Local classification of twodimensional affine spheres with constant curvature metric. (English) Zbl 0784.53002
Summary: We extend results of Radon, Li-Penn and Magid-Ryan and give a local classification of affine spheres with constant curvature metric in affine 3-space.
MSC:
| 53A15 | Affine differential geometry |
References:
| [1] | Blaschke, W., Vorlesungen über Differentialgeometrie II: Affine Differentialgeometrie (1923), J. Springer: J. Springer Berlin · JFM 49.0499.01 |
| [2] | Bokan, N.; Nomizu, K.; Simon, U., Affine hypersurfaces with parallel cubic forms, Tôhoku Math. J. (1989), to appear. |
| [3] | Calabi, E., Complete affine hyperspheres I, Symposia Math., 10, 19-38 (1972) · Zbl 0252.53008 |
| [4] | A.-M. Li, Some theorems in affine differential geometry Acta Math. Sinica; A.-M. Li, Some theorems in affine differential geometry Acta Math. Sinica · Zbl 0694.53008 |
| [5] | A.-M. Li, Calabi conjecture on hyperbolic affine hyperspheres, Mathematische Zeitschrift; A.-M. Li, Calabi conjecture on hyperbolic affine hyperspheres, Mathematische Zeitschrift · Zbl 0696.53004 |
| [6] | Li, A.-M.; Penn, G., Uniqueness theorems in affine differential geometry II, Results in Math., 13, 308-317 (1988) · Zbl 0646.53009 |
| [7] | Magid, M.; Ryan, P., Flat affine spheres in \(R^3 (1989)\), Preprint |
| [8] | Nomizu, K.; Pinkall, U., Cayley surfaces in affine differential geometry, Tôhoku Math. J. (1988), to appear. |
| [9] | Radon, J., Zur Affingeometrie der Regelflächen, Leipziger Berichte, 70, 147-155 (1918) · JFM 46.1114.05 |
| [10] | Schirokov, P. A.; Schirokov, A. P., Affine Differentialgeometrie (1962), Teubner: Teubner Leipzig · Zbl 0106.14703 |
| [11] | Schneider, R., Zur Differentialgeometrie im Groβen I, Math. Z., 101, 375-406 (1967) · Zbl 0156.20101 |
| [12] | Simon, U., The Pick invariant in equiaffine differential geometry, Abh. Math. Sem. Hamburg, 53, 225-228 (1983) · Zbl 0508.53006 |
| [13] | Simon, U., Zur Entwicklung der affinen Differentialgeometrie nach Blaschke, (W. Blaschke, Gesammelte Werke IV (1985), Thales Verlag: Thales Verlag Essen) |
| [14] | Simon, U., The fundamental theorem in affine hypersurface theory, Geom. Dedicata, 26, 125-137 (1988) · Zbl 0641.53011 |
| [15] | Tzitzéica, G., Sur une nouvelle classe de surfaces, Comptes Rendus Acad. Paris, 145, 132-133 (1907) · JFM 38.0642.01 |
| [16] | Tzitzéica, G., Sur une nouvelle classe de surfaces, Comptes Rendus Acad. Paris, 146, 165-166 (1908) · JFM 39.0685.04 |
| [17] | Tzitzéica, G., Sur une nouvelle classe de surfaces, Rendiconti di Palermo, 25, 180-187 (1908) · JFM 39.0685.05 |
| [18] | Tzitzéica, G., Sur une nouvelle classe de surfaces, Rendiconti di Palermo, 28, 210-216 (1909) · JFM 40.0668.04 |
| [19] | J.-H. Yu, Affine hyperspheres with constant sectional curvature in \(A^4\); J.-H. Yu, Affine hyperspheres with constant sectional curvature in \(A^4\) |
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.
