Skip to main content
SpringerLink
Log in

Structure tensor of a field of geometric objects

  • Published:
Geometriae Dedicata Aims and scope Submit manuscript

Abstract

A construction of a kth structure tensor of a field of geometric objects is presented here (k is a non-negative integer). For a given field σ we construct a vector bundle H k,2(σ). The kth structure tensor is defined as a section of H k,2(σ) generated by the torsion of σ. It is then shown that vanishing of the kth structure tensor is a necessary and sufficient condition for the field to be (k+1)-flat.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. Aczel, J. and Gołlab, S., Funktionalgleichungen der Geometrischen Objekte, Warszawa, 1960.

  2. Albert, C. and Molino, P., Pseudogroupes de Lie et structures différentiables, tome I. Univ. de Scien. et Techn. du Languedoc, 1980.

  3. Bauer, M., ‘Sur les G-Structures k-Plates’, Ann. Inst. Fourier, 24, 1 (1974), 297–310.

    Google Scholar 

  4. Bernard, D., ‘Sur la Géometrie Différentielle des G-Structures’, Ann. Inst. Fourier 10, (1960), 151–270.

    Google Scholar 

  5. Ehresmann, C., ‘Catégories Topologiques et Catégorie Différentiables’, Coll. de Géom. Différ. Globale. Bruxelles, 1959.

  6. Epstein, D. B. A., and Thurston, W. P., ‘Transformation Groups and Natural Bundles’, Proc. London Math. Soc. XXXVIII (1979)

  7. Guillemin, V., ‘The Integrability Problem for G-Structures’, Trans. Amer. Math. Soc. 116 (1965), 544–560.

    Google Scholar 

  8. Konderak, J., ‘On Homogeneity and Transitivity of Fields of Geometric Objects’, Publ. Univ. Aut. de Barcelona. 30,1 (1986), 1–5.

    Google Scholar 

  9. Konderak, J., ‘Fibre Bundles Associated with Fields of Geometric Objects and Structure Tensors’, (Preprint, 1987).

  10. Matsushima, Y., ‘Pseudogroupes de Lie transitif’, Seminare Bourbaki Exposé, Vol. 118, 1954/55.

  11. Nijenhuis, A., ‘Natural Bundles and their General Properties’, J. Diff. Geom. (in honour of K. Yano Kinokuniya, Tokyo, 1972.

  12. Palais, R. S., and Terng, C. L., Natural Bundles have a Finite Order', Topology 16 (1978), 271–277.

    Google Scholar 

  13. Salvioli, S. E., ‘On the Theory of Geometric Objects’, J. Diff. Geom. 7 (1972), 257–278.

    Google Scholar 

  14. Singer, I. M. and Sternberg, S., ‘The Infinite Groups of Lie and Cartan. Part I, J. Anal. Math. XV (1967), 1–114.

    Google Scholar 

  15. Zajtz, A., Foundations of Differential Geometry on Natural Bundles. Univ. of Caracas, 1984.

Download references

Author information

Authors and Affiliations

  1. Institute of Mathematics, Jagiellonian University, ul. Reymonta 4, 30-059, Kraków, Poland

    J. Konderak

Authors
  1. J. Konderak
    View author publications

    You can also search for this author in PubMed Google Scholar

Rights and permissions

Reprints and permissions

About this article

Cite this article

Konderak, J. Structure tensor of a field of geometric objects. Geom Dedicata 27, 171–177 (1988). https://doi.org/10.1007/BF00151347

Download citation

  • Received:

  • Revised:

  • Issue Date:

  • DOI: https://doi.org/10.1007/BF00151347

Keywords

Search

Navigation