What is Calabi-Yau manifold?
It is said that there are 10 dimensions in [[String theory|String Theory]], out of which 4 are our ordinary Length, Width, Height and [[Time]]. The other 6 dimensions are said to be closely packed together in a [[manifold]] called the Calabi-Yau manifold.<ref>{{Citation|title=Calabi–Yau manifold|date=2025-01-22|url=https://en.wikipedia.org/wiki/Calabi%E2%80%93Yau_manifold|work=Wikipedia|language=en|access-date=2025-01-29}}</ref> <references /> [[File:CalabiYau5.jpg|thumb|A 2D slice of the 6D Calabi-Yau quintic manifold. ]] A '''Calabi&ndash;Yau manifold''', or 'Calabi&ndash;Yau space', is a special type of [[manifold]]. It is described in certain branches of [[mathematics]] such as [[algebraic geometry]]. The Calabi–Yau manifold's properties, such as [[Ricci flat]]ness(condition when the average curvature of every point on a space cancels out to be '''zero'''),<ref>http://sgovindarajan.wikidot.com/ricciflatmetrics</ref> have applications in [[theoretical physics]]. Particularly in [[superstring theory]], the extra dimensions of [[spacetime]] might take the form of a 6-dimensional Calabi&ndash;Yau manifold. This led to the idea of [[string theory]] mirror [[symmetry]]. The Calabi-Yau manifolds are complex manifolds are extensions or broader forms of [[:en:K3_surface|K3 Surfaces]] in any number of complex dimensions (1 complex dimension = 2 real dimensions). K3 surfaces is a 4-dimensional (4D) complex surface which is smooth, compact, and Ricci Flat (look at the former paragraph). There are different types of such manifolds, depending on the number of complex dimensions they have. For example, Calabi-Yau 2-fold has 2 complex dimensions, Calabi-Yau 3-fold has 3, and so on. == Use in String Theory == The various properties of the Calabi-Yau manifold, like ricci-flatness, supersymmetry make it very useful in string theory. # '''The Ricci Flatness property''' of the Calabi-Yau manifold makes it useful in String Theory. String Theory requires the background spacetime to be flat at ''large scales'' Compactifying