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.