Hessian Manifolds and Convexity

  • Hirohiko Shima
Part of the Progress in Mathematics book series (PM, volume 14)


Let M be a flat affine manifold, that is, M admits open charts (Ui, x i 1 ,..., x i n such that M =U Ui and whose coordinate changes are all affine functions. Such local coordinate systems { i n ,...,x i n } will be called affine local coordinate systems. Throughout this note the local expressions for geometric concepts on M will be given in terms of affine local coordinate systems. A Riemannian metric g on M is said to be Hessian if for each point p∈M there exists a C-function (⌽ defined on a neighborhood of p such that gij=∂2ø/∂xi ∂xj. Such a function ⌽ is called a primitive of g on a neighborhood of p. Using the flat affine structure we define the exterior differentiation dl for tensor bundle valued forms on M. Let g be the cotangent bundle valued 1-form on M corresponding to a Riemannian metric g on M. Then we know that g is Hessian if and only if g0 is dl-closed. A flat affine manifold provided with a Hessian metric is called a Hessian manifold[4], [5]. Koszul dealt with the case where g0 is dl-exact [1], [2], [3].


Local Coordinate System Convex Domain Affine Function Open Chart Exterior Differentiation 
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    K. Yagi, “On Hessian structures on an affine manifold,” in Manifolds and Lie Groups, Papers in Honor of Yozô Matsushima, Progress in Mathematics, Vol. 14, Birkhäuser, Boston, Basel, Stuttgart, 1981, 449–459.Google Scholar

Copyright information

© Springer Science+Business Media New York 1981

Authors and Affiliations

  • Hirohiko Shima
    • 1
  1. 1.Yamaguchi UniversityYamaguchi 753Japan

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