![]() ![]() This section summarizes some of the more common definitions and the relations between them.Ī Calabi–Yau n equipped with a complex manifold structure. There are many other definitions of a Calabi–Yau manifold used by different authors, some inequivalent. The motivational definition given by Shing-Tung Yau is of a compact Kähler manifold with a vanishing first Chern class, that is also Ricci flat. 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. (1985), after Eugenio Calabi ( 1954, 1957) who first conjectured that such surfaces might exist, and Shing-Tung Yau ( 1978) who proved the Calabi conjecture.Ĭalabi–Yau manifolds are complex manifolds that are generalizations of K3 surfaces in any number of complex dimensions (i.e. ![]() 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. In algebraic geometry, a Calabi–Yau manifold, also known as a Calabi–Yau space, is a particular type of manifold which has properties, such as Ricci flatness, yielding applications in theoretical physics. ![]() A 2D slice of a 6D Calabi–Yau quintic manifold. ![]()
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