> In Riemannian manifolds, we take ℝ^n as the building block We take ℝ^n as the building block for any smooth manifold, not just Riemannian ones. Note that the

> In Riemannian manifolds, we take ℝ^n as the building block We take ℝ^n as the building block for any smooth manifold, not just Riemannian ones. Note that there is a difference between ℝ^n and Euclidean space. ℝ^n is a smooth manifold without a metric structure, so it is not a Riemannian manifold automatically. In contrast, Euclidean space is ℝ^n with a metric structure, namely the one given by the standard Euclidean metric δ. So Euclidean space is a Riemannian manifold and it is often denoted by the tuple (ℝ^n, δ). Finally, if we denote by η the (n-1, 1) Minkowski metric on ℝ^n, then (ℝ^n, η) is just Minkowski space (in particular, it is a Lorentzian / pseudo-Riemannian manifold). > Just as flatness in the Riemannian case means "globally like ℝ^n" This is not correct. There are many flat Riemannian manifolds that are not globally isometric (or even diffeomorphic) to Euclidean space (ℝ^n, δ). Take e.g. the torus T^n = S¹ × … × S¹ of any dimension. > On the other hand, one can take Minkowski space as a given, and then build spaces where we have space that is locally hyperbolic Again, I think you're confusing the terms "Lorentzian" and "hyperbolic" here. As I said (and you then yourself said), a hyperbolic manifold is a Riemannian manifold, not a Lorentzian one. A Lorentzian manifold is locally Lorentzian, not locally hyperbolic. You might want to read pa7x1's excellent explanation again. Also note that the quotes you gave from Wikipedia don't contradict what I (or pa7x1) said in the slightest. > As for _why_ we do this, my understanding is that we want to generalize SR(special relativity). For SR, we use a single space of signature (3,1). For GR, we want to 'put together' many such spaces, which are then distorted by gravity (this is curvature of space-time). Hence, we take the local space to be (3, 1), and the global space connects versions of (3, 1). This is certainly one of way of thinking about it, yes: Locally, we see Minkowski space but globally the geometry (and topology) might be more complicated and this is what GR allows us to implement through a general Lorentzian manifold.