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Calabi-Yau manifolds are a certain class of smooth [[complex analysis|complex]] [[manifolds]] that have proved to play an important role in several areas of mathematics and physics.  Calabi-Yau manifolds have proved to occupy a central place in [[string theory]] and are the subject of the important mirror symmetry conjecture.  Research here has in turn led to surprising results in the study of [[algebraic geometry]] and related fields.  

Calabi-Yau manifolds are a certain class of smooth [[complex analysis|complex]] [[manifolds]] that have proved to play an important role in several areas of mathematics and physics.  Calabi-Yau manifolds have proved to occupy a central place in [[string theory]] and are the subject of the important mirror symmetry conjecture.  Research here has in turn led to surprising results in the study of [[algebraic geometry]] and related fields.  

To be Calabi-Yau, a manifold must first have the structure of [[Kähler manifold]] -- this means that it is a manifold with some notion of distance provided by a [[metric|Riemannian manifold]] (called the Kähler metric), satisfying certain additional conditions.  Then a Kähler manifold <math>M</math> is said to be '''Calabi-Yau''' if it carries a non-vanishing [[complex analysis|holomorphic]] [[Calc3.10|differential form]] of maximal dimension (in more sophisticated language, one says that "the canonical bundle is trivial", and this is turn is equivalent to the vanishing of the first Chern class).  In the case that the manifold in question is [[compact space|compact]], this has many several equivalent statements, which may be interpreted directly as conditions imposed on the Kähler metric: <math>M</math> is Calabi-Yau if and only if the Kähler metric has zero [[curvature|Ricci curvature]].

To be Calabi-Yau, a manifold must first have the structure of [[Kähler manifold]]—this means that it is a manifold with some notion of distance provided by a [[metric|Riemannian manifold]] (called the Kähler metric), satisfying certain additional conditions.  Then a Kähler manifold <math>M</math> is said to be '''Calabi-Yau''' if it carries a non-vanishing [[complex analysis|holomorphic]] [[Calc3.10|differential form]] of maximal dimension (in more sophisticated language, one says that "the canonical bundle is trivial", and this is turn is equivalent to the vanishing of the first Chern class).  In the case that the manifold in question is [[compact space|compact]], this has many several equivalent statements, which may be interpreted directly as conditions imposed on the Kähler metric: <math>M</math> is Calabi-Yau if and only if the Kähler metric has zero [[curvature|Ricci curvature]].

==Examples==

==Examples==