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Encyclopedia > Complex manifold

In differential geometry, a complex manifold is a manifold such that every neighborhood looks like the complex n-space in a coherent way. More precisely, a complex manifold has an atlas of charts to Cn, such that the change of coordinates between charts are holomorphic. In mathematics, differential topology is the field dealing with differentiable functions on differentiable manifolds. ... On a sphere, the sum of the angles of a triangle is not equal to 180Â° (see spherical trigonometry). ... This is a glossary of some terms used in the branch of mathematics known as topology. ... In topology, a branch of mathematics, an atlas describes how a complicated space called a manifold is glued together from simpler pieces. ... In topology, an atlas describes how a complicated space is glued together from simpler pieces. ... Holomorphic functions are the central object of study of complex analysis; they are functions defined on an open subset of the complex number plane C with values in C that are complex-differentiable at every point. ...

## Implications of complex structure GA_googleFillSlot("encyclopedia_square");

Since complex analytic functions are much more rigid than smooth functions, the theories of smooth and complex manifolds have very different flavors. For example, the Whitney embedding theorem tells us that every smooth manifold can be embedded as a smooth submanifold of Rn, whereas it is "rare" for a complex manifold to have a holomorphic embedding into Cn. Consider for example any compact, connected complex manifold M: any holomorphic function on it is locally constant by Liouville's theorem. Now if we had a holomorphic embedding of M into Cn, then the coordinate functions of Cn would restrict to nonconstant holomorphic functions on M, contradicting compactness, except in the case that M is just a point. Complex manifolds that can be embedded in Cn are called Stein manifolds and form a very special class of manifolds including, for example, smooth complex affine algebraic varieties. In mathematical analysis, a differentiability class is a classification of functions according to the properties of their derivatives. ... In mathematics, particularly in differential topology, the strong Whitney embedding theorem states that any connected smooth m-dimensional manifold (required also to be Hausdorff and second-countable) can be smoothly embedded in Euclidean -space. ... Compact as a general noun can refer to: Look up Compact on Wiktionary, the free dictionary a diplomatic contract or covenant among parties, sometimes known as a pact, treaty, or an interstate compact; a British term for a newspaper format; In mathematics, it can refer to various concepts: Mostly commonly... In mathematics, a function f from a topological space A to a set B is called locally constant, iff for every a in A there exists a neighborhood U of a, such that f is constant on U. Every constant function is locally constant. ... Liouvilles theorem in complex analysis states that every bounded (i. ... In mathematics, a Stein manifold in the theory of several complex variables and complex manifolds is a closed, complex submanifold of the vector space of n complex dimensions. ...

Since the transition maps between charts are holomorphic, complex manifolds are, in particular, smooth and canonically oriented. The classification of complex manifolds is much more subtle than that of differentiable manifolds. For example, while in dimensions other than four, a given toplogical manifold has at most finitely many smooth structures, a topological manifold supporting a complex structure can and often does support uncountably many complex structures. Riemann surfaces, two dimensional manifolds equipped with a complex structure, which are topologically classified by the genus, are an important example of this phenomenon. The set of complex structures on a given orientable surface, modulo biholomorphic equivalence, itself forms a complex algebraic variety called a moduli space, the structure of which remains an area of active research. In algebraic geometry, a moduli space is a parameter space for families of algebraic objects (such as algebraic varieties, morphisms, vector bundles). ...

## Examples of complex manifolds

In mathematics, complex projective space, or CPn, is the projective space of (complex) lines in Cn+1. ... In mathematics, a Grassmannian is the space of all k-dimensional subspaces of an n-dimensional vector space V, often denoted Gk(V) or simply Gk,n. ... In mathematics, particularly in complex analysis, a Riemann surface is a one-dimensional complex manifold. ... In mathematics, a Lie group is an analytic real or complex manifold that is also a group such that the group operations multiplication and inversion are analytic maps. ... This article is about algebraic varieties. ...

## Almost-complex structures

An almost complex structure on any manifold (for instance, a real manifold as opposed to a complex one) is an endomorphism of the tangent bundle whose square is −Id. In mathematics, an almost complex manifold is a smooth manifold equipped with a structure that, roughly speaking, defines a multiplication by i on each tangent space. ... In mathematics, an endomorphism is a morphism (or homomorphism) from a mathematical object to itself. ... In mathematics, the tangent bundle of a differentiable manifold M, denoted by T(M) or just TM, is the disjoint union of the tangent spaces to each point of M An element of T(M) is a pair (x,v) where x âˆˆ M and v âˆˆ Tx(M), the tangent space...

