On the projective Finsler metrizability and the integrability of Rapcsák equation

Tamás Milkovszki; Zoltán Muzsnay

Czechoslovak Mathematical Journal (2017)

  • Volume: 67, Issue: 2, page 469-495
  • ISSN: 0011-4642

Abstract

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A. Rapcsák obtained necessary and sufficient conditions for the projective Finsler metrizability in terms of a second order partial differential system. In this paper we investigate the integrability of the Rapcsák system and the extended Rapcsák system, by using the Spencer version of the Cartan-Kähler theorem. We also consider the extended Rapcsák system completed with the curvature condition. We prove that in the non-isotropic case there is a nontrivial Spencer cohomology group in the sequences determining the 2-acyclicity of the symbol of the corresponding differential operator. Therefore the system is not integrable and higher order obstruction exists.

How to cite

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Milkovszki, Tamás, and Muzsnay, Zoltán. "On the projective Finsler metrizability and the integrability of Rapcsák equation." Czechoslovak Mathematical Journal 67.2 (2017): 469-495. <http://eudml.org/doc/288197>.

@article{Milkovszki2017,
abstract = {A. Rapcsák obtained necessary and sufficient conditions for the projective Finsler metrizability in terms of a second order partial differential system. In this paper we investigate the integrability of the Rapcsák system and the extended Rapcsák system, by using the Spencer version of the Cartan-Kähler theorem. We also consider the extended Rapcsák system completed with the curvature condition. We prove that in the non-isotropic case there is a nontrivial Spencer cohomology group in the sequences determining the 2-acyclicity of the symbol of the corresponding differential operator. Therefore the system is not integrable and higher order obstruction exists.},
author = {Milkovszki, Tamás, Muzsnay, Zoltán},
journal = {Czechoslovak Mathematical Journal},
keywords = {Euler-Lagrange equation; metrizability; projective metrizability; geodesics; spray; formal integrability},
language = {eng},
number = {2},
pages = {469-495},
publisher = {Institute of Mathematics, Academy of Sciences of the Czech Republic},
title = {On the projective Finsler metrizability and the integrability of Rapcsák equation},
url = {http://eudml.org/doc/288197},
volume = {67},
year = {2017},
}

TY - JOUR
AU - Milkovszki, Tamás
AU - Muzsnay, Zoltán
TI - On the projective Finsler metrizability and the integrability of Rapcsák equation
JO - Czechoslovak Mathematical Journal
PY - 2017
PB - Institute of Mathematics, Academy of Sciences of the Czech Republic
VL - 67
IS - 2
SP - 469
EP - 495
AB - A. Rapcsák obtained necessary and sufficient conditions for the projective Finsler metrizability in terms of a second order partial differential system. In this paper we investigate the integrability of the Rapcsák system and the extended Rapcsák system, by using the Spencer version of the Cartan-Kähler theorem. We also consider the extended Rapcsák system completed with the curvature condition. We prove that in the non-isotropic case there is a nontrivial Spencer cohomology group in the sequences determining the 2-acyclicity of the symbol of the corresponding differential operator. Therefore the system is not integrable and higher order obstruction exists.
LA - eng
KW - Euler-Lagrange equation; metrizability; projective metrizability; geodesics; spray; formal integrability
UR - http://eudml.org/doc/288197
ER -

