Differentiable structures-II

In the previous post we have defined a differentiable structure.The definition allows us to talk of differentiable functions i.e., we gave meaning to the term ” f is differentiable function on the manifold”. So basically, differentiable structure is the topological manifold together with sets of differentiable functions where the sets satisfy some properties. For instance a differentiable function on a open set has to be differentiable on any subset of this set. These properties hence formalise things like local nature of differentiability, fact that one can glue local informations to get the complete information about a function etc. Here we gave the information of differentiable function on the manifolds by gluing together the informations about the differentiable function locally on the open subsets in a compatible manner. So more generally one can glue together many other types of informations . The precise framework for all this is the theory of sheaves. To get a better appreciation of sheaves one better has to look at algebraic geometry.

Anyway lets now try to look at some examples of differentiable manifolds.

1. The euclidean spaces $\mathbb{R}^n$ are equipped with a standard differentiable structure. Any open set in $\mathbb{R}^n$ is a differentiable manifold.One just needs a single coordinate chart.
2. Any sphere $S^n$ can be given a differentiable structure. One needs two coordiante charts.
3. Quotient spaces of manifolds like $\mathbb{R}^n$ by discrete group actions can be given differentiable structures.
4. The real projective space $RP^n$ can be given a differentiable structure.

But here is a very non intuitive thing that can happen. A manifold can admit two very different differentiable structures!. Infact it is known that a 7 dimensional sphere admits 28 different differentiable structure. $\mathbb{R}^4$ admits uncountable many differentiable structures!. I don’t have an understanding of these facts. But I am planning to understand them and write about them some day. Our intuitive understanding that smoothness seems to be not very complete. There are manifolds which don’t admit differentiable structures. I don’t understand why one can’t smoothen out rough edges to remove singularities. One may say there may be fractal kind of roughness lying around. But I heard its not actually the case. Anyway keep thinking about this and let me know what it means if you get something.

Lets get back to the examples.

5. Lie groups can be given a differentiable structure. Theory of lie groups is very rich. I will try to write about it sometime later. The tangent space( we are going define it in a latter post) at the identity gives a lie algebra. This is very crucial in understand the lie group.
6. Just like quotients we can also get differentiable manifolds by usual constructions like the products.

Implicit function theorem. This theorem basically says that the level sets of the functions are also differentiable manifolds of lesser dimensions. Most differentiable manifolds used in practise come out of such a construction from some nice functions. For instance unit sphere $S^2$ is a the level set of the function $F(x,y,z)=x^2+y^2+z^2$ at the level 1. If $f:R^n+k \to R^k$ then level sets corresponding to nice values in $\mathbb{R}^k$ (we will later see that these values be regular values) are of dimension n. If k=1 the dimension of the level set is one less. So it says that the level constrains the manifold to a lesser dimension. That is in physicists terms the number of degrees of freedom goes down by one. So for k dimensions the degrees of freedom goes by k and the manifold has a dimension n.
Here if n=0 then we get nothing but the 0-dimensional spaces which are objects. But in this case we actually have a local homemorphism which is also a diffeomorphism.(Inverse function theorem)
In the next post we shall see what a tangent space and we will define derivative of a functions on a abstract differentiable manifold.