Abstract
A Riemannian metric g on a flat manifold M with flat connection D is called a Hessian metric if it is locally expressed by the Hessian of local functions ϕ with respect to the affine coordinate systems, that is, g = Ddϕ Such pair (D, g), g, and M are called a Hessian structure, a Hessian metric, and a Hessian manifold, respectively [S7].
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Preview
Unable to display preview. Download preview PDF.
References
Amari, S.: Differential-geometrical methods in statistics. Springer Lecture Notes in Statistics (1985)
Amari, S., Nagaoka, H.: Methods of information geometry. Translation of Mathematical Monographs. AMS, Oxford Univ. Press (2000)
Cheng, S.Y., Yau, S.T.: The real Monge-Ampére equation and affine flat structures. In: Proc. the 1980 Beijing Symposium of Differential Geometry and Differential Equations, pp. 339–370. Science Press, Beijing, Gordon and Breach, Science Publishers, Inc., New York (1982)
Delanoë, P.: Remarques sur les variëtës localement hessiennes. Osaka J. Math., 65–69 (1989)
Furuhata, H., Kurose, T.: Hessian manifolds of nonpositive constant Hessian sectionl curvature. Tohoku Math. J., 31–42 (2013)
Koszul, J.L.: Domaines bornés homogènes et orbites de groupes de transformations affines. Bull. Soc. Math. France 89, 515–533 (1961)
Koszul, J.L.: Ouvert convexes homogènes des espaces affines. Math. Zeitschr. 79, 254–259 (1962)
Koszul, J.L.: Variétés localement plates et convexité. Osaka J. Math. 2, 285–290 (1965)
Nguiffo Boyom, M.: The cohomology of Koszul-Vinberg algebras. Pacific J. Math. 225, 119–153 (2006)
Nomizu, K., Sasaki, T.: Affine Differential Geometry. Cambridge Univ. Press (1994)
Shima, H.: Symmetric spaces with invariant locally Hessian structures. J. Math. Soc. Japan, 581–589 (1977)
Shima, H.: Homogeneous Hessian manifolds. Ann. Inst. Fourier, 91–128 (1980)
Shima, H.: Vanishing theorems for compact Hessian manifolds. Ann. Inst. Fourier, 183–205 (1986)
Shima, H.: Harmonicity of gradient mappings of level surfaces in a real affine space. Geometriae Dedicata, 177–184 (1995)
Shima, H.: Hessian manifolds of constant Hessian sectional curvature. J. Math. Soc. Japan, 735–753 (1995)
Shima, H.: Homogeneous spaces with invariant projectively flat affine connections. Trans. Amer. Math. Soc., 4713–4726 (1999)
Shima, H.: The Geometry of Hessian Structures. World Scientific (2007)
Vinberg, E.B.: The Theory of convex homogeneous cones. Trans. Moscow Math. Soc., 340–403 (1963)
Author information
Authors and Affiliations
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2013 Springer-Verlag Berlin Heidelberg
About this paper
Cite this paper
Shima, H. (2013). Geometry of Hessian Structures. In: Nielsen, F., Barbaresco, F. (eds) Geometric Science of Information. GSI 2013. Lecture Notes in Computer Science, vol 8085. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-40020-9_4
Download citation
DOI: https://doi.org/10.1007/978-3-642-40020-9_4
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-642-40019-3
Online ISBN: 978-3-642-40020-9
eBook Packages: Computer ScienceComputer Science (R0)