Manifold

Manifold and Coordinate Functions

A topological manifold M of dimension n is a topological space that is locally homeomorphic to Rⁿ. We may say that a manifold is an object that looks locally like a little piece of Rⁿ. An example would be a torus, the two dimensional surface of a "doughnut" in R³, which looks locally (but not globally) like R². This means that for each point m in M , there is a neighborhood of U of m and a one-to-one, continuous map Φ of U into Rⁿ onto some open set  Φ(U) in Rⁿ such that inverse map Φ^-1 : Φ(U) --> U is also continuous.

The map Φ can be considered as local coordinate functions x₁, ..., x_n, where each x_k is the continuous function from U into R given by x_k(m) = Φ(m)_k, be the k-th component of Φ(m). 

If Ψ is another homeomorphism of another neighborhood V of m, and y_k(m) = Ψ(m)_k is the associated coordinate system, then both coordinate systems are defined in the neighborhood U ∩ V of m. The "change of coordinates" map for m in U ∩ V is the map Ψ o Φ^-1 that maps the set Φ(U ∩ V) onto the set Ψ(U ∩ V), i.e.,

(y₁(m), ..., y_n(m)) = (Ψ o Φ^-1) (x₁(m), ..., x_n(m)). This is continuous since both Ψ and Φ^-1 are continuous.

Smooth Manifold

Smooth manifold is made by choosing a collection of local coordinate systems that cover the whole manifold. For any two coordinate systems that are defined in overlapping regions, the expression for one set of coordinates in terms of the other is always smooth. These coordinate systems were chosen in order to give a smooth structure to the topological manifold M . Some manifolds do not admit a smooth structure. When a smooth structure exists, it is not unique.  

Once a smooth structure is chosen, we define a smooth local coordinate system to be any local coordinate system (U, Φ) such that Φ o Φ_α^-1 is smooth for each (U_α, Φ_α). A function f : M --> R is called smooth if for each smooth coordinate system (U, Φ), the function f o Φ^-1 is a smooth function on the set Φ(U).

Complex Manifold

A complex manifold is a smooth manifold of dimension 2n such that the basic coordinate patches (U_α, Φ_α) have the property that the change-of-coordinates map Φ_β o Φ_α^-1 is holomorphic for each β and α. Here, R²ⁿ is identified with Cⁿ and holomorphic means the same as complex analytic.  

If V is a complex vector space, then a subset M of V is called an embedded complex submanifold of dimension k if, given any m₀ in M , there exists a holomorphic local coordinate system  (Φ, U) defined in the neighborhood U of m₀ such that for any m ∈ U, m is in U ∩ M if and only if Φ(m) is in C^k ⊂ Cⁿ.