Minkowski space is not curved.

Contrary to a common misunderstanding, Minkowski space refers to a four dimensional flat space, the one consistent with the special theory of relativity.

Space, that is length, breadth, and depth are Euclidean dimensions, which means when we rotate something from one 3D direction to another we use sines and cosines in our formulae. When we add the dimension of time, we add it into our "metric" with a factor of i (the square root of -1). This means that in addition to rotations in 3 dimensions, we can talk about 4D rotations, which encompasses the usual rotations plus "Lorentz boosts". The latter is what leads, through special relativity, to phenomena such as length contraction and time dilation. When a rotation is made in all 4 dimensions, the Minkowski metric means that some of the sines and cosines become hyperbolic: sinh and cosh.

To summarize in mathematical terms: when we measure the norm (length) of a position vector in 3D: it is given by

sqrt( x2 + y2 + z2 )

If our 4 dimensions were Euclidean, a postion vector's norm would be

sqrt( x2 + y2 + z2 + (ct)2 )

However, special relativity says that our 4 dimensions are Minkowskian, so a position vector's norm is

sqrt( x2 + y2 + z2 - (ct)2 )

That minus sign makes a big difference! But this is still a flat space. I.e. the metric (and so the formula for the norm) is the same, independent of where you are. In curved space, there are position-depedent coefficients in the formula for the norm.