Results in Mathematics

, Volume 19, Issue 1–2, pp 147–156 | Cite as

Affine Kähler hypersurfaces satisfying certain conditions on the curvature tensor

  • Fabio Podestà


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.


Vector Field Curvature Tensor Ricci Tensor Shape Operator Codazzi Equation 
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  1. [N-Pi-Po]
    K. Nomizu, U.Pinkall and F.Podestà On the geometry of affine Kähler immersions Preprint Scuola Normale Superiore (1989)Google Scholar
  2. [N-Po]
    K. Nomizu and F. Podestà On the Cartan-Norden Theorem for affine Kähler immersions Preprint Scuola Normale Superiore (1990)Google Scholar
  3. [N-S]
    K. Nomizu and B. Smyth Differential Geometry for complex Hypersurfaces, II J. Math. Soc. Japan 20 (1968) 498–521MathSciNetCrossRefMATHGoogle Scholar

Copyright information

© Birkhäuser Verlag, Basel 1991

Authors and Affiliations

  • Fabio Podestà
    • 1
  1. 1.Pisa

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