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Spin (7)-structures in principal fibre bundles over Riemannian manifolds withG 2-structure

  • Francisco M. Cabrera
Article

Abstract

We consider principal fibre bundles with one-dimensional fiber over manifolds withG 2-structure. We define twoSpin(7)-structures on the total bundle space and find relations between the respective structures on the total space and the base. Finally we construct examples ofSpin(7)-structures using the results previously proved.

1991 Mathematics Subject Classification

53C10 53C21 55R10 

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References

  1. [AS] Ambrose W., Singer I.M.,A theorem on holonomy, Trans. Amer. Math. Soc.,79 (1953), 428–443.CrossRefMathSciNetGoogle Scholar
  2. [Be] Berger M.,Sur les groupes d'holonomie homogène des variétés à connexion affine et des variétés riemanniennes, Bull. Soc. Math. France,83 (1955), 279–330.MATHMathSciNetGoogle Scholar
  3. [Bo] Bonan E.,Sur les variétés riemaniennes à groupe d'holonomie G 2 ou Spin(7), C. R. Acad. Sci. Paris,262 (1966), 127–129.MATHMathSciNetGoogle Scholar
  4. [BG] Brown R. B., Gray A.,Vector cross products, Comment. Math. Helv.,42 (1967), 222–236.MATHCrossRefMathSciNetGoogle Scholar
  5. [Br] Bryant R. L.,Metrics with exceptional holonomy, Ann. of Math.,126 (1987), 525–576.CrossRefMathSciNetGoogle Scholar
  6. [BS] Bryant R. L., Salamon S. M.,On the constructions of some complete metrics with exceptional holonomy, Duke Math. J.,58 (1989), 829–850.MATHCrossRefMathSciNetGoogle Scholar
  7. [C1] Cabrera F. M.,On Riemannian manifolds with G 2-structure, Bollettino U.M.I., to appear.Google Scholar
  8. [C2] Cabrera F. M.,On Riemannian manifolds with Spin(7)-structure, Publ. Math. Dehecen, to appear.Google Scholar
  9. [CMS] Cabrera F. M., Monar M. D., Swann A. F.,Classification of G 2-structures, J. Lond. Math. Soc., to appear.Google Scholar
  10. [F1] Fernández M.,A Classification of Riemannian Manifolds with Structure Group Spin(7), Ann. Mat. Pura Appl. (IV),148 (1986), 101–122.CrossRefGoogle Scholar
  11. [F2] Fernández M.,A new example of a compact Riemannian mmanifold with structure group Spin(7), Port. Math.,44 (1987), 161–165.MATHGoogle Scholar
  12. [F3] Fernández M.,An example of a compact calibrated manifold associated with the exceptional Lie Group G 2, J. Diff. Geom.,26 (1987), 367–370.MATHGoogle Scholar
  13. [FG] Fernández M., Gray A.,Riemannian manifolds with structure group G 2. Ann. Mat. Pura Appl. (IV),32 (1982), 19–45.CrossRefGoogle Scholar
  14. [FI] Fernández M., Iglesias T.,New examples of Riemannian manifolds with structure group G 2. Rend. Circ. Mat. Palermo,35 (1986), 276–290.MATHCrossRefMathSciNetGoogle Scholar
  15. [Gr1] Gray A.,Some examples of almost Hermitian manifolds, Ill. J. Math.,10 (1969), 353–366.Google Scholar
  16. [Gr2] Gray A.,Vector cross products on manifolds, Trans. Amer. Soc.,141 (1969), pp. 463–504. Correction,148 (1970), 625.CrossRefGoogle Scholar
  17. [Gr3] Gray A.,Vector cross products, Rend. Sem. Mat. Univ. Politec. Torino,35 (1976–77), 69–75.Google Scholar
  18. [J] Joyce D.,Compact Riemannian 7-manifolds with holonomy G 2, I and II, J. Diff. Geom., to appear.Google Scholar
  19. [K] Kobayashi S.,Principal Fibre Bundles with 1-dimensional Toroidal Group, Tôhoku Math. J.,2 (1956), 29–45.CrossRefGoogle Scholar
  20. [KN] Kobayashi S., Nomizu K.,Foundations of Differential Geometry, 2 volumes, Intersciences Pub., New York (1963), (1969).MATHGoogle Scholar
  21. [Ko] Kostant B.,Quantization and unitary representations, Lecture Notes in Math., Soc.,170, Springer-Verlag, Berlin and New York (1970), 87–207.Google Scholar
  22. [M] Marchiafava S.,Alcune osservazioni riguardanti i gruppi di Lie G 2 e Spin(7), candidati a gruppi di olonomia, Ann. Mat. Pura appl.,129 (1981), 247–264.MATHCrossRefMathSciNetGoogle Scholar
  23. [Z] Zvengrowski A.,A 3-fold vector cross product in IR8, Comment Math. Helv.40, 216–231.Google Scholar

Copyright information

© Springer 1995

Authors and Affiliations

  • Francisco M. Cabrera
    • 1
  1. 1.Departamento de Matemática FundamentalUniversidad de la LagunaTenerife, Canary IslandsSpain

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