Abstract
The notion of affine Kähler immersions has been recently introduced by Nomizu-Pinkall-Podestà ([N-Pi-Po]). This work is aimed at giving some results towards the classification of non degenerate affine Kähler hypersurfaces with symmetric and parallel Ricci tensor; this problem generalizes the classical results due to Nomizu-Smyth ([N-S]) in the theory of Kählerian hypersurfaces. In a second section we deal with the case of “semisymmetric” affine Kähler immersions, when the curvature tensor R satisfies R · R = 0 and the Ricci tensor is symmetric, providing a complete classification; for affine Kähler curves we prove that the conditions above are actually equivalent to saying that the immersion is isometric for a suitable Kähler metric in C2.
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References
K. Nomizu, U.Pinkall and F.Podestà On the geometry of affine Kähler immersions Preprint Scuola Normale Superiore (1989)
K. Nomizu and F. Podestà On the Cartan-Norden Theorem for affine Kähler immersions Preprint Scuola Normale Superiore (1990)
K. Nomizu and B. Smyth Differential Geometry for complex Hypersurfaces, II J. Math. Soc. Japan 20 (1968) 498–521
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Podestà, F. Affine Kähler hypersurfaces satisfying certain conditions on the curvature tensor. Results. Math. 19, 147–156 (1991). https://doi.org/10.1007/BF03322423
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DOI: https://doi.org/10.1007/BF03322423