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1998 | 45 | 1 | 155-167
Tytuł artykułu

The geometry of a closed form

Treść / Zawartość
Warianty tytułu
Języki publikacji
EN
Abstrakty
EN
It is proved that a closed r-form ω on a manifold M defines a cohomology (called ω-coeffective) on M. A general algebraic machinery is developed to extract some topological information contained in the ω-coeffective cohomology. The cases of 1-forms, symplectic forms, fundamental 2-forms on almost contact manifolds, fundamental 3-forms on $G_{2}$-manifolds and fundamental 4-forms in quaternionic manifolds are discussed.
Słowa kluczowe
Rocznik
Tom
45
Numer
1
Strony
155-167
Opis fizyczny
Daty
wydano
1998
Twórcy
  • Departamento de Matemáticas, Universidad del Pais Vasco, Apartado 644, 48080 Bilbao, Spain
  • Departamento de Matemáticas, Universidad del Pais Vasco, Apartado 644, 48080 Bilbao, Spain
  • Instituto de Matemáticas y Física Fundamental, Consejo Superior de Investigaciones Científicas, Serrano 123, 28006 Madrid, Spain
Bibliografia
  • [1] L. C. de Andrés, M. Fernández, M. de León, R. Ibáñez and J. Mencía, On the coeffective cohomology of compact symplectic manifolds, C. R. Acad. Sci. Paris, 318, Série I, (1994), 231-236.
  • [2] M. Berger, Sur les groupes d'holonomie des variétés à connexion affine et des variétés riemannienes, Bull. Soc. Math. France, 83 (1955), 279-330.
  • [3] D. E. Blair, Contact Manifolds in Riemannian Geometry, Lecture Notes in Math. 509, Springer-Verlag, Berlin, 1976.
  • [4] E. Bonan, Sur l'algèbre extérieure d'une variété presque hermitienne quaternionique, C. R. Acad. Sci. Paris, 295, Série I (1982), 115-118.
  • [5] E. Bonan, Isomorphismes sur une variété presque hermitienne quaternionique, in: Proc. of the Meeting on Quaternionic Structures in Math. and Physics, Trieste, SISSA, (1994), pp. 1-6.
  • [6] T. Bouché, La cohomologie coeffective d'une variété symplectique, Bull. Sci. Math., 114 (2) (1990), 115-122.
  • [7] G. B. Brown and A. Gray, Vector cross products, Comment. Math. Helv. 42 (1967), 222-236.
  • [8] F. Cantrijn, L. Ibort and M. de León, On the geometry of multisymplectic manifolds, to appear in Journal of the Australian Mathematical Society.
  • [9] D. Chinea, M. de León and J. C. Marrero, Topology of cosymplectic manifolds, J. Math. Pures et Appl., 72 (6) (1993), 567-591.
  • [10] D. Chinea, M. de León and J. C. Marrero, Coeffective cohomology on cosymplectic manifolds, Bull. Sci. Math., 119 (1) (1995), 3-20.
  • [11] P. Deligne, Ph. Griffiths, J. Morgan and D. Sullivan, Real homotopy theory of Kähler manifolds, Invent. Math. 29 (1975), 245-274.
  • [12] M. Fernández and A. Gray, Riemannian manifolds with structure group $G_2$, Ann. Mat. Pura Appl. (IV) 32 (1982), 19-45.
  • [13] M. Fernández, R. Ibáñez and M. de León, A Nomizu's theorem for the coeffective cohomology, Math. Z. 226 (1997), 11-23.
  • [14] M. Fernández, R. Ibáñez and M. de León, The coeffective cohomology for compact symplectic nilmanifolds, in: Proceedings of the III Fall Workshop Differential Geometry and its Applications, Granada, Sept. 26-27, 1994, Anales de Física, Monografías 2, CIEMAT-RSFE, Madrid, 1995, pp. 131-144.
  • [15] M. Fernández, R. Ibáñez and M. de León, Coeffective and de Rham cohomologies of symplectic manifolds, to appear in J. of Geometry and Physics 491 (1998).
  • [16] M. Fernández, R. Ibáñez and M. de León, Coeffective and de Rham cohomologies on almost contact manifolds, Differential Geometry and Its Applications 8 (1998), 285-303.
  • [17] M. Fernández, R. Ibáñez and M. de León, Coeffective cohomology of quaternionic Kähler manifolds, Conference on Differential Geometry, Budapest, July 27-30, 1996.
  • [18] M. Fernández and L. Ugarte, Dolbeault cohomology for $G_2$-manifolds, Geometriae Dedicata 70 (1998), 57-86.
  • [19] M. Fernández and L. Ugarte, A differential complex for locally conformal calibrated $G_2$-manifolds, preprint 1996.
  • [20] Ph. Griffiths and J. Morgan, Rational homotopy theory and differential forms, Progress in Math. 16, Birkhäuser, 1981.
  • [21] R. Ibáñez, Coeffective-Dolbeault cohomology of compact indefinite Kähler manifolds, Osaka J. Math. 34 (1997), 553-571.
  • [22] V. Kraines, Topology of quaternionic manifolds, Trans. Amer. Math. Soc. 122 (1966), 357-367.
  • [23] P. Libermann and Ch. M. Marle, Symplectic geometry and analytical mechanics, Kluwer, Dordrecht, 1987.
  • [24] G. Lupton and J. Oprea, private communication.
  • [25] G. Lupton and J. Oprea, Symplectic manifolds and formality, J. Pure and Appl. Algebra 91 (1994), 193-207.
  • [26] D. McDuff and D. Salamon, Introduction to symplectic topology, Oxford Math. Monographs, Oxford Univ. Press, 1995.
  • [27] J. Moser, On the volume elements on manifolds, Trans. Amer. Soc. Math., 120 (1965), 286-295.
  • [28] K. Nomizu, On the cohomology of compact homogeneous spaces of nilpotent Lie group, Annals of Math. 59 (2) (1954), 531-538.
  • [29] J. A. Oubiña, New classes of almost contact metric structures, Publicationes Mathematicae 32 (3-4) (1985), 187-193.
  • [30] M. S. Raghunatan, Discrete subgroups of Lie groups, Ergebnisse der Mathematik 68, Springer-Verlag, Berlin, 1972.
  • [31] R. Reyes, Some special geometries defined by Lie groups, Thesis, Oxford, 1993.
  • [32] S. Salamon, Riemannian Geometry and Holonomy Groups, Pitman Research Notes in Math. Series 201, Longman, Boston, 1989.
  • [33] A. Swann, Hyperkähler and quaternionic Kähler geometry, Math. Ann. 289 (1991), 421-450.
  • [34] A. Tralle and J. Oprea, Symplectic Manifolds with no Kähler Structure, Lecture Notes in Math. 1661, Springer, Berlin, 1997.
  • [35] I. Vaisman, Locally conformal symplectic manifolds, Internat. J. Math. & Math. Sci. 8 (1985), 3, 521-536.
  • [36] A. Weinstein, Lectures on symplectic manifolds, CBMS, Amer. Math. Soc. 29, Providence (R.I.), 1977.
Typ dokumentu
Bibliografia
Identyfikatory
Identyfikator YADDA
bwmeta1.element.bwnjournal-article-bcpv45i1p155bwm
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