Ivancevic_Applied-Diff-Geom

430 Applied Differential Geometry: A Modern Introductionholomorphic. It is not hard to show that this necessarily implies that φ −1is holomorphic as well and hence φ is known as a biholomorphism. Such amap allows us to identify the complex structures on M and N and hencethey are isomorphic as complex manifolds.These definitions are important because there are pairs of smooth manifoldsM and N which are homeomorphic but not diffeomorphic, as well as,there are complex manifolds M and N which are diffeomorphic but not biholomorphic.This means that if one simply ignored the fact that M and Nadmit local complex coordinates (with holomorphic transition functions),and one only worked in real coordinates, there would be no distinction betweenM and N. The difference between them only arises from the way inwhich complex coordinates have been laid down upon them.Again, recall that a tangent space to a manifold M at a point p isthe closest flat approximation to M at that point. A convenient basis forthe tangent space of M at p consists of the n linearly independent partialderivatives,T p M : {∂ x 1| p , ..., ∂ x n| p }. (3.237)A vector v ∈ T p M can then be expressed as v = v α ∂ x α| p .Also, a convenient basis for the dual, cotangent space T ∗ p M, is the basisof one–forms, which is dual to (3.237) and usually denoted byT ∗ p M : {dx 1 | p , ..., dx n | p }, (3.238)where, by definition, dx i : T p M → R is a linear map with dx i p(∂ x j | p ) = δ i j.Now, if M is a complex manifold of complex dimension d = n/2, there isa notion of the complexified tangent space of M, denoted by T p M C , whichis the same as the real tangent space T p M except that we allow complexcoefficients to be used in the vector space manipulations. This is oftendenoted by writing T p M C = T p M ⊗ C. We can still take our basis tobe as in (3.237) with an arbitrary vector v ∈ T p M C being expressed asv = v α ∂∂x| α p , where the v α can now be complex numbers. In fact, it isconvenient to rearrange the basis vectors in (3.237) to more directly reflectthe underlying complex structure. Specifically, we take the following linearcombinations of basis vectors in (3.237) to be our new basis vectors:T p M C : {(∂ x 1 + i∂ x d+1)| p , ..., (3.239)(∂ x d + i∂ x 2D)| p , (∂ x 1 − i∂ x d+1)| p , ..., (∂ x d − i∂ x 2D)| p }.