Generalized Geometry of Norden and Para Norden Manifolds

  • Antonella NanniciniEmail author
Part of the Springer INdAM Series book series (SINDAMS, volume 21)


We study complex structures \(\widehat{J}\) on the generalized tangent bundle of a smooth manifold M compatible with the standard symplectic structure. In particular we describe the class of such generalized complex structures defined by a pseudo Riemannian metric g and a g-symmetric operator H such that H2 = μI, \(\mu \in \mathbb{R}\). These structures include the case of complex Norden manifolds for μ = −1 and the case of Para Norden manifolds for μ = 1 (Nannicini, J Geom Phys 99:244–255, 2016; Nannicini, On a class of pseudo calibrated generalized complex structures: from Norden to para Norden through statistical manifolds, preprint, 2016). We describe integrability conditions of \(\widehat{J}\) with respect to a linear connection ∇ and we give examples of geometric structures that naturally give rise to integrable generalized complex structures. We define the concept of generalized \(\bar{\partial }\mbox{ -}operator\) of (M, H, g, ∇), and we describe certain holomorphic sections. We survey several results appearing in a series of author’s previous papers, (Nannicini, J Geom Phys 56:903–916, 2006; Nannicini, J Geom Phys 60:1781–1791, 2010; Nannicini, Differ Geom Appl 31:230–238, 2013; Adv Geom 16(2):165–173, 2016; Nannicini, Adv Geom 16(2):165–173, 2016; Nannicini, J Geom Phys 99:244–255, 2016; Nannicini, On a class of pseudo calibrated generalized complex structures: from Norden to para Norden through statistical manifolds, preprint, 2016), with special attention to recent results on the generalized geometry of Norden and Para Norden manifolds (Nannicini, J Geom Phys 99:244–255, 2016; Nannicini, Balkan J Geom Appl 22:51–69, 2017).


Generalized geometry Hessian manifolds Lie algebroids Norden and Para Norden manifolds Statistical manifolds 



The author’s research was partially supported by the following grants of the Italian Ministry of Education (MIUR): PRIN Varietà reali e complesse (2010NNBZ78) and by GNSAGA of INDAM.


