Advertisement

On Hessian Structures on an Affine Manifold

  • Katsumi Yagi
Part of the Progress in Mathematics book series (PM, volume 14)

Abstract

On a smooth manifold, an affine connection whose torsion and curvature vanish identically is called an affine structure. A smooth manifold provided with an affine structure is called an affine manifold. Let M be an affine manifold with an affine structure D. The co-variant differentiation by D will be also denoted by D. A Riemannian metric h on M is called a hessian metric if for each point x∈M there exist a neighborhood U of x and a smooth function ⌽ on U such that g = D2⌽ on U [5]. In this note we shall give an example of an affine manifold which does not admit any hessian metric and then determine the structure of A-Lie algebras which admit hessian metrics. For these purposes, we shall also establish a vanishing theorem of a certain cohomology group. The author would like to thank Professor H. Shima who introduced him to the problem discussed here.

Keywords

Cohomology Group Associative Algebra Smooth Manifold Affine Structure Hopf Manifold 
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.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. [1]
    W. Greub, S. Halperin, and R. Vanstone, Connection, Curvature and Cohomology, Vol. It, Academic Press, New York and London, 1973.Google Scholar
  2. [2]
    J.L. Koszul, “Variétés localement plates et convexité,” Osaka J. Math. 2 (1965), 285–290.MathSciNetMATHGoogle Scholar
  3. [3]
    J.L. Koszul, “Déformations de connexions localement plates,” Ann. Inst. Fourier, Grenoble 18, 1 (1968), 103–114.MathSciNetMATHCrossRefGoogle Scholar
  4. [4]
    T. Nagano and K. Yagi, “The affine structures on the real two-torus (1),” Osaka J. Math. 11 (1974), 181–210.MathSciNetMATHGoogle Scholar
  5. [5]
    H. Shima, “On certain locally flat homogeneous manifolds of solvable Lie groups,” Osaka J. Math. 13 (1976), 213–229.MathSciNetMATHGoogle Scholar
  6. [6]
    H. Shima, “Homogeneous hessian manifolds,” in Manifolds and Lie Groups, Payers in Honor of Yozô Matsushima, Progress in Mathematics, Vol. 14, Birkhäuser, Boston, Basel, Stuttgart, 1981, 385–392.Google Scholar
  7. [7]
    K. Yagi, “On compact homogeneous affine manifolds,” Osaka J. Math. 7 (1970), 457–475.MathSciNetMATHGoogle Scholar

Copyright information

© Springer Science+Business Media New York 1981

Authors and Affiliations

  • Katsumi Yagi
    • 1
  1. 1.Osaka UniversityToyonaka, Osaka 560Japan

Personalised recommendations