A splitting theorem for Kahler manifolds whose Ricci tensors have constant eigenvalues

Details A splitting theorem for Kahler manifolds whose Ricci tensors have constant eigenvalues A splitting theorem for Kahler manifolds whose Ricci tensors have constant eigenvalues

2000-07-20

arxiv.org/abs/math/0007122v3

International Journal of Mathematics 12 (2001), 769-789

Mathematics

Differential Geometry

18 pages; final version; accepted for publication in International Journal of Mathematics

Scientific paper

10.1142/S0129167X01001052

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. Irreducible Kahler manifolds with two distinct, constant eigenvalues of the Ricci tensor are shown to exist in various situations: there are homogeneous examples of any complex dimension n > 1, if one eigenvalue is negative and the other positive or zero, and of any complex dimension n > 2, if the both eigenvalues are negative; there are non-homogeneous examples of complex dimension 2, if one of the eigenvalues is zero. The problem of existence of Kahler metrics whose Ricci tensor has two distinct, constant eigenvalues is related to the celebrated (still open) Goldberg conjecture. 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.

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