Remarks on the behaviour of higher-order derivations on the gluing of differential spaces

Krzysztof Drachal

Czechoslovak Mathematical Journal (2015)

  • Volume: 65, Issue: 4, page 1137-1154
  • ISSN: 0011-4642

Abstract

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This paper is about some geometric properties of the gluing of order k in the category of Sikorski differential spaces, where k is assumed to be an arbitrary natural number. Differential spaces are one of possible generalizations of the concept of an infinitely differentiable manifold. It is known that in many (very important) mathematical models, the manifold structure breaks down. Therefore it is important to introduce a more general concept. In this paper, in particular, the behaviour of k th order tangent spaces, their dimensions, and other geometric properties, are described in the context of the process of gluing differential spaces. At the end some examples are given. The paper is self-consistent, i.e., a short review of the differential spaces theory is presented at the beginning.

How to cite

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Drachal, Krzysztof. "Remarks on the behaviour of higher-order derivations on the gluing of differential spaces." Czechoslovak Mathematical Journal 65.4 (2015): 1137-1154. <http://eudml.org/doc/276054>.

@article{Drachal2015,
abstract = {This paper is about some geometric properties of the gluing of order $k$ in the category of Sikorski differential spaces, where $k$ is assumed to be an arbitrary natural number. Differential spaces are one of possible generalizations of the concept of an infinitely differentiable manifold. It is known that in many (very important) mathematical models, the manifold structure breaks down. Therefore it is important to introduce a more general concept. In this paper, in particular, the behaviour of $k^\{\rm th\}$ order tangent spaces, their dimensions, and other geometric properties, are described in the context of the process of gluing differential spaces. At the end some examples are given. The paper is self-consistent, i.e., a short review of the differential spaces theory is presented at the beginning.},
author = {Drachal, Krzysztof},
journal = {Czechoslovak Mathematical Journal},
keywords = {gluing of differential space; higher-order differential geometry; Sikorski differential space},
language = {eng},
number = {4},
pages = {1137-1154},
publisher = {Institute of Mathematics, Academy of Sciences of the Czech Republic},
title = {Remarks on the behaviour of higher-order derivations on the gluing of differential spaces},
url = {http://eudml.org/doc/276054},
volume = {65},
year = {2015},
}

TY - JOUR
AU - Drachal, Krzysztof
TI - Remarks on the behaviour of higher-order derivations on the gluing of differential spaces
JO - Czechoslovak Mathematical Journal
PY - 2015
PB - Institute of Mathematics, Academy of Sciences of the Czech Republic
VL - 65
IS - 4
SP - 1137
EP - 1154
AB - This paper is about some geometric properties of the gluing of order $k$ in the category of Sikorski differential spaces, where $k$ is assumed to be an arbitrary natural number. Differential spaces are one of possible generalizations of the concept of an infinitely differentiable manifold. It is known that in many (very important) mathematical models, the manifold structure breaks down. Therefore it is important to introduce a more general concept. In this paper, in particular, the behaviour of $k^{\rm th}$ order tangent spaces, their dimensions, and other geometric properties, are described in the context of the process of gluing differential spaces. At the end some examples are given. The paper is self-consistent, i.e., a short review of the differential spaces theory is presented at the beginning.
LA - eng
KW - gluing of differential space; higher-order differential geometry; Sikorski differential space
UR - http://eudml.org/doc/276054
ER -

