets on Sub-Lorentzian Structures

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Abstract

We prove a sub-Lorentzian analog of the area formula for intrinsically Lipschitz mappings of open subsets of Carnot groups of arbitrary depth with a sub-Lorentzian structure introduced on the image space.

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REFERENCES

  1. I. Bars and J. Terning, Extra Dimensions in Space and Time (Springer, New York, 2010).

    Book  Google Scholar 

  2. V. N. Berestovskiĭ and V. M. Gichev, “Metrized left invariant orders on topological groups,” Algebra Anal. bf 11(4), 1 (1999) [St. Petersbg. Math. J. 11:4, 543 (2000)].

    MathSciNet  Google Scholar 

  3. W. Craig and S. Weinstein, “On determinism and well-posedness in multiple time dimensions,” Proc. R. Soc. Lond., Ser. A, Math. Phys. Eng. Sci. 465:2110, 3023 (2008).

    MathSciNet  ADS  Google Scholar 

  4. G. B. Folland and E. M. Stein, Hardy Spaces on Homogeneous Groups (Princeton Univ. Press, Princeton, 1982).

    Google Scholar 

  5. M. Grochowski, “Geodesics in the sub-Lorentzian geometry,” Bull. Polish Acad. Sci. Math. 50, 161 (2002).

    MathSciNet  Google Scholar 

  6. M. Grochowski, “Reachable sets for the Heisenberg sub-Lorentzian structure on \(\mathbb R^3 \). An estimate for the distance function,” J. Dyn. Control Syst. 12, 145 (2006).

    Article  MathSciNet  Google Scholar 

  7. M. Grochowski, “Properties of reachable sets in the sub-Lorentzian geometry,” J. Geom. Phys. 59, 885 (2009).

    Article  MathSciNet  ADS  Google Scholar 

  8. M. Grochowski, “Normal forms and reachable sets for analytic Martinet sub-Lorentzian structures of Hamiltonian type,” J. Dyn. Control Syst. 17, 49 (2011).

    Article  MathSciNet  Google Scholar 

  9. M. Grochowski, “Reachable sets for Contact sub-Lorentzian metrics on \(\mathbb {R}^3 \). Application to control affine systems with the scalar input,” Sovrem. Mat. Prilozh. 70 (2011) [J. Math. Sci., New York 177:3, 383 (2011)].

    Article  MathSciNet  Google Scholar 

  10. M. Grochowski, “The structure of reachable sets for affine control systems induced by generalized Martinet sub-Lorentzian metrics,” ESAIM Control Optim. Calc. Var. 18, 1150 (2012).

    Article  MathSciNet  Google Scholar 

  11. M. Grochowski, “Remarks on the global sub-Lorentzian geometry,” Anal. Math. Phys. 3, 295 (2013).

    Article  MathSciNet  Google Scholar 

  12. M. Grochowski, “The structure of reachable sets and geometric optimality of singular trajectories for certain affine control systems in \(\mathbb {R}^3 \). The sub-Lorentzian approach,” J. Dyn. Control Syst. 20, 59 (2014).

    Article  MathSciNet  Google Scholar 

  13. M. B. Karmanova, “An area formula for Lipschitz mappings of Carnot-Carathéodory spaces,” Izv. Ross. Akad. Nauk, Ser. Mat. 78(3), 53 (2014) [Izv. Math. 78, 475 (2014)].

    Article  MathSciNet  Google Scholar 

  14. M. B. Karmanova, “The area of graphs on arbitrary Carnot groups with sub-Lorentzian structure,” Sib. Mat. Zh. 61, 823 (2020) [Sib. Math. J. 61, 648 (2020)].

    Article  MathSciNet  Google Scholar 

  15. M. B.Karmanova, “Two-step sub-Lorentzian structures and graph surfaces,” Izv. Ross. Akad. Nauk, Ser. Mat. 84(1), 60 (2020) [Izv. Math. 84, 52 (2020)].

    Article  MathSciNet  Google Scholar 

  16. M. B. Karmanova, “The coarea formula for vector functions on Carnot groups with sub-Lorentzian structure,” Sib. Mat. Zh. 62, 298 (2021) [Sib. Math. J. 62, 239 (2021)].

    Article  MathSciNet  Google Scholar 

  17. M. B. Karmanova, “On the approximability and parametrization of preimages of elements of Carnot groups on sub-Lorentzian structures,” Mat. Zametki 111 (1), 140 [Math. Notes 111:1, 152 (2022)].

    Article  MathSciNet  Google Scholar 

  18. M. B. Karmanova, “Measure of images of contact mappings on two-step sub-Lorentzian structures,” Mat. Zametki 113 (1), 149 [Math. Notes 113:1, 154 (2023)].

