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A splitting theorem for Kähler manifolds with constant eigenvalues of the Ricci tensor

Domenii publicaţii > Matematica + Tipuri publicaţii > Articol în revistã ştiinţificã

Autori: V. Apostolov, T. Draghici, A. Moroianu

Editorial: Int. J. Math., 12, p.769-789, 2001.

Rezumat:

It is proved that a compact K{„a}hler manifold whose Ricci
tensor has two distinct constant non-negative eigenvalues is
locally the product of two K{„a}hler-Einstein manifolds. A
stronger result is established for the case of K”ahler surfaces.
Without the compactness assumption, irreducible K”ahler manifolds
with Ricci tensor having two distinct constant eigenvalues are
shown to exist in various situations: there are homogeneous
examples of any complex dimension $nge 2$ with one eigenvalue
negative and the other one positive or zero; there are
homogeneous examples of any complex dimension $nge 3$ with two
negative eigenvalues; there are non-homogeneous examples of
complex dimension 2 with one of the eigenvalues zero. The problem
of existence of K{„a}hler metrics whose Ricci tensor has two
distinct constant eigenvalues is related to the celebrated (still
open) conjecture of Goldberg cite{goldberg}. Consequently, the
irreducible homogeneous examples with negative eigenvalues give
rise to {it complete} Einstein strictly almost K”ahler metrics
of any even real dimension greater than 4.

Cuvinte cheie: Ricci tensor, Goldberg conjecture, almosta Kaehler manifolds

URL: http://math.polytechnique.fr/cmat/moroianu/publi.html