I am confused about the different Ricci-named objects in complex and specifically Kahler geometry: We have the Ricci curvature tensor, which we get by contracting the holomorphic indices of the Riemann tensor. We have the Ricci scalar Ric, which we get by contracting the Ricci tensor. Then there is a Ricci form, and I don't see how it is at all different from the Ricci tensor (apart from it not having any indices - how can it be a 2-form without having indices?), and finally the *other* Ricci scalar R, which we get by contracting the Ricci form.(adsbygoogle = window.adsbygoogle || []).push({});

Finally, we have the Calabi-Yau manifold, which we get by taking a "Ricci-flat" Kahler manifold. and I can't figure out for the life of me is which of these Ricci-named objects vanishes for a Ricci-flat manifold. Help in unraveling this ridiculous confusion is much appreciated!

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# Ricci form and Kahler manifolds

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