In general relativity, four-dimensional spacetime is considered and curvature is calculated for spacetime, not only space alone. However, looking deeper into the equations, many sources of symmetry can be found regarding space and time:
time runs slower where there is more mass / near mass: Where there is time dilation, there is length contraction.
in the general spherically symmetric metric and in mass-free space (exterior Schwarzschild-metric), $A=1/B$ can be derived.
You can't have curvature in 1 dimension: it is a line and always locally flat, with $R^a{}_{bcd} = R^0{}_{000} =0$. So it doesn't make sense to ask about 'time being curved'.
If you want to consider spacetimes where the time-coordinate has a privileged position (as is often done in calculations in GR), take a look at the 3+1 ADM forumulation of GR. Here, the 4D spacetime is split into 3D spacelike hypersurfaces at a given constant $t$. This is especially useful for initial value problems, and considering the evolution of 'spatial configurations' in 'time'.
Are there cases where time is curved independent from space?
To my knowledge not. Einstein field equations (EFE) are just about this dependency. In case of Schwarzschild exterior solution the metric components $g_{00}$ and $g_{rr}$ describing time dilation and space contraction fulfill indeed the relation $g_{00}=g_{rr}^{-1}$ but in other cases their relation is quite different. For example, in Schwarzschild interior solution $g_{00}=\frac{3}{2}\sqrt{1-\alpha}-\frac{1}{2}\sqrt{1-\alpha~(r/R)^2}$ and $g_{rr}^{-1}=\sqrt{1-\alpha~(r/R)^2}$. The corresponding relation reads then $g_{00}=\frac{3}{2}\sqrt{1-\alpha}-\frac{1}{2}g_{rr}^{-1}$.
Generally, in case of spacetime of static spherically symmetric perfect fluid sphere the relation between that two metric components is determined by the linear differential equation (see https://physics.stackexchange.com/a/679431/281096 ), which is for a given $g_{rr}^{-1}$ second order on $\sqrt{g_{00}}$ and for a given $\sqrt{g_{00}}$ first order on $g_{rr}^{-1}$. Thus, the resulting relation between time dilation and space contraction can be quite complex.
