What is a smooth manifold anyway?

Manifolds, both smooth and otherwise, are important topological constructs. This post aims to give a sense of understanding for smooth manifolds.

 

Intuitively, a smooth manifold is a space that locally looks like some Euclidean space. Thus we can carry out all the usual nice mathematical things we look to do, find limits of sequences, do calculus, etc, etc. So smooth manifolds seem like a nice generalization of Euclidean space, different terrain, same ideas.

 

Now to make this definition of a smooth manifold we need to define a few terms.

Firstly, it’s probably good to know what smooth means! A smooth function F is a function of an open set U \subset R^{N} to R^{M} and which is an element of C^{\infty}, that is F has continuos partial derivatives of all orders. For example: F (x) = e^{x} is smooth, as well as F(x) = x, the identity function. An example of a function which is not smooth would be F = |x|, the absolute value function. Note that \frac{d}{dx}|x| does not exist at 0.

 

You might be wondering, “well that definition of smooth seems pretty restricted, only functions of open sets of R^{n} to R^{m} can be smooth?” Good point. We can extend the notion of smoothness to arbitrary maps f : X \to R^{m}, for any arbitrary subset X \subset R^{n}. Such a function is smooth if f can be locally extended to a smooth function, in the above sense, F : U \to R^{m} . More precisely, this means that for every x \in X we can find a neighborhood U \subset R^{n} and F : U \to R^{m} such that F|_{U \cap X} = f|_{U \cap X}. Note that this is purely a local quality.

 

With the notion of smoothness, we only need one more idea to define a smooth manifold. As most fields of mathematics do, differential topology has a notion of equivalence. We say that subsets of Euclidean space X, Y are diffeomorphic if there exists a diffeomorphism between them. So…what’s a diffeomorphism?

 

A diffeomorphism is, in a sense, an extension of the notion of a homeomorphism. A homeomorphism is simply a continuous function between topological spaces that has a continuos inverse. To make a diffeomorphism out of a homeomorphism, we simply tag on the extra condition that the function and its inverse both be smooth. Just as homeomorphisms are sometimes referred to as bi-continous functions, we could refer to a diffeomorphism as a bi-smooth function (thought that doesn’t sound quite as nice).

 

Therefore we can say the subsets X, Y are essentially equivalent if there exists a diffeomorphism between them. Intuitively, this means that X can be smoothly deformed into Y, so we can stretch, squish, and bend X into Y as long as we don’t tear any new holes (or remove any) in X. For example, a circle and an ellipse are diffeomorphic, because we can deform a circle into an ellipse just by smooshing it a bit. However a circle is not diffeomorphic to triangle, as we would have to pinch the circle at the vertices, creating sharp (non-differentiable) points.

 

Diffeomorphism

 

Now that have all the required tool set, we can finally define a smooth manifold. A k-dimensional smooth manifold is a subset X \subset R^{N} which is locally diffeomorphic to R^{k}. That is, about each point in X we can find a neighborhood which maps smoothly onto an open set in the Euclidean space. Such a map \phi : U_{x} \to R^{k} where U_{x} is a neighborhood about x is a called a parameterization.

 

So a smooth manifold is just a subset of some Euclidean space where we can smooth out the wrinkles so that it looks like the Euclidean space. That’s it for this post, the next will focus on some interesting properties of smooth manifolds.

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