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# Article

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Keywords:
gluing of differential space; higher-order differential geometry; Sikorski differential space
Summary:
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.
References:
[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). MR 0674117 | Zbl 0486.57001
[3] Bucataru, I.: Linear connections for systems of higher order differential equations. Houston J. Math. 31 (2005), 315-332. MR 2132839 | Zbl 1078.58005
[4] Chen, K. T.: Iterated path integrals. Bull. Am. Math. Soc. 83 (1977), 831-879. DOI 10.1090/S0002-9904-1977-14320-6 | MR 0454968 | Zbl 0389.58001
[5] Dodson, C. T. J., Galanis, G. N.: Second order tangent bundles of infinite dimensional manifolds. J. Geom. Phys. 52 (2004), 127-136. DOI 10.1016/j.geomphys.2004.02.005 | MR 2088972 | Zbl 1076.58002
[6] Drachal, K.: Introduction to $d$-spaces theory. Math. Aeterna 3 (2013), 753-770. MR 3157564 | Zbl 1298.58008
[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. MR 1726003 | Zbl 0955.58001
[8] Epstein, M., Śniatycki, J.: The Koch curve as a smooth manifold. Chaos Solitons Fractals 38 (2008), 334-338. DOI 10.1016/j.chaos.2006.11.036 | MR 2415937 | Zbl 1146.28300
[9] G\_infinity ({\tt http://mathoverflow.net/users/22606/g-infinity}), : Extending derivations to the superposition closure (version: 2014-10-23). http://mathoverflow.net/q/182778</b>
[10] Gillman, L., Jerison, M.: Rings of Continuous Functions. Graduate Texts in Mathematics 43 Springer, Berlin (1976). MR 0407579 | Zbl 0327.46040
[11] Gruszczak, J., Heller, M., Sasin, W.: Quasiregular singularity of a cosmic string. Acta Cosmologica 18 (1992), 45-55.
[12] Heller, M., Multarzynski, P., Sasin, W., Zekanowski, Z.: Local differential dimension of space-time. Acta Cosmologica 17 (1991), 19-26.
[13] Heller, M., Sasin, W.: Origin of classical singularities. Gen. Relativ. Gravitation 31 (1999), 555-570. DOI 10.1023/A:1026650424098 | MR 1679416 | Zbl 0932.83036
[14] Kolář, I., Michor, P. W., Slovák, J.: Natural Operations in Differential Geometry. Springer, Berlin (1993). MR 1202431
[15] Kriegl, A., Michor, P. W.: The convenient setting of global analysis. Mathematical Surveys and Monographs 53 American Mathematical Society, Providence (1997). DOI 10.1090/surv/053 | MR 1471480 | Zbl 0889.58001
[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. MR 1855570 | Zbl 1020.58002
[17] Kuratowski, C.: Topologie. I. Panstwowe Wydawnictwo Naukowe 13, Warszawa French (1958). MR 0090795
[18] Mallios, A., Rosinger, E. E.: Space-time foam dense singularities and de Rham cohomology. Acta Appl. Math. 67 (2001), 59-89. DOI 10.1023/A:1010663502915 | MR 1847884 | Zbl 1005.46020
[19] Mallios, A., Rosinger, E. E.: Abstract differential geometry, differential algebras of generalized functions, and de Rham cohomology. Acta Appl. Math. 55 (1999), 231-250. DOI 10.1023/A:1006106718337 | MR 1686596
[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. MR 2801347 | Zbl 1219.83037
[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). MR 1437362 | Zbl 0877.53001
[22] Moreno, G.: On the canonical connection for smooth envelopes. Demonstr. Math. (electronic only) 47 (2014), 459-464. MR 3217741 | Zbl 1293.58002
[23] Morimoto, A.: Liftings of tensor fields and connections to tangent bundles of higher order. Nagoya Math. J. 40 (1970), 99-120. MR 0279719 | Zbl 0208.50201
[24] Mostow, M. A.: The differentiable space structures of Milnor classifying spaces, simplicial complexes, and geometric realizations. J. Differ. Geom. 14 (1979), 255-293. MR 0587553 | Zbl 0427.58005
[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. MR 1156985 | Zbl 0808.58008
[26] Nestruev, J.: Smooth Manifolds and Observables. Graduate Texts in Mathematics 220 Springer, New York (2003). MR 1930277 | Zbl 1021.58001
[27] Newns, W. F., Walker, A. G.: Tangent planes to a differentiable manifold. J. Lond. Math. Soc. 31 (1956), 400-407. DOI 10.1112/jlms/s1-31.4.400 | MR 0084163 | Zbl 0071.15303
[28] Pohl, W. F.: Differential geometry of higher order. Topology 1 (1962), 169-211. DOI 10.1016/0040-9383(62)90103-9 | MR 0154293 | Zbl 0112.36605
[29] Sardanashvily, G.: Lectures on Differential Geometry of Modules and Rings. Application to Quantum Theory. Lambert Academic Publishing, Saarbrucken (2012).
[30] Sasin, W.: Gluing of differential spaces. Demonstr. Math. (electronic only) 25 (1992), 361-384. MR 1170697 | Zbl 0759.58005
[31] Sasin, W.: Geometrical properties of gluing of differential spaces. Demonstr. Math. (electronic only) 24 (1991), 635-656. MR 1156988 | Zbl 0759.58004
[32] Sasin, W.: On equivalence relations on a differential space. Commentat. Math. Univ. Carol. 29 (1988), 529-539. MR 0972834 | Zbl 0679.58001
[33] Sasin, W., Spallek, K.: Gluing of differentiable spaces and applications. Math. Ann. 292 (1992), 85-102. DOI 10.1007/BF01444610 | MR 1141786 | Zbl 0735.32020
[34] Sikorski, R.: An Introduction to Differential Geometry. Biblioteka matematyczna 42 Panstwowe Wydawnictwo Naukowe, Warszawa Polish (1972). MR 0467544 | Zbl 0255.53001
[35] Sikorski, R.: Differential modules. Colloq. Math. 24 (1971), 45-79. MR 0482794 | Zbl 0226.53004
[36] Sikorski, R.: Abstract covariant derivative. Colloq. Math. 18 (1967), 251-272. MR 0222799 | Zbl 0162.25101
[37] Śniatycki, J.: Reduction of symmetries of Dirac structures. J. Fixed Point Theory Appl. 10 (2011), 339-358. DOI 10.1007/s11784-011-0063-y | MR 2861565 | Zbl 1252.53094
[38] Śniatycki, J.: Geometric quantization, reduction and decomposition of group representations. J. Fixed Point Theory Appl. 3 (2008), 307-315. DOI 10.1007/s11784-008-0081-6 | MR 2434450 | Zbl 1149.53323
[39] Śniatycki, J.: Orbits of families of vector fields on subcartesian spaces. Ann. Inst. Fourier 53 (2003), 2257-2296. DOI 10.5802/aif.2006 | MR 2044173 | Zbl 1048.53060
[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. MR 0607688 | Zbl 0501.58010
[41] Spallek, K.: Differenzierbare Räume. Math. Ann. 180 German (1969), 269-296. MR 0261035 | Zbl 0169.52901
[42] Vassiliou, E.: Topological algebras and abstract differential geometry. J. Math. Sci., New York 95 (1999), 2669-2680. DOI 10.1007/BF02169286 | MR 1712993 | Zbl 0936.53022
[43] Warner, F. W.: Foundations of Differentiable Manifolds and Lie Groups. Graduate Texts in Mathematics 94 Springer, New York (1983). DOI 10.1007/978-1-4757-1799-0 | MR 0722297 | Zbl 0516.58001

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