Calabi–Eckmann manifold

In complex geometry, a part of mathematics, a Calabi–Eckmann manifold (or, often, Calabi–Eckmann space), named after Eugenio Calabi and Beno Eckmann, is a complex, homogeneous, non-Kähler manifold, homeomorphic to a product of two odd-dimensional spheres of dimension ≥ 3.

The Calabi–Eckmann manifold is constructed as follows. Consider the space ${\displaystyle {\mathbb {C} }^{n}\backslash 0\times {\mathbb {C} }^{m}\backslash 0}$, m,n > 1, equipped with an action of a group ${\displaystyle {\mathbb {C}}}$:

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