Czechoslovak Mathematical Journal, Vol. 67, No. 2, pp. 297-316, 2017

The general rigidity result for bundles of $A$-covelocities and $A$-jets

Jiří Tomáš

Received October 23, 2015.   First published March 1, 2017.

Jiří Tomáš, Department of Mathematics, Faculty of Mechanical Engineering, Brno University of Technology, Technická 2, Brno, Czech Republic, e-mail: Tomas@fme.vutbr.cz

Abstract: Let $M$ be an $m$-dimensional manifold and $A=\mathbb D^r_k /I=\mathbb R \oplus N_A$ a Weil algebra of height $r$. We prove that any $A$-covelocity $T^A_x f \in T^{A*}_x M$, $x \in M$ is determined by its values over arbitrary $\max\{\mathop{\rm width}A, m \}$ regular and under the first jet projection linearly independent elements of $T^A_xM$. Further, we prove the rigidity of the so-called universally reparametrizable Weil algebras. Applying essentially those partial results we give the proof of the general rigidity result $T^{A*}M \simeq T^{r*}M$ without coordinate computations, which improves and generalizes the partial result obtained in Tomáš (2009) from $m \ge k$ to all cases of $m$. We also introduce the space $J^A(M,N)$ of $A$-jets and prove its rigidity in the sense of its coincidence with the classical jet space $J^r(M,N)$.

Keywords: $r$-jet; bundle functor; Weil functor; Lie group; jet group; $B$-admissible $A$-velocity

Classification (MSC 2010): 58A20, 58A32, 53C24

DOI: 10.21136/CMJ.2017.0566-15

Full text available as PDF.


References:
  [1] R. J. Alonso: Jet manifolds associated to a Weil bundle. Arch. Math., Brno 36 (2000), 195-209. MR 1785036 | Zbl 1049.58007
  [2] R. J. Alonso-Blanco, D. Blázquez-Sanz: The only global contact transformations of order two or more are point transformations. J. Lie Theory 15 (2005), 135-143. MR 2115233 | Zbl 1073.58006
  [3] W. Bertram: Differential geometry, Lie groups and symmetric spaces over general base fields and rings. Mem. Am. Math. Soc. 192 (2008), 202 pages. DOI 10.1090/memo/0900 | MR 2369581 | Zbl 1144.58002
  [4] G. N. Bushueva, V. V. Shurygin: On the higher order geometry of Weil bundles over smooth manifolds and over parameter-dependent manifolds. Lobachevskii J. Math. (electronic only) 18 (2005), 53-105. MR 2169080 | Zbl 1083.58005
  [5] D. J. Eck: Product-preserving functors on smooth manifolds. J. Pure Appl. Algebra 42 (1986), 133-140. DOI 10.1016/0022-4049(86)90076-9 | MR 0857563 | Zbl 0615.57019
  [6] G. Kainz, P. W. Michor: Natural transformations in differential geometry. Czech. Math. J. 37 (1987), 584-607. MR 0913992 | Zbl 0654.58001
  [7] I. Kolář: Covariant approach to natural transformations of Weil functors. Commentat. Math. Univ. Carol. 27 (1986), 723-729. DOI 10.1016/0022-4049(86)90076-9 | MR 0874666 | Zbl 0615.57019
  [8] I. Kolář, P. W. Michor, J. Slovák: Natural Operations in Differential Geometry. Springer, Berlin (1993). DOI 10.1007/978-3-662-02950-3 | MR 1202431 | Zbl 0782.53013
  [9] I. Kolář, W. M. Mikulski: On the fiber product preserving bundle functors. Differ. Geom. Appl. 11 (1999), 105-115. DOI 10.1016/S0926-2245(99)00022-4 | MR 1712139 | Zbl 0935.58001
  [10] M. Kureš: Weil algebras associated to functors of third order semiholonomic velocities. Math. J. Okayama Univ. 56 (2014), 117-127. MR 3155085 | Zbl 1315.58003
  [11] O. O. Luciano: Categories of multiplicative functors and Weil's infinitely near points. Nagoya Math. J. 109 (1988), 63-89. DOI 10.1017/s0027763000002774 | MR 0931952 | Zbl 0661.58007
  [12] W. M. Mikulski: Product preserving bundle functors on fibered manifolds. Arch. Math., Brno 32 (1996), 307-316. MR 1441401 | Zbl 0881.58002
  [13] J. Mu noz, J. Rodriguez, F. J. Muriel: Weil bundles and jet spaces. Czech. Math. J. 50 (2000), 721-748. DOI 10.1023/A:1022408527395 | MR 1792967 | Zbl 1079.58500
  [14] H. Nishimura: Axiomatic differential geometry I-1 - towards model categories of differential geometry. Math. Appl., Brno 1 (2012), 171-182. DOI 10.13164/ma.2012.11 | MR 3275606 | Zbl 1285.51009
  [15] V. V. Shurygin: The structure of smooth mappings over Weil algebras and the category of manifolds over algebras. Lobachevskii J. Math. 5 (1999), 29-55. MR 1752307 | Zbl 0985.58001
  [16] V. V. Shurygin: Some aspects of the theory of manifolds over algebras and Weil bundles. J. Math. Sci., New York 169 (2010), 315-341; translation from Itogi Nauki Tekh., Ser. Sovrem. Mat. Prilozh., Temat. Obz. 123 (2009), 211–255. DOI 10.1007/s10958-010-0051-6 | Zbl 1226.53032
  [18] J. Tomáš: Some results on bundles of covelocities. J. Appl. Math., Aplimat V 4 (2011), 297-306.
  [19] J. Tomáš: Bundles of $(p, A)$-covelocities and $(p, A)$-jets. Miskolc Math. Notes 14 (2013), 547-555. MR 3144090 | Zbl 1299.58010
  [20] A. Weil: Théorie des points proches sur les variétés des différentiables. Colloques internat. Centre nat. Rech. Sci. 52 (1953), 111-117. (In French) MR 0061455 | Zbl 0053.24903


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