  In the framework of twisted-diffeomorphism approach to noncommutative gravity
with canonical/Moyal-Weyl type noncommutative (NC) coordinate structure, we
show that the NC Minkowski spacetime parametrized either with spherical polar
coordinates or with parabolic coordinates has nontrivial NC corrections to
Riemann curvature tensor, Ricci tensor and curvature scalar. Apparently, there
are no such corrections if we choose rectilinear coordinates or even
cylindrical coordinates, for which the metric in the commutative-counterpart is
dependent on at most one coordinate only. We present both first order and
second order calculations. The emergent curvature corrections might seem to
raise the question of whether the statement of curvature is
coordinate-dependent, but note, for example, that the transformation from
spherical polar system to Cartesian system is not a diffeomorphism as such
since its non-injective nature makes it a local diffeomorphism. In other words,
if the flat-spacetime metric tensor in the commutative case depends on more
than one curvilinear coordinates, the introduction of noncommutativity among
these coordinates can possibly make the spacetime curved. It is worth remarking
that such a curvature emerges in the context of NC Minkowski spacetime in the
absence of any gauge or matter fields. It is purely an NC geometric effect.
