The gravity that we (as slow moving objects) experience here one the Earth (far away from any black holes) is almost entirely due to "curvature of time", i.e. the time dilation field around the Earth. The easiest way to see this is to assume a spherical non-rotating Earth, so we can use the Schwarzschild metric:
ds2 = -(1 - 2GM/r)c2dt2 + (1 - 2GM/r)-1dr2 + r2dΩ2
In the weak-field limit where GM/r is small (i.e. far from any black holes, r >> rs = 2GM/c2) this is approximately:
ds2 = -c2dt2 + (1 - 2GM/r)-1dr2 + r2dΩ2 + (2GM/r)c2dt2 + (2GM/r)dr2
The 2nd and 3rd terms together are just dx2 + dy2 + dz2 (in polar coordinates); they give the length of flat Euclidean space. The first 3 terms together give flat Minkowski spacetime, which has no gravity. For particles that are slow (v = dr/dt << c), dr << cdt so we can ignore the last term which corresponds to the space curvature, and get simply:
ds2 = -c2dt2 + (1 - 2GM/r)-1dr2 + r2dΩ2 + (2GM/r)c2dt2 = flat_spacetime + time_dilation
This tells us that space is extremely flat near the Earth, and almost ALL the gravity is coming from the time term, which merely describes the time dilation field. In fact the geodesics of this metric give Newtonian gravity; note that GM/r = Φ(r) = the Newtonian gravitational potential. "The reason that your ass is being pressed into your seat is that time is moving faster at your head than at your feet." - Landman's Mantra
(As a curious aside, chapter 12 of Misner-Thorne-Wheeler Gravitation states that it is IMPOSSIBLE to have a metric for Newtonian gravity, but their proof makes assumptions that don't apply to this metric, so their conclusion is false despite the fact that the proof is perfectly valid. See their Exercise 12.10 for more details.)
Swansont Bird11dog asks "do we have proof that matter warps space?" Yes we do. The bending of light by the Sun is twice as great as predicted by Newtonian gravity. The time dilation term gives the Newtonian prediction, and the other half comes from the space term which we ignored for slow-moving objects. Light is not slow-moving, so we can't ignore that term for light. To light, time and space appear about equally curved by the Sun. GR also predicts that space will be severely warped near a black hole event horizon, but we may need to wait a few years before we can test that directly. In the mean time, gravitational waves might tell us something.
Swansont Bird11dog also asks how time can bend light. The effect is similar to a graded index of refraction (GRIN) lens. If time is flowing faster at the top than at the bottom of a wave packet moving right, then then the speed of light is different, and a wave packet that starts out like ||||||||||| will eventually start to look like /////////// and the packet will be traveling slightly downward.
Machapungo complains that "[nothing] has ever shown that time is a physical reality", but certainly many experiments have shown that time dilation is a physical reality, so I'm not sure what his point is. It's kind of like quantum phase; absolute phase may or may not be real, but phase frequency = energy and certainly is real. Similarly, absolute time is probably meaningless since you can set T0 to be anywhere you want; but the passage of time is demonstrably real and measurable.