A Splitting Theorem for Kähler Manifolds Whose Ricci Tensors Have Constant Eigenvalues

Abstract

It is proved that a compact Kähler manifold whose Ricci tensor has two distinct constant non-negative eigenvalues is locally the product of two Kähler-Einstein manifolds. A stronger result is established for the case of Kähler surfaces. Without the compactness assumption, irreducible Kähler 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 Kähler 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 Kähler metrics of any even real dimension greater than 4.

Kählerian twistor spinors

“…We first show that on a compact Kähler spin manifold of constant scalar curvature all Kählerian twistor spinors are special Kählerian twistor spinors; then we show that the existence of such a nontrivial spinor imposes strong restrictions on the Ricci tensor, namely it only has two constant eigenvalues. This has already been proven by A. Moroianu, [24], in the special case of limiting manifolds of Kirchberg's inequality for even complex dimension and we notice that his method works for any bundle Σ r M. By a result of V. Apostolov, T. Drȃghici and A. Moroianu, [2], we derive that the Ricci tensor must be parallel. Thus, assuming that the manifold is simply connected, it must be, by de Rham's decomposition theorem, a product of irreducible Kähler-Einstein manifolds.…”

The curvature and the integrability of almost-Kähler manifolds: A survey

“…It is erroneously stated in [10] and [7,Example 3] that the homogeneous Kähler surface M = (SU (2) · Sol 2 )/U (1) appearing in the classification of Shima [82] is deRham irreducible. In fact, a more careful investigation of the construction in [82] shows that any invariant Kähler metric (g, I) on M = (SU (2)·Sol 2 )/U (1) is the Kähler product of a metric of constant Gauss curvature on CP 1 with a metric of constant Gauss curvature on CH 1 .…”

Unit Killing Vector Fields on Nearly Kähler Manifolds

“…The geometry of the situation is completely understood in terms of this data. Moreover, if the nearly Kähler manifold is compact, a Sekigawatype argument from [2] shows that the almost-Kähler structure is actually integrable, allowing us to prove the main result of this paper. …”

“…We will denote the projection of a 2-form α onto Λ inv by α (1,1) and the projection onto Λ anti by α (2,0) . This is motivated by the isomorphisms…”

“…Despite of all these interesting features, very little is known about these manifolds. In particular, apart from the 3-symmetric spaces S 6 , S 3 × S 3 , CP 3 , F (1,2) no compact example is available and moreover these is are the only compact homogeneous examples [5].…”