What is a conformal structure?

In this post I want to introduce the notion of a conformal structure  on a smooth manifold, as this would be convenient to have for references later. A manifold endowed with a conformal structure will be termed as a conformal manifold.

I suppose that we work with a smooth (C^{\infty}-differentiable) manifold M which we also choose to be closed (i.e. without boundary). These restriction are applied in order to avoid dealing with some technical peculiarities. Later on I will refine the discussion to the case of manifolds with boundaries. It is also appropriate to remind here that manifolds are understood as Hausdorff second-countable topological spaces that are locally homeomorphic to Euclidean space \mathbb{R}^n. Here n is the dimension of the manifold: \dim{M}=n. I would recommend the book of Loring Tu “An Introduction to Manifolds” if the idea of the manifold does not sound very attractive to you.

Moreover, our manifolds are smooth, in other words, they carry a smooth structure. Well, there are more structures coming, and it would be a good idea to discuss what a structure is, but I would do this at some other time.

A smooth structure gives rise to the tangent bundle on the manifold, and from here we get the whole world of different things such as vector fields, tensors, differential forms etc. These all are sections of some vector bundles.

An inner product smoothly defined in the tangent space at each point of manifold M can be expressed as a section g \in \bigodot^2 T^*M of the symmetrized tensor square of the cotangent bundle of manifold M. This is what we call a Riemannian metric g in M.

How do we know that such a thing exists? This is the whole point about the definition of the manifold, more precisely, about the requirements of Hausdorficity and second-countability. The latter ensure that we have partitions of unity, so in fact one can prove that in any vector bundle there is a fiber metric (use the fact that vector bundles are locally trivial and glue up the obvious choices of metrics using the partition of unity subordinated to the trivializing atlas).

In fact. we can see that there is a plenty of, say, Riemannian metrics in a manifold. If the manifold is given abstractly (not as a submanifold of something standard) then there is no particular reason to prefer one metric with regards to another. This suggests that some metrics are in a sense better that others. This can be seen in the light of the Nash embedding theorem that we will discuss at some stage later.

As one would expect, a manifold M endowed with a choice of a Riemannian metric g in it is called a Riemannian manifold (M,\,g).

This g is also called a Riemannian structure. It defines, besides all, the lengths of tangent vectors \| u \| = \sqrt{g(u,\,u)} and the angles between them via

\cos (u,v) = \frac{g(u, \, v)}{\| u \| \cdot \|v\|}.

A conformal structure disregards the lengths and only respects the angles between the tangent vectors. Equivalently, one can consider two metrics g , \, \hat{g} on a manifold M such that the define the same angles by the above formula, and show that in fact these two metrics are related to each other as \hat{g} = f g where f \in C^{\infty} (M, \mathbb{R}) is some strictly positive smooth function. We will stick to the convention to write \Omega ^2 instead of f.  Hence we arrive to our first important

Definition 1. Metrics g , \, \hat{g} are called conformally equivalent if there is a smooth function \Omega such that \hat{g} = \Omega ^2 g.

It is straightforward to check that we have an equivalence relation in the space of all metrics in M.  The class of equivalence [g] of a metric g is called a conformal structure on manifold M.

Definition 2. A conformal manifold is a pair (M, \, [g]) where M is a smooth manifold, and [g] is a conformal structure on M.

More about motivation and applications of conformal structure is planned to be discussed in the future posts. I will be glad to see comments and questions related to the topic here.