The Riemann curvature tensor is a tensor field defined on a $4$-dimensional manifold. However, the curvature I was talking about is the Riemann curvature tensor defined by the induced metric on a $3$-dimensional sub-manifold. The sphere is the $3$-dimensional sub-manifold that is described by $t=const.$ in their chart, while the disc in the second observer's coordinate is described by $t^\prime=const$ where in general are different. So, the $3$-dimensional curvature need not be invariant under Poincare transformations in $4$ dimension.