I often hear that spacetime corresponds to a hyperbolic geometry. The distance metric is not Euclidean, it is hyperbolic, and Lorentz transformations correspond to hyperbolic rotations. Hyperbolic geometry has a negative Gaussian curvature. At the same time, I also hear that Minkowski is flat and has vanishing Riemann tensor. How do I reconcile these two apparently contradictory statements about Minkowski curvature ?
I'm not sure where you got that impression, as it's simply not right.
I'm guessing you read a 3D hyperbola embedded in 4D Minkowski space has constant negative curvature. That's true, but it doesn't mean Minkowski space itself has curvature. Similarly, you can embed a 3D sphere in a 4D Euclidean space, but that doesn't mean the Euclidean space has curvature.