Almost complex manifold

In mathematics, an almost complex manifold is a smooth manifold equipped with a smooth linear complex structure on each tangent space. The existence of this structure is a necessary, but not sufficient, condition for a manifold to be a complex manifold. That is, every complex manifold is an almost complex manifold, but not vice versa. Almost complex structures have important applications in symplectic geometry. In mathematics, an almost complex manifold is a smooth manifold equipped with a smooth linear complex structure on each tangent space. The existence of this structure is a necessary, but not sufficient, condition for a manifold to be a complex manifold. That is, every complex manifold is an almost complex manifold, but not vice versa. Almost complex structures have important applications in symplectic geometry. The concept is due to Charles Ehresmann and Heinz Hopf in the 1940s. Let M be a smooth manifold. An almost complex structure J on M is a linear complex structure (that is, a linear map which squares to −1) on each tangent space of the manifold, which varies smoothly on the manifold. In other words, we have a smooth tensor field J of degree (1, 1) such that J 2 = − 1 {displaystyle J^{2}=-1} when regarded as a vector bundle isomorphism J : T M → T M {displaystyle Jcolon TM o TM} on the tangent bundle. 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. Suppose M is n-dimensional, and let J : TM → TM be an almost complex structure. If J2 = −1 then (det J)2 = (−1)n. 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. Therefore, an even dimensional manifold always admits a (1, 1)-rank tensor pointwise (which is just a linear transformation on each tangent space) such that Jp2 = −1 at each point p. 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(2n, R) to GL(n, C). The existence question is then a purely algebraic topological one and is fairly well understood. For every integer n, the flat space R2n admits an almost complex structure. An example for such an almost complex structure is (1 ≤ i, j ≤ 2n): J i j = − δ i , j − 1 {displaystyle J_{ij}=-delta _{i,j-1}} for odd i, J i j = δ i , j + 1 {displaystyle J_{ij}=delta _{i,j+1}} for even i. The only spheres which admit almost complex structures are S2 and S6 (Borel & Serre (1953)). In particular, S4 cannot be given an almost complex structure (Ehresmann and Hopf). In the case of S2, the almost complex structure comes from an honest complex structure on the Riemann sphere. The 6-sphere, S6, 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 a complex structure on a vector space V allows a decomposition of VC into V+ and V− (the eigenspaces of J corresponding to +i and −i, respectively), so an almost complex structure on M allows a decomposition of the complexified tangent bundle TMC (which is the vector bundle of complexified tangent spaces at each point) into TM+ and TM−. A section of TM+ is called a vector field of type (1, 0), while a section of TM− is a vector field of type (0, 1). Thus J corresponds to multiplication by i on the (1, 0)-vector fields of the complexified tangent bundle, and multiplication by −i on the (0, 1)-vector fields. 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