A SPLITTING THEOREM FOR KÄHLER MANIFOLDS WHOSE RICCI TENSORS HAVE CONSTANT EIGENVALUES

By V. Apostolov, Tedi Drăghici, Andrei Moroianu. 2000

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It is proved that a compact Kahler manifold whose Ricci tensor has two distinct constant non-negative eigenvalues is locally the product of two Kahler–Einstein manifolds. A stronger result is established for the case of Kahler surfaces. Without the compactness assumption, irreducible Kahler manifolds with Ricci tensor having two distinct constant eigenvalues are shown to exist in various situations: there are homogeneous examples of any complex dimension n ≥ 2 with one eigenvalue negative and the other one positive or zero; there are homogeneous examples of any complex dimension n ≥ 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 Kahler metrics whose Ricci tensor has two distinct constant eigenvalues is related to the celebrated (still open) conjecture of Goldberg [24]. Consequently, the irreducible homogeneous examples with negative eigenvalues give rise to complete Einstein strictly almost Kahler metrics of any even real dimension greater than 4.

Subfields: Applied Mathematics, Geometry and Topology

View Paper: https://www.semanticscholar.org/paper/e4d0ba491cadd2693d52cacc09c5f0b84c15db51

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In Applied Mathematics

In Geometry and Topology

Citations in the corpus

The curvature and the integrability of almost-Kahler manifolds: a survey (2003, background): “The examples given in this section are studied in [10], and rely on the following two simple lemmas.”
The Srní lectures on non-integrable geometries with torsion (2006, result): “r manifold [NP97], J. Armstrong showed some non-existence results [Arm98], while V. Apostolov, T. Dra˘ghici and A. Moroianu constructed non-compact counterexamples to the conjecture in dimensions ≥ 6 [ADM01]. Diﬀerent partial results with variousadditional curvature assumptions are now available. Th”