Consider the Minkowski space time
\begin{equation} ds^2= -dt^2 + dx^2 + dy^2 + dz^2$ \end{equation}
Can we such a coordinate transformation on just the spatial part of the above metric (and not on the time part) such that the metric changes to something like
\begin{equation} ds^2= -dt^2 + “ curved { } { } spatial { } { } part” \end{equation}
such that the original flat (Riemann Tensor vanishing indentically) 4d Manifold have a curved( non vanishing Riemann Tensor) 3d spatial submanifold.
Is such a case possible. Can a flat (Riemann Tensor vanishing indentically) manifold have a curved ( non vanishing Riemann Tensor) submanifold?
Can a flat 4d manifold be foliated with curved ( non vanishing Riemann Tensor) hypersurfaces ?
Curvature is independent of coordinate system. If a metric is flat, it is flat. (For example, in $\Bbb R^2$, we have the standard metric $ds^2=dx^2+dy^2$; when we switch to polar coordinates, we have $ds^2=dr^2+r^2\,d\theta^2$. Christoffel symbols may be nonzero, but the curvature is still $0$.)
Of course flat spaces have submanifolds of all sorts of curvature. Just take any surfaces you wish sitting in $\Bbb R^3$; the same holds in all dimensions. It all then depends on the second fundamental form, which is not intrinsic to the submanifold. And you can certainly foliate flat space by non-flat hypersurfaces, yes.