## 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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### References:

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