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The Ricci tensor of an almost homogeneous Kähler manifold. (English) Zbl 1045.53049

A \(K\)-manifold is a pair consisting of a compact Kähler manifold \(M\) with vanishing first Betti number and a compact semisimple Lie group \(G\) that acts almost effectively and isometrically, and so biholomorphically, on \(M\) such that \(M\) has cohomogeneity one with respect to the action of \(G\), that is, the regular \(G\)-orbits have codimension one in \(M\). For such a \(K\)-manifold \(M\), the author shows an explicit expression for the Ricci tensor and proves that, up to few exceptions, the Kähler form \(\omega\) and the Ricci form \(\rho\) of \(M\) are uniquely determined by two special curves \(Z_{\omega}\) and \(Z_{\rho}\) with values in the Lie algebra of \(G\). Moreover, he shows how the curve \(Z_{\rho}\) is determined by \(Z_{\omega}\).

MSC:

53C55 Global differential geometry of Hermitian and Kählerian manifolds
57S15 Compact Lie groups of differentiable transformations
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