Abstract
In this paper, continuing with Hu–Li–Vrancken and the recent work of Antić–Dillen- Schoels–Vrancken, we obtain a decomposition theorem which settled the problem of how to determine whether a given locally strongly convex affine hypersurface can be decomposed as a generalized Calabi composition of two affine hyperspheres, based on the properties of its difference tensor K and its affine shape operator S.
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Antić, M., Dillen, F., Schoels, K., et al.: Decomposable affine hypersurfaces. Kyushu J. Math., 68, 93–103 (2014)
Binder, T., Simon, U.: Progress in affine differential geometry-problem list and continued bibliography. In: Geometry and Topology of Submanifolds, X (Beijing/Berlin, 1999), World Sci. Publ., River Edge, NJ, 2000, 1–17
Calabi, E.: Complete affine hyperspheres, I. Sympos. Math., 10, 19–38 (1972)
Dillen, F., Vrancken, L.: Quasi-umblical, locally strongly convex homogeneous affine hypersurfaces. J. Math. Soc. Japan, 46, 477–502 (1994)
Dillen, F., Vrancken, L.: Homogeneous affine hypersurfaces with rank one shape operators. Math. Z., 212, 61–72 (1993)
Dillen, F., Vrancken, L.: Calabi-type composition of affine spheres. Differential Geom. Appl., 4, 303–328 (1994)
Hu, Z. J., Li, C. C., Li, H., et al.: Lorentzian affine hypersurfaces with parallel cubic form. Result. Math., 59, 577–620 (2011)
Hu, Z. J., Li, H., Vrancken, L.: Characterizations of the Calabi product of hyperbolic affine hyperspheres. Results. Math., 52, 299–314 (2008)
Hu, Z. J., Li, H., Vrancken, L.: Locally strongly convex affine hypersurfaces with parallel cubic form. J. Diff. Geom., 87, 239–307 (2011)
Hu, Z. J., Li, C. C., Zhang, C.: On quasi-umbilical locally strongly convex homogeneous affine hypersurfaces. Diff. Geom. Appl., 33, 46–74 (2014)
Kriele, M., Vrancken, L.: An extremal class of three-dimensional hyperbolic affine spheres. Geom. Dedicata, 77, 239–252 (1999)
Kriele, M., Scharlach, C., Vrancken, L.: An extremal class of 3-dimensional elliptic affine spheres. Hokkaido Math. J., 30, 1–23 (2001)
Li, A.-M., Simon, U., Zhao, G. S.: Global Affine Differential Geometry of Hypersurfaces, W. de Gruyter, Berlin, 1993
Nölker, S.: Isometric immersions of warped products. Differential Geom. Appl., 6, 1–30 (1996)
Nomizu, K., Sasaki, T.: Affine differential geometry. In: Geometry of Affine Immersions, Cambridge University Press, Cambridge, 1994
Vrancken, L., Li, A.-M., Simon, U.: Affine spheres with constant affine sectional curvature. Math. Z., 206, 651–658 (1991)
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The first author is supported by the Ministry of Science and Technological Development of Serbia, Project 174012; the second author is supported by NSFC (Grant No. 11371330) and the third author is supported by NSFC (Grant Nos. 11326072 and 11401173)
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Antić, M., Hu, Z.J., Li, C.C. et al. Characterization of the generalized Calabi composition of affine hyperspheres. Acta. Math. Sin.-English Ser. 31, 1531–1554 (2015). https://doi.org/10.1007/s10114-015-4431-1
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DOI: https://doi.org/10.1007/s10114-015-4431-1