    Article  MathSciNet  Google Scholar 

  19. M. B. Karmanova, “The area of surfaces on sub-Lorentzian structures of depth two,” Mat. Tr. 26(1), 93 (2023) [Sib. Adv. Math. 33, 214 (2023)].

    Article  MathSciNet  Google Scholar 

  20. M. Karmanova and S. Vodopyanov, “An area formula for contact \(C^1 \)-mappings of Carnot manifolds,” Complex Var. Elliptic Equ. 55, 317 (2010).

    Article  MathSciNet  Google Scholar 

  21. A. Korolko and I. Markina, “Nonholonomic Lorentzian geometry on some \(\mathbb {H} \)-type groups,” J. Geom. Anal. 19, 864 (2009).

    Article  MathSciNet  Google Scholar 

  22. A. Korolko and I. Markina, “Geodesics on \(\mathbb {H} \)-type quaternion groups with sub-Lorentzian metric and their physical interpretation,” Complex Anal. Oper. Theory. 4, 589 (2010).

    Article  MathSciNet  Google Scholar 

  23. V. R. Krym and N. N. Petrov, “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,” Vestn. St-Peterbg. Univ., Ser. I, Mat. Mekh. Astron. 2007, No. 1, 62 (2007) [Vestn. St. Petersbg. Univ., Math. 40:1, 52 (2007)].

    Article  MathSciNet  Google Scholar 

  24. V. R. Krym and N. N. Petrov, “The curvature tensor and the Einstein equations for a four-dimensional nonholonomic distribution,” Vestn. St-Peterbg. Univ., Ser. I, Mat. Mekh. Astron. 2008, No. 3, 68 (2008) [Vestn. St. Petersbg. Univ., Math. 41:3, 256 (2008)].

    Article  MathSciNet  Google Scholar 

  25. V. M. Miklyunov, A. A.Klyachin, and V. A. Klyachin, Maximal Surfaces in the Minkowski Spacetime, http://www.uchimsya.info/maxsurf.pdf (2011).

  26. G. L. Naber, The Geometry of Minkowski Spacetime. An Introduction to the Mathematics of the Special Theory of Relativity. Applied Mathematical Sciences, 92 (Springer–Verlag, Berlin, 1992).

  27. B. Nielsen, “Minimal immersion, Einstein’s equations and Mach’s principle,” J. Geom. Phys. 4, 1 (1987).

    Article  MathSciNet  ADS  Google Scholar 

  28. P. Pansu, “Métriques de Carnot–Carathéodory et quasi-isométries des espaces symétriques de rang 1,” Math. Ann. 129, 1 (1989).

    Article  Google Scholar 

  29. S. K. Vodopyanov, “\(\mathcal P\) -differentiability on Carnot groups in different topologies and related topics,” Proceedings on Analysis and Geometry. International Conference in honor of the 70th Birthday of Professor Yu. G. Reshetnyak, Novosibirsk, Russia, August 30 – September 3, 1999 , 603 (Izdatel’stvo Instituta Matematiki Im. S. L. Soboleva SO RAN, Novosibirsk, 2000).

  30. S. Vodopyanov, “Geometry of Carnot–Carathéodory Spaces and Differentiability of Mappings,” In: The Interaction of Analysis and Geometry. Contemporary Mathematics. 424 (Amer. Math. Soc., Providence, RI; 2007), 247–301.

  31. S. K. Vodopyanov and A. D. Ukhlov, “Set functions and their applications in the theory of Lebesgue and Sobolev spaces. I,” Sib. Adv. Math. 14:4, 78 (2004).

    Google Scholar 

  32. S. K. Vodopyanov and A. D. Ukhlov, “Set functions and their applications in the theory of Lebesgue and Sobolev spaces. II,” Sib. Adv. Math. 15:1, 91 (2005).

    Google Scholar 

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Funding

The study was carried out in the framework of the State Task to the Sobolev Institute of Mathematics (project no. FWNF–2022–0006).

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Authors and Affiliations

  1. Sobolev Institute of Mathematics, Novosibirsk, 630090, Russia

    M. B. Karmanova

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  1. M. B. Karmanova
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Correspondence to M. B. Karmanova.

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CONFLICT OF INTEREST. The author of this work declares that she has no conflicts of interest.

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Karmanova, M.B. Lipschitz Images of Open Sets on Sub-Lorentzian Structures. Sib. Adv. Math. 34, 67–79 (2024). https://doi.org/10.1134/S1055134424010036

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  • DOI: https://doi.org/10.1134/S1055134424010036

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