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The Pontryagin Forms of Hessian Manifolds

  • J. ArmstrongEmail author
  • S. Amari
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 9389)

Abstract

We show that Hessian manifolds of dimensions 4 and above must have vanishing Pontryagin forms. This gives a topological obstruction to the existence of Hessian metrics. We find an additional explicit curvature identity for Hessian 4-manifolds. By contrast, we show that all analytic Riemannian 2-manifolds are Hessian.

Keywords

Riemannian Metrics Symmetric Power Flat Structure Flat Connection Topological Obstruction 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

References

  1. 1.
    Amari, S., Armstrong, J.: Curvature of Hessian manifolds. Differ. Geom. Appl. 33, 1–12 (2014)MathSciNetCrossRefzbMATHGoogle Scholar
  2. 2.
    Ay, N., Tuschmann, W.: Dually flat manifolds and global information geometry. Open. Syst. Inf. Dyn. 9(2), 195–200 (2002)MathSciNetCrossRefzbMATHGoogle Scholar
  3. 3.
  4. 4.
    Cartan, E.: Les systèmes différentiels extérieurs et leur applications géométriques, vol. 994. Hermann & cie, Paris (1945)Google Scholar
  5. 5.
    Goldschmidt, H.: Integrability criteria for systems of nonlinear partial differential equations. J. Differ. Geom. 1(3–4), 269–307 (1967)MathSciNetzbMATHGoogle Scholar
  6. 6.
    Guillemin, V.W., Sternberg, S.: An algebraic model of transitive differential geometry. Bull. Am. Math. Soc. 70(1), 16–47 (1964)MathSciNetCrossRefzbMATHGoogle Scholar
  7. 7.
    Penrose, R., Rindler, W.: Spinors and Space-Time: Volume 1, Two-Spinor Calculus and Relativistic Fields. Cambridge Monographs on Mathematical Physics. Cambridge University Press, Cambridge (1987)zbMATHGoogle Scholar
  8. 8.
    Shima, H.: The Geometry of Hessian Structures, vol. 1. World Scientific, Singapore (2007)CrossRefzbMATHGoogle Scholar

Copyright information

© Springer International Publishing Switzerland 2015

Authors and Affiliations

  1. 1.King’s College LondonLondonUK
  2. 2.RIKEN Brain Science InstituteSaitamaJapan

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