Any complex manifold has an almost complex structure, but not every almost complex structure comes from a complex structure. For example, the 6 dimensional sphere has a natural almost complex structure arising from the fact that it is the orthogonal complement of i in the unit sphere of the octonions, but this is not a complex structure. (It is not currently known whether or not the 6-sphere has a complex structure.) Using an almost complex structure we can make sense of holomorphic maps and ask about the existence of holomorphic coordinates on the manifold. The existence of holomorphic coordinates is equivalent to saying the manifold is complex (which is what the chart definition says). 2-sphere wireframe as an orthogonal projection Just as a stereographic projection can project a spheres surface to a plane, it can also project a 3-spheres surface into 3-space. ... In the mathematical fields of linear algebra and functional analysis, the orthogonal complement of a subspace W of an inner product space V is the set of all vectors in V that are orthogonal to every vector in W, i. ... In mathematics, the octonions are a nonassociative extension of the quaternions. ...

Tensoring the tangent bundle with the complex numbers we get the complexified tangent bundle, on which multiplication by complex numbers makes sense (even if we started with a real manifold). The eigenvalues of an almost complex structure are $pm i$ and the eigenspaces form sub-bundles denoted by T0,1M and T1,0M. The Newlander-Niremberg theorem shows that an almost complex structure is actually a complex structure precisely when these subbundles are involutive, i.e., closed under the Lie bracket of vector fields. When this happens, we say that the almost complex structure is integrable. In mathematics, an almost complex manifold is a smooth manifold equipped with smooth linear complex structure on each tangent space. ... Integrability is a mathematical concept used in different areas. ...

The Nijenhuis tensor is defined on pairs of vector fields,

X,Y

by

NJ(X,Y) = [X,Y] + J[JX,Y] + J[X,JY] − [JX,JY].

## Kähler and Calabi-Yau manifolds

One can define an analogue of a Riemannian metric for complex manifolds, called a Hermitian metric. Like a Riemannian metric, a Hermitian metric consists of a smoothly varying, positive definite inner product on the tangent bundle, which is Hermitian with respect to the complex structure on the tangent space at each point. As in the Riemannian case, such metrics always exist in abundance on any complex manifold. If the skew symmetric part of such a metric is symplectic, i.e. closed and nondegenerate, then the metric is called Kähler. Kähler structures are much more difficult to come by and are much more rigid. In mathematics, in Riemannian geometry, the metric tensor is a tensor of rank 2 that is used to measure distance and angle in a space. ... A hermitian metric on a complex vector bundle E over a smooth manifold M, is a positive-definite, hermitian inner product on each fiber Ep, that varies smoothly with the point p in M. An important special case is that of a hermitian metric on the complexified tangent bundle of...

Examples of Kähler manifolds include smooth projective varieties and more generally any complex submanifold of a Kähler manifold. The Hopf manifolds are examples of complex manifolds that are not Kähler. To construct one, take a complex vector space minus the origin and consider the action of the group of integers on this space by multiplication by exp(n). The quotient is a complex manifold whose first Betti number is one, so by the Hodge theorem, it cannot be Kähler. This article is about algebraic varieties. ... In algebraic topology, the Betti number of a topological space is, in intuitive terms, a way of counting the maximum number of cuts that can be made without dividing the space into two pieces. ... For the Hodge theorem in mathematics, see Hodge theory Hodge index theorem This is a disambiguation pageâ€”a list of articles associated with the same title. ...

A Calabi-Yau manifold is a compact Ricci-flat Kähler manifold or equivalently one whose first Chern class vanishes. Calabi-Yau manifold (an artists impression) In mathematics, a Calabi-Yau manifold is a compact KÃ¤hler manifold with a vanishing first Chern class. ... This article needs to be cleaned up to conform to a higher standard of quality. ... Results from FactBites:

 PlanetMath: complex analytic manifold (221 words) Examples of complex analytic manifolds are for example the Stein manifolds or the Riemann surfaces. Complex analytic manifolds can also be considered as a special case of CR manifolds where the CR dimension is maximal. This is version 2 of complex analytic manifold, born on 2005-02-22, modified 2005-03-05.
 Lie group (1429 words) In mathematics, a Lie group (pronounced "lee", named after Sophus Lie) is an analytic real or complex manifold that is also a group such that the group operations multiplication and inversion are analytic maps. Lie groups are important in mathematical analysis, physics and geometry because they serve to describe the symmetry of analytical structures. The set of all vector fields on an analytic manifold is a Lie algebra over F.
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