# NONDEGENERATE AFFINE HOMOGENEOUS DOMAIN OVER A GRAPH

• Choi, Yun-Cherl (Division of General Education-Mathematics Kwangwoon University)
• Published : 2006.11.01

#### Abstract

The affine homogeneous hypersurface in ${\mathbb{R}}^{n+1}$, which is a graph of a function $F:{\mathbb{R}}^n{\rightarrow}{\mathbb{R}}$ with |det DdF|=1, corresponds to a complete unimodular left symmetric algebra with a nondegenerate Hessian type inner product. We will investigate the condition for the domain over the homogeneous hypersurface to be homogeneous through an extension of the complete unimodular left symmetric algebra, which is called the graph extension.

#### References

1. Y. Choi, Left invariant flat affine structure on Lie group, master's thesis, Seoul National Univ., 1995
2. Y. Choi, Cayley hypersurface and left symmetric algebra, PhD thesis, Seoul National Univ., 2002
3. Y. Choi and H. Kim, A characterization of Cayley Hypersurface and Eastwood and Ezhov conjecture, Internat. J. Math. 16 (2005) no. 8, 841-862 https://doi.org/10.1142/S0129167X05003168
4. F. Dillen and L. Vrancken, Hypersurfaces with parallel difference tensor, Japan. J. Math. (N.S.) 24 (1998), no. 1, 43-60
5. M. Eastwood and V. Ezhov, On affine normal forms and a classification of homogeneous surfaces in affine three-space, Geom. Dedicata 77 (1999) no. 1, 11-69 https://doi.org/10.1023/A:1005083518793
6. M. Eastwood and V. Ezhov, Homogeneous hypersurfaces with isotropy in affine four-space, Tr. Mat. Inst. Steklova 253 (2001), Anal. i Geom. Vopr. Kompleks. Analiza, 57-70
7. W. M. Goldman and M. W. Hirsch, Affine manifolds and orbits of algebraic groups, Trans. Amer. Math. Soc. 295 (1986) no. 1, 175-198 https://doi.org/10.2307/2000152
8. J. H. 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
9. H. Kim, Complete left invariant affine structures on nilpotent Lie groups, J. Differential Geom. 24 (1986), 373-394 https://doi.org/10.4310/jdg/1214440553
10. H. Kim, Developing maps of affinely flat Lie groups, Bull. Korean Math. Soc. 34 (1997), no. 4, 509-518
11. K. Kim, Left invariant affinely flat structures on solvable Lie group, PhD thesis, Seoul National Univ., 1998
12. J.-L. Koszul, Domaines bornes homogenes et orbites de groupes de transformations affines, Bull. Soc. Math. France 89 (1961), no. 26, 515-533
13. A. Mizuhara, On left symmetric algebras with a principal idempotent, Math. Japon. 49 (1999), no. 1, 39-50
14. K. Nomizu and T. Sasaki, Affine differential geometry, Cambridge Tracts in Mathematics, 111, Cambridge University Press, 1994
15. B. O'Neill, Semi-Riemannian geometry, Pure and Applied Mathematics 103, Academic Press Inc., 1983
16. D. Segal, The structure of complete left-symmetric algebras, Math. Ann. 293 (1992), no. 3, 569-578 https://doi.org/10.1007/BF01444735
17. H. Shima, On certain locally flat homogeneous manifolds of solvable Lie groups, Osaka J. Math. 13 (1976), no. 2, 213-229
18. H. Shima, Homogeneous Hessian manifolds, Ann. Inst. Fourier (Grenoble) 30 (1976) no. 3, 91-128
19. E. B. Vinberg, The theory of convex homogeneous cones, Trans. Moscow Math. Soc. 12 (1963), 340-403

#### Cited by

1. Left-symmetric algebras and homogeneous improper affine spheres pp.1572-9060, 2017, https://doi.org/10.1007/s10455-017-9582-0