Let M be a smooth manifold.
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A manifold equipped with an almost complex structure is called an almost complex manifold. If M admits an almost complex structure, it must be even-dimensional. This can be seen as follows. But if M is a real manifold, then det J is a real number — thus n must be even if M has an almost complex structure. One can show that it must be orientable as well.
An easy exercise in linear algebra shows that any even dimensional vector space admits a linear complex structure. Only when this local tensor can be patched together to be defined globally does the pointwise linear complex structure yield an almost complex structure, which is then uniquely determined. The possibility of this patching, and therefore existence of an almost complex structure on a manifold M is equivalent to a reduction of the structure group of the tangent bundle from GL 2 n , R to GL n , C.
Almost complex manifold
The existence question is then a purely algebraic topological one and is fairly well understood. For every integer n, the flat space R 2 n admits an almost complex structure.
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In particular, S 4 cannot be given an almost complex structure Ehresmann and Hopf. In the case of S 2 , the almost complex structure comes from an honest complex structure on the Riemann sphere.
The 6-sphere, S 6 , when considered as the set of unit norm imaginary octonions , inherits an almost complex structure from the octonion multiplication; the question of whether it has a complex structure is known as the Hopf problem, after Heinz Hopf. Just as we build differential forms out of exterior powers of the cotangent bundle , we can build exterior powers of the complexified cotangent bundle which is canonically isomorphic to the bundle of dual spaces of the complexified tangent bundle.
The almost complex structure induces the decomposition of each space of r -forms. Thus we may use the almost complex structure to refine the action of the exterior derivative to the forms of definite type. These operators are called the Dolbeault operators. Since the sum of all the projections must be the identity map , we note that the exterior derivative can be written.
Every complex manifold is itself an almost complex manifold. One easily checks that this map defines an almost complex structure.
Thus any complex structure on a manifold yields an almost complex structure, which is said to be 'induced' by the complex structure, and the complex structure is said to be 'compatible with' the almost complex structure. The converse question, whether the almost complex structure implies the existence of a complex structure is much less trivial, and not true in general.
On an arbitrary almost complex manifold one can always find coordinates for which the almost complex structure takes the above canonical form at any given point p.
In general, however, it is not possible to find coordinates so that J takes the canonical form on an entire neighborhood of p. Such coordinates, if they exist, are called 'local holomorphic coordinates for J'. If M admits local holomorphic coordinates for J around every point then these patch together to form a holomorphic atlas for M giving it a complex structure, which moreover induces J.
J is then said to be ' integrable '. If J is induced by a complex structure, then it is induced by a unique complex structure. Narasimhan Hardback, Be the first to write a review About this product. All listings for this product Buy it now Buy it now. Any condition Any condition.
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Sard's Theorem and Functional Dependence. Borel's Theorem on Taylor Series. Whitney's Approximation Theorem. An Approximation Theorem for Holomorphic Functions. Ordinary Differential Equations. Basic Definitions.
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The Tangent and Cotangent Bundles. Grassmann Manifolds. Vector Fields and Differential Forms. Exterior Differentiation. Manifolds with Boundary. One Parameter Groups. The Frobenius Theorem. Almost Complex Manifolds. The Lemmata of Poincare and Grothendieck. Immersions and Imbeddings: Whitney's Theorems. Thom's Transversality Theorem. Linear Elliptic Differential Operators. Vector Bundles. Fourier Transforms.