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Annali di Matematica Pura ed Applicata

, Volume 107, Issue 1, pp 131–157 | Cite as

Sui fibrati con struttura quaternionale generalizzata

  • Stefano Marchiafava
  • Giuliano Romani
Article

Summary

Quaternion generalized fiber bundles\(\tilde E_n^\mathbb{Q} \to X\) are studied, both isomorphic to global tensorial product\(E_n^\mathbb{Q} \otimes _\mathbb{Q} E_1^\mathbb{Q} (E_n^\mathbb{Q} , E_1^\mathbb{Q}\) ordinary quaternion fiber bundles right and left respectively) and quite general ones. A cohomology class\(\varepsilon (\tilde E_n^\mathbb{Q} ) \in H^2 (X;\mathbb{Z}_2 )\) is considered which represents the obstruction in order the fiber bundle be a tensorial product. Several properties and a splitting principle are proved for bundles\(\tilde E_n^\mathbb{Q}\). On this ground and founding on a convenient bundle BE → X associated to jaz (that we call Bonan's bundle and for which ɛ(\(\tilde E_n^\mathbb{Q}\)=ɛ(BE)) relations are stated among Stiefel-Whitney classes of\(\tilde E_n^\mathbb{Q}\), BE and the class ɛ.

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Bibliografia

  1. [1]
    E. Bonan,Sur les G-structures du type quaternionien, Cahiers de Topologie etGéometrie différentielle,9, fasc. 4 (1967).Google Scholar
  2. [2]
    A. Borel,La cohomologie mod 2de certains espaces homogènes, Commentarii Mathematici Helvetici,27 (1953).Google Scholar
  3. [3]
    A. Dold,Metodi moderni di topologia algebrica, Quaderni dei gruppi di ricerca matematica del C.N.R., Roma, 1973.Google Scholar
  4. [4]
    P. Du Val,Homographies quaternions and rotations, Oxford University Press, 1964.Google Scholar
  5. [5]
    A. Gray,A note on manifolds whose holonomy groups is a subgroup of Sp(n)·Sp(1), Michigan Math. J.,19 (1969).Google Scholar
  6. [6]
    A. Gray,Errata, Michigan Math. J.,17 (1970).Google Scholar
  7. [7]
    Th. Hangan,Tensor product tangent bundles, Archiv. der Mathematick,19, fasc. 4 (1968), pp. 436–448.CrossRefzbMATHMathSciNetGoogle Scholar
  8. [8]
    D. Husemoller,Fibre bundles, McGraw-Hill, New York, 1966.Google Scholar
  9. [9]
    E. Martinelli,Varietà a struttura quaternionale generalizzata, Atti Accad. Naz. Lincei,26 (1959), pp. 353–362.zbMATHMathSciNetGoogle Scholar
  10. [10]
    E. Martinelli,Metriche hermitiane sulle varietà a struttura quasi quaternionale generalizzata, Atti Accad. Naz. Lincei,39 (1965), pp. 400–407.zbMATHMathSciNetGoogle Scholar
  11. [11]
    J. T. Schwartz,Differential Geometry and Topology, Gordon and Breach, New York, 1968.Google Scholar
  12. [12]
    R. H. Szczarba —W. C. Hsiang,On the tangent bundle of a Grassmann Manifold, Amer. J. Math.,86 (1964), pp. 698–704.MathSciNetGoogle Scholar
  13. [13]
    E. M. Spanier,Algebraic Topology, McGraw-Hill, New York, 1966.Google Scholar
  14. [14]
    N. Steenrod,The Topology of Fibre Bundles, Princeton University Press, Princeton, 1951.Google Scholar
  15. [15]
    A. Weil,Sur les théorèmes de de Rham, Commentarii Mathematici Helvetici,26 (1952), pp. 313–339.MathSciNetGoogle Scholar
  16. [16]
    J. A. Wolf,Complex homogeneous contact manifolds and quaternionic symmetric spaces, Journal of Math. and Mech.,14 (1965), p. 166.Google Scholar

Copyright information

© Fondazione Annali di Matematica Pura ed Applicata 1975

Authors and Affiliations

  • Stefano Marchiafava
    • 1
  • Giuliano Romani
    • 1
  1. 1.Roma

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