Riemann manifolds in relation to other classes of differentiable manifolds

Question

I am trying to get an overview over the different categories of manifolds. In particular i have the following chain of inclusions: Riemann surfaces $\subset$ complex manifolds $\subset$ orientable manifolds $\subset$ conformal manifolds $\subset$ differentiable manifolds

I hope this is correct, in case there is an error please hint me in the direction of an instructive counter-example. My confusion now is the following: Where do Riemann manifolds fit into this picture? Despite the name, Riemann surfaces are defined as complex manifolds (of real dimension 2) and therefore are not necessarily Riemann manifolds. Now, every differentiable manifold (paracompact) can be equipped with a Riemann metric. What does it then mean, when i read somewhere that a manifold is not a Riemann manifold? For example: the Riemann sphere $\widehat C$ is said not to be a Riemann manifold in the article http://en.wikipedia.org/wiki/Riemann_sphere. As i understand it the reason is that every comlex manifold is orientable while the Riemann sphere $\widehat C = \mathbb R \mathbb P^2$ is not. But it is also said that it can be given a metric g that is isometric to the sphere $S^2 \subset \mathbb R^3$. But the spaces are not homeomorphic, so i guess what is meant here is 'local isometry', i.e. map that pulls back the metric and not 'isometry'(diffeomorphism which is a local isometry), or am i wrong? Now does it mean to say a differentiable manifold is a Riemann manifold iff it admits up to isometry only one metric?

Answer 1

Some of the inclusions in your chain are a little misleading. For example, you write "conformal manifolds $\subset$ differentiable manifolds." It is true that every conformal manifold is a differentiable manifold, because that's part of the definition of conformal manifolds. But it's not true that the class of all conformal manifolds is a "subclass" of the class of differentiable manifolds. In fact, every differentiable manifold can be given many different conformal structures. A conformal manifold is actually a pair: a differentiable manifold together with a particular choice of conformal structure. To say that a differentiable manifold is "not a conformal manifold" is just to say that no specific conformal structure has been chosen.

Something similar but more complicated applies to "complex manifolds $\subset$ orientable manifolds." Again, it is true that every complex manifold is orientable; but in this case, not every orientable manifold can be given a complex structure, and some that do have complex structures have many of them.

A better way to think about these relationships is by using the language of category theory. There is a functor from the category of conformal manifolds to the category of differentiable manifolds, which simply takes each conformal manifold to its underlying differentiable manifold. Such a functor is called a forgetful functor, because it "forgets" some of the additional structure of a conformal manifold. Each of the "inclusions" in your chain except the first can be interpreted as a forgetful functor. In some cases, they are surjective on objects, but none of them are injective on objects.

Now to Riemannian manifolds, which are differentiable manifolds with the extra structure of a Riemannian metric. There are forgetful functors from the category of Riemannian manifolds to those of conformal manifolds and differentiable manifolds, both of which are surjective on objects. But these functors don't fit neatly into your chain, because there's no forgetful functor from the category of complex manifolds to the category of Riemannian manifolds.

The first inclusion, "Riemann surfaces $\subset$ complex manifolds," is a little different. It really is an inclusion, because the category of Riemann surfaces is a full subcategory of the category of complex manifolds.

Answer 2

The Riemann sphere is diffeomorphic to $S^2$, it is definitely not diffeomorphic to $\mathbb{R}P^2$. It is constructed via stereographic projection, but is not the projective plane.

The Riemann sphere is also definitely orientable. When they say that a Riemann surface "isn't a Riemannian manifold" they just mean that there isn't an obvious metric arising from the definition. This shouldn't be read to mean that there can be no Riemannian metric structure on it, nor even that there aren't ones that play well with the structure.