  1. 1.
    S. Amari, Differential-Geometrical Methods in Statistics. Lectures Notes in Statistics, vol. 28 (Springer, New York, 1985)Google Scholar
  2. 2.
    A.M. Blaga, M. Crasmareanu, A class of almost tangent structures in generalized geometry. Balkan J. Geom. Appl. 19(2), 23–35 (2014)MathSciNetzbMATHGoogle Scholar
  3. 3.
    T. Courant, Dirac manifolds. Trans. Am. Math. Soc. 319, 631–661 (1990)MathSciNetCrossRefzbMATHGoogle Scholar
  4. 4.
    L. David, On cotangent manifolds, complex structures and generalized geometry. Ann. Inst. Fourier 66(1), 1–28 (2016)MathSciNetCrossRefzbMATHGoogle Scholar
  5. 5.
    J. Diustermaat, On Hessian Riemannian structures. Asian J. Math. 5, 79–91 (2001)MathSciNetCrossRefGoogle Scholar
  6. 6.
    D.S. Freed, Special Kähler manifolds. Commun. Math. Phys. 203(1), 31–52 (1999)CrossRefzbMATHGoogle Scholar
  7. 7.
    G. Ganchev, V. Mihova, Canonical connection and the canonical conformal group on an almost complex manifold with B-metric. Ann. Univ. Sofia Fac. Math. Inf. 81, 195–206 (1987)MathSciNetzbMATHGoogle Scholar
  8. 8.
    M. Gualtieri, Generalized complex geometry. Ann. Math. (2) 174, 75–123 (2011). arXiv:math.DG/0703298Google Scholar
  9. 9.
    N. Hitchin, The moduli space of special Lagrangian submanifolds. Ann. Sc. Norm. Super. Pisa Cl. Sci. (4) 25, 503–515 (1997)Google Scholar
  10. 10.
    N. Hitchin, Generalized Calabi-Yau manifolds. Q. J. Math. Oxford 54, 281–308 (2003)MathSciNetCrossRefzbMATHGoogle Scholar
  11. 11.
    N. Hitchin, Lectures on generalized geometry, in Survey in Differential Geometry. Geometry of Special Holonomy and Related Topics, vol. XVI (International Press, Somerville, MA, 2011), pp. 79–124Google Scholar
  12. 12.
    M. Iscan, A. Salimov, On the curvature of Kähler Norden manifolds. Proc. Indian Acad. Sci. Math. Sci. 119(1), 71–80 (2009)MathSciNetCrossRefzbMATHGoogle Scholar
  13. 13.
    Y. Kosmann-Schwarzbach, Exact Gerstenhaber algebras and Lie bialgebroids. Acta Appl. Math. 41, 153–165 (1995)MathSciNetCrossRefzbMATHGoogle Scholar
  14. 14.
    J. Loftin, S.T. Yau, E. Zaslow, Affine manifolds, SYZ geometry and the “Y” vertex. J. Differ. Geom. 71(1), 129–158 (2005)MathSciNetCrossRefzbMATHGoogle Scholar
  15. 15.
    A. Nannicini, Calibrated complex structures on the generalized tangent bundle of a Riemannian manifold. J. Geom. Phys. 56, 903–916 (2006)MathSciNetCrossRefzbMATHGoogle Scholar
  16. 16.
    A. Nannicini, Almost complex structures on cotangent bundles and generalized geometry. J. Geom. Phys. 60, 1781–1791 (2010)Google Scholar
  17. 17.
    A. Nannicini, Special Kähler manifolds and generalized geometry. Differ. Geom. Appl. 31, 230–238 (2013)CrossRefzbMATHGoogle Scholar
  18. 18.
    A. Nannicini, Generalized geometry of pseudo Riemannian manifolds and generalized \(\bar{\partial }\)-operator. Adv. Geom. 16(2), 165–173 (2016)MathSciNetCrossRefzbMATHGoogle Scholar
  19. 19.
    A. Nannicini, Generalized geometry of Norden manifolds. J. Geom. Phys. 99, 244–255 (2016)MathSciNetCrossRefzbMATHGoogle Scholar
  20. 20.
    A. Nannicini, On a class of pseudo calibrated generalized complex structures related to Norden, para-Norden and statistical manifolds. Balkan J. Geom. Appl. 22, 51–69 (2017)MathSciNetzbMATHGoogle Scholar
  21. 21.
    M. Nogushi, Geometry of statistical manifolds. Differ. Geom. Appl. 2, 197–222 (1992)MathSciNetCrossRefGoogle Scholar
  22. 22.
    A.P. Norden, On a class of four-dimensional A-spaces. Russian Math. (Izv VUZ) 17(4), 145–157 (1960)Google Scholar
  23. 23.
    J. Pradines, Théorie de Lie pour les groupoides différentiable. Calcul différentiel dans la categorie des groupoides infinitésimaux. C.R.Acad. Sci. Paris, Sér. A 264, 245–248 (1967)Google Scholar
  24. 24.
    A. Salimov, M. Iscan, F. Etayo, Paraholomorphic B-manifolds and its properties. Topol. Appl. 154, 925–933 (2007)MathSciNetCrossRefzbMATHGoogle Scholar
  25. 25.
    H. Shima, K. Yagi, Geometry of Hessian manifolds. Differ. Geom. Appl. 7, 277–290 (1997)MathSciNetCrossRefzbMATHGoogle Scholar
  26. 26.
    B. Totaro, The curvature of a Hessian manifold. Int. J. Math. (4) 15, 369–391 (2004)Google Scholar

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Authors and Affiliations

  1. 1.Dipartimento di Matematica e Informatica “U. Dini”Università di FirenzeFirenzeItaly

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