References

top
  1. Batubenge, A., Iglesias-Zemmour, P., Karshon, Y., Watts, J., Diffeological, Frölicher, and differential spaces. Preprint (2013). http://www.math.illinois.edu/ {jawatts/papers/reflexive.pdf}, . 
  2. Bröcker, T., Jänich, K., Introduction to Differential Topology, Cambridge University Press, Cambridge (1982). (1982) Zbl0486.57001MR0674117
  3. Bucataru, I., Linear connections for systems of higher order differential equations, Houston J. Math. 31 (2005), 315-332. (2005) Zbl1078.58005MR2132839
  4. Chen, K. T., 10.1090/S0002-9904-1977-14320-6, Bull. Am. Math. Soc. 83 (1977), 831-879. (1977) Zbl0389.58001MR0454968DOI10.1090/S0002-9904-1977-14320-6
  5. Dodson, C. T. J., Galanis, G. N., 10.1016/j.geomphys.2004.02.005, J. Geom. Phys. 52 (2004), 127-136. (2004) Zbl1076.58002MR2088972DOI10.1016/j.geomphys.2004.02.005
  6. Drachal, K., Introduction to d -spaces theory, Math. Aeterna 3 (2013), 753-770. (2013) Zbl1298.58008MR3157564
  7. Ebrahim, E., Mhehdi, N., The tangent bundle of higher order, Proc. of 2nd World Congress of Nonlinear Analysts, Nonlinear Anal., Theory Methods Appl. 30 (1997), 5003-5007. (1997) Zbl0955.58001MR1726003
  8. Epstein, M., Śniatycki, J., 10.1016/j.chaos.2006.11.036, Chaos Solitons Fractals 38 (2008), 334-338. (2008) Zbl1146.28300MR2415937DOI10.1016/j.chaos.2006.11.036
  9. G_infinity (),, Extending derivations to the superposition closure (version: 2014-10-23). http://mathoverflow.net/q/182778, . 
  10. Gillman, L., Jerison, M., Rings of Continuous Functions, Graduate Texts in Mathematics 43 Springer, Berlin (1976). (1976) Zbl0327.46040MR0407579
  11. Gruszczak, J., Heller, M., Sasin, W., Quasiregular singularity of a cosmic string, Acta Cosmologica 18 (1992), 45-55. (1992) 
  12. Heller, M., Multarzynski, P., Sasin, W., Zekanowski, Z., Local differential dimension of space-time, Acta Cosmologica 17 (1991), 19-26. (1991) 
  13. Heller, M., Sasin, W., 10.1023/A:1026650424098, Gen. Relativ. Gravitation 31 (1999), 555-570. (1999) Zbl0932.83036MR1679416DOI10.1023/A:1026650424098
  14. Kolář, I., Michor, P. W., Slovák, J., Natural Operations in Differential Geometry, Springer, Berlin (1993). (1993) MR1202431
  15. Kriegl, A., Michor, P. W., 10.1090/surv/053, Mathematical Surveys and Monographs 53 American Mathematical Society, Providence (1997). (1997) Zbl0889.58001MR1471480DOI10.1090/surv/053
  16. Krupka, D., Krupka, M., Jets and contact elements, Proceedings of the Seminar on Differential Geometry, Opava, Czech Republic, 2000 Mathematical Publications 2 Silesian University at Opava, Opava (2000), 39-85 D. Krupka. (2000) Zbl1020.58002MR1855570
  17. Kuratowski, C., Topologie. I, Panstwowe Wydawnictwo Naukowe 13, Warszawa French (1958). (1958) MR0090795
  18. Mallios, A., Rosinger, E. E., 10.1023/A:1010663502915, Acta Appl. Math. 67 (2001), 59-89. (2001) Zbl1005.46020MR1847884DOI10.1023/A:1010663502915
  19. Mallios, A., Rosinger, E. E., 10.1023/A:1006106718337, Acta Appl. Math. 55 (1999), 231-250. (1999) MR1686596DOI10.1023/A:1006106718337
  20. Mallios, A., Zafiris, E., The homological Kähler-de Rham differential mechanism I: Application in general theory of relativity, Adv. Math. Phys. 2011 (2011), Article ID 191083, 14 pages. (2011) Zbl1219.83037MR2801347
  21. Miron, R., The Geometry of Higher-Order Lagrange Spaces. Applications to Mechanics and Physics, Fundamental Theories of Physics 82 Kluwer Academic Publishers, Dordrecht (1997). (1997) Zbl0877.53001MR1437362
