Calabi-Yau spaces

A Calabi–Yau manifold, also known as a Calabi–Yau space, is a special type of manifold that is described in certain branches of mathematics such as algebraic geometry. The Calabi–Yau manifold's properties, such as Ricci flatness, also yield applications in theoretical physics. Particularly in superstring theory, the extra dimensions of spacetime are sometimes conjectured to take the form of a 6-dimensional Calabi–Yau manifold, which led to the idea of mirror symmetry.

Calabi–Yau manifolds are complex manifolds that are generalizations of K3 surfaces in any number of complex dimensions (i.e. any even number of real dimensions). They were originally defined as compact Kähler manifolds with a vanishing first Chern class and a Ricci-flat metric, though many other similar but inequivalent definitions are sometimes used. They were named "Calabi–Yau spaces" by Candelas et al. (1985) after Eugenio Calabi (1954, 1957) who first conjectured that such surfaces might exist, and Shing-Tung Yau (1978) who proved the Calabi conjecture.

The motivational definition given by Yau is of a compact Kähler manifold with a vanishing first Chern class, that is also Ricci flat. Calabi conjectured their existence and Yau proved the conjecture.

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