References

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  1. Bácsó, S., Szilasi, Z., On the projective theory of sprays, Acta Math. Acad. Paedagog. Nyházi. (N.S.) (electronic only) 26 (2010), 171-207. (2010) Zbl1240.53047MR2754415
  2. Bryant, R. L., Chern, S. S., Gardner, R. B., Goldschmidt, H. L., Griffiths, P. A., 10.1007/978-1-4613-9714-4, Mathematical Sciences Research Institute Publications 18, Springer, New York (1991). (1991) Zbl0726.58002MR1083148DOI10.1007/978-1-4613-9714-4
  3. Bucataru, I., Milkovszki, T., Muzsnay, Z., 10.1007/s00009-016-0762-0, Mediterr. J. Math. (2016), 4567-4580. (2016) Zbl1356.53021MR3564521DOI10.1007/s00009-016-0762-0
  4. Bucataru, I., Muzsnay, Z., 10.3842/SIGMA.2011.114, SIGMA, Symmetry Integrability Geom. Methods Appl. (electronic only) 7 (2011), Paper 114, 22 pages. (2011) Zbl1244.49072MR2861227DOI10.3842/SIGMA.2011.114
  5. Bucataru, I., Muzsnay, Z., 10.1142/S0129167X12500991, Int. J. Math. 23 (2012), 1250099, 15 pages. (2012) Zbl1263.53070MR2959445DOI10.1142/S0129167X12500991
  6. Crampin, M., Isotropic and R-flat sprays, Houston J. Math. 33 (2007), 451-459. (2007) Zbl1125.53012MR2308989
  7. Crampin, M., On the inverse problem for sprays, Publ. Math. 70 (2007), 319-335. (2007) Zbl1127.53015MR2310654
  8. Crampin, M., Some remarks on the Finslerian version of Hilbert's fourth problem, Houston J. Math. 37 (2011), 369-391. (2011) Zbl1228.53085MR2794554
  9. Crampin, M., Mestdag, T., Saunders, D. J., 10.1016/j.difgeo.2012.07.004, Differ. Geom. Appl. 30 (2012), 604-621. (2012) Zbl1257.53105MR2996856DOI10.1016/j.difgeo.2012.07.004
  10. Crampin, M., Mestdag, T., Saunders, D. J., 10.1016/j.difgeo.2012.10.012, Differ. Geom. Appl. 31 (2013), 63-79. (2013) Zbl1262.53064MR3010078DOI10.1016/j.difgeo.2012.10.012
  11. Do, T., Prince, G., 10.1016/j.difgeo.2016.01.005, Differ. Geom. Appl. 45 (2016), 148-179. (2016) Zbl1333.37070MR3457392DOI10.1016/j.difgeo.2016.01.005
  12. Frölicher, A., Nijenhuis, A., Theory of vector-valued differential forms. I: Derivations in the graded ring of differential forms, Nederl. Akad. Wet., Proc., Ser. A. 59 (1956), 338-359. (1956) Zbl0079.37502MR0082554
  13. Grifone, J., Muzsnay, Z., 10.1142/9789812813596, World Scientific Publishing, Singapore (2000). (2000) Zbl1023.49027MR1769337DOI10.1142/9789812813596
  14. Klein, J., Voutier, A., 10.5802/aif.282, Ann. Inst. Fourier 18 French (1968), 241-260. (1968) Zbl0181.49902MR0247599DOI10.5802/aif.282
  15. Matveev, V. S., 10.1142/S0129167X12500930, Int. J. Math. 23 (2012), 1250093, 14 pages. (2012) Zbl1253.53018MR2959439DOI10.1142/S0129167X12500930
  16. Mestdag, T., 10.1007/s00009-014-0505-z, Mediterr. J. Math. 13 (2016), 825-839. (2016) Zbl1338.53103MR3483865DOI10.1007/s00009-014-0505-z
  17. Muzsnay, Z., The Euler-Lagrange PDE and Finsler metrizability, Houston J. Math. 32 (2006), 79-98. (2006) Zbl1113.53049MR2202354
  18. Rapcsák, A., Über die bahntreuen Abbildungen metrischer Räume, Publ. Math. German 8 (1961), 285-290. (1961) Zbl0101.39901MR0138079
  19. Sarlet, W., Thompson, G., Prince, G. E., 10.1090/S0002-9947-02-02994-X, Trans. Am. Math. Soc. 354 (2002), 2897-2919. (2002) Zbl1038.37044MR1895208DOI10.1090/S0002-9947-02-02994-X
  20. Shen, Z., 10.1007/978-94-015-9727-2, Kluwer Academic Publishers, Dordrecht (2001). (2001) Zbl1009.53004MR1967666DOI10.1007/978-94-015-9727-2
  21. Szilasi, J., Lovas, R. L., Kertész, D. C., Connections, Sprays and Finsler Structures, World Scientific Publishing, Hackensack (2014). (2014) Zbl06171673MR3156183
  22. Szilasi, J., Vattamány, S., 10.1023/A:1014928103275, Period. Math. Hung. 44 (2002), 81-100. (2002) Zbl0997.53056MR1892276DOI10.1023/A:1014928103275

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