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Complete affine hypersurfaces. I: The completeness of affine metrics. (English) Zbl 0623.53002
The authors consider the problem of completeness for affine hypersurfaces, proving that in the case of affine hyperspheres Euclidean completeness \((=closedness)\) implies completeness of the Berwald- Blaschke, unimodular affine invariant Riemannian metric. They acknowledge that their result has also been obtained by E. Calabi and L. Nirenberg, by somehow different methods (unpublished). This problem has also been recently treated by K. Nomizu [On completeness in affine differential geometry, Prepr. SFB 40 Univ. Bonn/MPI Bonn, No.84-33]. The authors also prove, by the Legendre transformation method the first part of Calabi’s conjecture as to the classification of hyperbolic affine hyperspheres. The full conjecture by the same method, has been treated, too, by T. Sasaki [Nagoya Math. J. 77, 107-123 (1980; Zbl 0404.53003)]. The use of the Legendre transformation was suggested by K. Calabi himself [Symp. Math. 10, 19-38 (1972; Zbl 0252.53008)]. This reviewer proved the second part of the conjecture, by using a different method [Geom. Dedicata 11, 387-396 (1981; Zbl 0475.53009)].
References:
| [1] | Vorlesungen über Differentialgeometrie. II, J. Springer, Berlin, 1923. |
| [2] | Calabi, NAZ Alta Mat. Sym. Mat. pp 19– (1972) |
| [3] | Calabi, Mich. Math. J. 5 pp 105– (1958) |
| [4] | Cheng, Comm. Pure Appl. Math. 28 pp 333– (1975) |
| [5] | Jörgens, = 1, Math. Ann. 127 pp 130– (1954) |
| [6] | Pogorelov, Geometriae Dedicata 1 pp 33– (1972) |
| [7] | Tzitzeica, Rend. Circ. Mat. Palermo 25 pp 180– (1908) |
| [8] | Rend. Circ. Mat. Palermo 28 pp 210– (1909) |
| [9] | Yau, Comm. Pure Appl. Math. 28 pp 201– (1975) |
| [10] | Cheng, Ann. Math. 104 pp 407– (1976) |
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