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HESSIAN GEOMETRY OF THE HOMOGENEOUS GRAPH DOMAIN

  • Published : 2007.07.31

Abstract

In this paper, we will investigate the Hessian geometry of the homogeneous domain over the hypersurface given by a function F : $\mathbb{R}^n{\rightarrow}\mathbb{R}$ with ${\mid}{\det}\;DdF\mid=1$.

Keywords

References

  1. N. Bokan, K. Nomizu, and U. Simon, Affine hypersurfaces with parallel cubic forms, Tohoku Math. J. 42 (1990), no. 2, 101-108 https://doi.org/10.2748/tmj/1178227697
  2. Y. Choi and H. Kim, A characterization of Cayley Hypersurface and Eastwood and Ezhov conjecture, Internat. J. Math. 16 (2005), no. 8, 841-861 https://doi.org/10.1142/S0129167X05003168
  3. Y. Choi, Nondegenerate affine homogeneous domain over a graph, J. Korean Math. Soc. 43 (2006), no. 6, 1301-1324 https://doi.org/10.4134/JKMS.2006.43.6.1301
  4. F. Dillen and L. Vrancken, Hypersurfaces with parallel difference tensor, Japan. J. Math. (N.S.) 24 (1998), no. 1, 43-60 https://doi.org/10.4099/math1924.24.43
  5. F. Dillen, L. Vrancken, and S. Yaprak, Affine hypersurfaces with parallel cubic form, Nagoya Math. J. 135 (1994), 153-164 https://doi.org/10.1017/S0027763000005006
  6. J. Hua Hao and H. Shima, Level surfaces of nondegenerate functions in $R^{n+1}$, Geom. Dedicata 50 (1994), 193-204 https://doi.org/10.1007/BF01265310
  7. H. Kim, Developing maps of affinely flat Lie groups, Bull. Korean Math. Soc. 34 (1997), no. 4, 509-518
  8. A. Mizuhara, On left symmetric algebras with a principal idempotent, Math. Japon. 49 (1999), 39-50
  9. K. Nomizu and T. Sasaki, Affine differential geometry, Cambridge University Press, 1994
  10. B. O'Neill, Semi-Riemannian geometry, Pure and Applied Mathematics 103, Academic Press Inc., 1983
  11. H. Shima, On certain locally flat homogeneous manifolds of solvable Lie groups, Osaka J. Math. 13 (1976), 213-229
  12. H. Shima, Homogeneous convex domains of negative sectional curvature J. Differential Geom. 12 (1977), no. 3, 327-332 https://doi.org/10.4310/jdg/1214434088
  13. H. Shima, Homogeneous Hessian manifolds, Ann. lnst. Fourier (Grenoble) 30 (1976), no. 3, 91-128
  14. H. Shima, The Geometry of Hessian Structures, World Scientific Publishing Co., 2007
  15. E. B. Vinberg, The theory of convex homogeneous cones, Trans. Moscow Math. Soc. 12 (1963), 340-403

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  2. Affine hypersurfaces with parallel difference tensor relative to affine α-connection vol.86, 2014, https://doi.org/10.1016/j.geomphys.2014.07.018
  3. Left-symmetric algebras and homogeneous improper affine spheres vol.53, pp.3, 2018, https://doi.org/10.1007/s10455-017-9582-0