  22. Moreno, G., On the canonical connection for smooth envelopes, Demonstr. Math. (electronic only) 47 (2014), 459-464. (2014) Zbl1293.58002MR3217741
  23. Morimoto, A., 10.1017/S002776300001388X, Nagoya Math. J. 40 (1970), 99-120. (1970) Zbl0208.50201MR0279719DOI10.1017/S002776300001388X
  24. Mostow, M. A., 10.4310/jdg/1214434974, J. Differ. Geom. 14 (1979), 255-293. (1979) Zbl0427.58005MR0587553DOI10.4310/jdg/1214434974
  25. Multarzyński, P., Sasin, W., Żekanowski, Z., Vectors and vector fields of k -th order on differential spaces, Demonstr. Math. (electronic only) 24 (1991), 557-572. (1991) Zbl0808.58008MR1156985
  26. Nestruev, J., Smooth Manifolds and Observables, Graduate Texts in Mathematics 220 Springer, New York (2003). (2003) Zbl1021.58001MR1930277
  27. Newns, W. F., Walker, A. G., 10.1112/jlms/s1-31.4.400, J. Lond. Math. Soc. 31 (1956), 400-407. (1956) Zbl0071.15303MR0084163DOI10.1112/jlms/s1-31.4.400
  28. Pohl, W. F., 10.1016/0040-9383(62)90103-9, Topology 1 (1962), 169-211. (1962) Zbl0112.36605MR0154293DOI10.1016/0040-9383(62)90103-9
  29. Sardanashvily, G., Lectures on Differential Geometry of Modules and Rings. Application to Quantum Theory, Lambert Academic Publishing, Saarbrucken (2012). (2012) 
  30. Sasin, W., Gluing of differential spaces, Demonstr. Math. (electronic only) 25 (1992), 361-384. (1992) Zbl0759.58005MR1170697
  31. Sasin, W., Geometrical properties of gluing of differential spaces, Demonstr. Math. (electronic only) 24 (1991), 635-656. (1991) Zbl0759.58004MR1156988
  32. Sasin, W., On equivalence relations on a differential space, Commentat. Math. Univ. Carol. 29 (1988), 529-539. (1988) Zbl0679.58001MR0972834
  33. Sasin, W., Spallek, K., 10.1007/BF01444610, Math. Ann. 292 (1992), 85-102. (1992) MR1141786DOI10.1007/BF01444610
  34. Sikorski, R., An Introduction to Differential Geometry, Biblioteka matematyczna 42 Panstwowe Wydawnictwo Naukowe, Warszawa Polish (1972). (1972) Zbl0255.53001MR0467544
  35. Sikorski, R., 10.4064/cm-24-1-45-79, Colloq. Math. 24 (1971), 45-79. (1971) Zbl0226.53004MR0482794DOI10.4064/cm-24-1-45-79
  36. Sikorski, R., 10.4064/cm-18-1-251-272, Colloq. Math. 18 (1967), 251-272. (1967) Zbl0162.25101MR0222799DOI10.4064/cm-18-1-251-272
  37. Śniatycki, J., 10.1007/s11784-011-0063-y, J. Fixed Point Theory Appl. 10 (2011), 339-358. (2011) Zbl1252.53094MR2861565DOI10.1007/s11784-011-0063-y
  38. Śniatycki, J., 10.1007/s11784-008-0081-6, J. Fixed Point Theory Appl. 3 (2008), 307-315. (2008) Zbl1149.53323MR2434450DOI10.1007/s11784-008-0081-6
  39. Śniatycki, J., 10.5802/aif.2006, Ann. Inst. Fourier 53 (2003), 2257-2296. (2003) Zbl1048.53060MR2044173DOI10.5802/aif.2006
  40. Souriau, J.-M., Groupes différentiels, Differential Geometrical Methods in Mathematical Physics. Proc. Conf. Aix-en-Provence and Salamanca, 1979 Lecture Notes in Math. 836 Springer, Berlin French (1980), 91-128. (1980) Zbl0501.58010MR0607688
  41. Spallek, K., Differenzierbare Räume, Math. Ann. 180 German (1969), 269-296. (1969) Zbl0169.52901MR0261035
  42. Vassiliou, E., 10.1007/BF02169286, J. Math. Sci., New York 95 (1999), 2669-2680. (1999) Zbl0936.53022MR1712993DOI10.1007/BF02169286
  43. Warner, F. W., 10.1007/978-1-4757-1799-0, Graduate Texts in Mathematics 94 Springer, New York (1983). (1983) Zbl0516.58001MR0722297DOI10.1007/978-1-4757-1799-0

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