Manifolds: Topological, Smooth, Complex, Kähler (Work in Progresss)
In this post, we aspire to define some main classes of manifolds, as well as to give an intuitive understanding of how they are related.
Topological manifolds
Manifolds are spaces that "locally" look like
An-dimensional topological manifold or just -manifold is a second countable Hausdorff topological space, such that each point has an open neighbourhood which is homeomorphic to .
Moreover, one can add structures to a manifold:
An atlas on an-manifold is a set such that each is open and they form a covering , and is a homeomorphism.
Using the unifying theme of atlases, one can then add structure to the maps
Smooth manifolds
Topological manifolds are good generalisations of the euclidean spaces
To circumvent this, we can restrict our attention to functions whose derivatives always exist, namely the smooth functions:
-
A function
is smooth or if it is continuous and all the derivatives exist for . -
Generalising,
is smooth if all the partial derivatives , , exist. -
Finally,
is smooth if all the coordinate functions are smooth.
Using these new types of functions, we can create manifolds with a lot more structure.
A-structure on an -manifold is an atlas of such that is a function, i.e. a smooth function, for all . A manifold with a -structure is called a or smooth manifold.
Are all topological manifolds also smooth manifolds? No, but the smallest example of a topological manifold
which does not have a smooth structure is in dimension
Complex Manifolds
A complex structure on a
A
Given a point
Now we want to talk about tangents on our manifold. There are three possible tangent space we could consider:
-
The real tangent space
is all the real directions along our manifold, imagine a real hyperplane lying tangent to the space. -
The complex tangent space
is all the complex directions along our manifold, imagine a complex hyperplane lying tangent to the space. -
The holomorphic tangent space
is all the holomorphic directions along our manifold, so that the differentials vanish on antiholomorphic functions.