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?