This is an extended comment on Valter's answer, so please upvote his answer not this one.
In Relativity (General and Special) there is no unique way to divide spacetime into space and time. Different observers, using different coordinate systems, will disagree about whether a four vector is just a displacement in time or just a displacement in space. So to this extent your question can't be answered because it's an artificial distinction.
A related question would be whether we can choose a coordinate system in which the curvature is just in time, or just in space. I asked a question along these lines in What makes a coordinate curved?. The answers are possibly a bit too deep for this discussion, but they boil down to that's a silly question :-)
But I'd like to pick up on Valter's last point because it's an interesting one. We have a tendancy to set $c = 1$ when writing down the metric, so we get nice symmetric looking equations like:
$$ ds^2 = -dt^2 + dx^2 + dy^2 + dz^2 $$
But written in the units we use for everyday observations the metric is really:
$$ ds^2 = -c^2dt^2 + dx^2 + dy^2 + dz^2 $$
where $c \approx 3 \times 10^8$m/s. So a displacement in time of 1 second contributes a factor of $3 \times 10^8$ more to the line element than does a displacement of 1 metre. What this means is that when considering weak gravitational fields, like the Earth's gravity, we get a good approximation by just ignoring the spatial curvature and only considering the curvature in the time coordinate.
This makes sense because experience tells us that thrown objects move in curves (parabolae) but also that space isn't obviously curved - if I draw a circle and measure the ratio of its circumference to its radius I always get the flat space value of $2\pi$. Time isn't obviously curved either because the curvature is small (though atomic clocks can measure it) but once you multiply by $3 \times 10^8$ the effect is big enough to make objects move in parabolae.
One day I will write a canonical Q/A explaining why objects accelerate towards the centre of the Earth. I have actually started this a couple of times, but finding a way of describing the physics that is universally accessible has proved challenging so far.