# Why do we say Minkowski space is flat when hyperbolic geometry has negative Gaussian curvature?

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 ?

• It has a vanishing Riemann tensor. Commented Sep 28, 2018 at 23:48
• Minkowski does not have negative gaussian curvature. Where did you get this idea? Commented Sep 28, 2018 at 23:55
• thanks for the answer. i made the "negative curvature" statement because distances in special relavity are hyperbolic distances, and lorentz transformations are hyperbolic rotations. thus, special relavity seems to correspond to hyperbolic geometry, which has negative constant curvature. i'm obviously confusing things, since that hyperbolic geometry apparently has nothing to do with the geometry of space-time (or of minkowski spacetime) itself ? Commented Sep 30, 2018 at 4:06
• Hi marjimbel welcome to StackExchange and Physics SE in particular. I upvoted your question since it is a fair question that you made in good faith. I also did it as a welcome gift to compensate the actions of 2 "superior" members who down voted your very first question, for no particular reason. They clearly violated our code of conduct. In Italian people use to say - citing Dante's Divine Commedy - "Non ti curar di lor ma guarda e passa " (Let us not speak of them, but look, and pass) Commented Sep 30, 2018 at 8:27
• thank you @magma :) . i'm honestly confused about this. people make statements like "the geometry of Minkowski 4-space is hyperbolic" all the time. perhaps those statements are wrong (if so i'd like to understand why). you see them all the time even in other SE questions ( ie. math.stackexchange.com/questions/497504/… ) . i'm just trying to understand the two things Commented Sep 30, 2018 at 13:33