Czechoslovak Mathematical Journal

, Volume 67, Issue 2, pp 297–316

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



Let M be an m-dimensional manifold and A = Dkr/I = R⊕NA a Weil algebra of height r. We prove that any A-covelocity TxAfTxA*M, xM is determined by its values over arbitrary max{width A,m} regular and under the first jet projection linearly independent elements of TxAM. 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 TA*MTr*M without coordinate computations, which improves and generalizes the partial result obtained in Tomáš (2009) from mk to all cases of m.

We also introduce the space JA(M,N) of A-jets and prove its rigidity in the sense of its coincidence with the classical jet space Jr(M,N).


r-jet bundle functor Weil functor Lie group jet group B-admissible A-velocity 

MSC 2010

58A20 58A32 53C24 


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  1. [1]
    R. J. Alonso: Jet manifolds associated to a Weil bundle. Arch. Math., Brno 36 (2000), 195–209.MathSciNetMATHGoogle Scholar
  2. [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.MathSciNetMATHGoogle Scholar
  3. [3]
    W. Bertram: Differential geometry, Lie groups and symmetric spaces over general base fields and rings. Mem. Am. Math. Soc. 192 (2008), 202 pages.MathSciNetMATHGoogle Scholar
  4. [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.MathSciNetMATHGoogle Scholar
  5. [5]
    D. J. Eck: Product-preserving functors on smooth manifolds. J. Pure Appl. Algebra 42 (1986), 133–140.MathSciNetCrossRefMATHGoogle Scholar
  6. [6]
    G. Kainz, P. W. Michor: Natural transformations in differential geometry. Czech. Math. J. 37 (1987), 584–607.MathSciNetMATHGoogle Scholar
  7. [7]
    I. Kolář: Covariant approach to natural transformations of Weil functors. Commentat. Math. Univ. Carol. 27 (1986), 723–729.MathSciNetMATHGoogle Scholar
  8. [8]
    I. Kolář, P. W. Michor, J. Slovák: Natural Operations in Differential Geometry, Springer, Berlin, 1993.CrossRefMATHGoogle Scholar
  9. [9]
    I. Kolář, W. M. Mikulski: On the fiber product preserving bundle functors. Differ. Geom. Appl. 11 (1999), 105–115.MathSciNetCrossRefMATHGoogle Scholar
  10. [10]
    M. Kureš: Weil algebras associated to functors of third order semiholonomic velocities. Math. J. Okayama Univ. 56 (2014), 117–127.MathSciNetMATHGoogle Scholar
  11. [11]
    O. O. Luciano: Categories of multiplicative functors and Weil’s infinitely near points. Nagoya Math. J. 109 (1988), 63–89.MathSciNetCrossRefMATHGoogle Scholar
  12. [12]
    W. M. Mikulski: Product preserving bundle functors on fibered manifolds. Arch. Math., Brno 32 (1996), 307–316.MathSciNetMATHGoogle Scholar
  13. [13]
    J. Mu˜noz, J. Rodriguez, F. J. Muriel: Weil bundles and jet spaces. Czech. Math. J. 50 (2000), 721–748.MathSciNetCrossRefMATHGoogle Scholar
  14. [14]
    H. Nishimura: Axiomatic differential geometry I–1—towards model categories of differential geometry. Math. Appl., Brno 1 (2012), 171–182.MathSciNetCrossRefMATHGoogle Scholar
  15. [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.MathSciNetMATHGoogle Scholar
  16. [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.CrossRefMATHGoogle Scholar
  17. [17]
    J. M. Tomáš: On bundles of covelocities. Lobachevskii J. Math. 30 (2009), 280–288.MathSciNetCrossRefMATHGoogle Scholar
  18. [18]
    J. Tomáš: Some results on bundles of covelocities. J. Appl. Math., Aplimat V 4 (2011), 297–306.Google Scholar
  19. [19]
    J. Tomáš: Bundles of (p, A)-covelocities and (p, A)-jets. Miskolc Math. Notes 14 (2013), 547–555.MathSciNetMATHGoogle Scholar
  20. [20]
    A. Weil: Théorie des points proches sur les variétés des différentiables. Colloques internat. Centre nat. Rech. Sci. 52 (1953), 111–117. (In French.)MATHGoogle Scholar

Copyright information

© Institute of Mathematics of the Academy of Sciences of the Czech Republic, Praha, Czech Republic 2017

Authors and Affiliations

  1. 1.Department of Mathematics, Faculty of Mechanical EngineeringBrno University of TechnologyBrnoCzech Republic

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