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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
Article

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.

Keywords

Vector Field Curvature Tensor Ricci Tensor Shape Operator Codazzi Equation 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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References

  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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