It is well known that gravity does distort space. However, it takes a lot of matter, and an awful lot of energy (the amount of energy locked into that matter according to E=mc2) to distort spacetime significantly. This is the real reason for Miguel Alcubierre's pessimistic calculations of how much negative energy a warp drive would require. There is hope though. One theory about why gravity is the weakest force is that gravitons are not confined to normal spacetime, but can dissipate into higher dimensions. The theory is supported by the fact that although gravity normally travels at the speed of light, it can and does escape black holes, meaning it must pass outside normal spacetime.A technique to force gravitons (and antigravitons) to stay in normal spacetime would cut energy requirements for manipulating spacetime in different ways to realistic levels.
Uses
Weakly manipulated spacetimes
A relatively weak manipulation of spacetime can be used for better launches that is much cheaper than traditional chemical rocket launches. It can also be used for creating artificial gravity in space colonies with very weak gravity, such as hollow asteroid/comet colonies or to create lower gravity in colonies on high-gravity worlds such as Jupiter or as a method of BTC travel.
Strongly manipulated spacetimes
Although nothing can move faster than the speed of light through spacetime, there is no relativistic constraints on spacetime itself. Strong manipulation of spacetime can therefore be used to create traversable wormholes and Alcubierre-like warp drives. Another note about Alcubierre drive. If the spacetime manipulation is generated by multiple generators, where each generator individually creates a slower than light effect but the collected effect of all the generators is faster than light, that would solve the problem about quantum effects since there would be no fixed event horizon to generate Hawking radiation. Arranging the generators in a steep slope configuration so that he Meisner effect makes most of the warping miss the spacecraft itself would also solve the problem about how to avoid the destruction of the spacecraft itself by the distorsion. Using "anti-gravity" for better launches can also be considered strongly manipulated spacetimes if the launch is from a black hole. Strongly manipulated spacetimes can also be used for preventing the end of the Universe. Recent observations show that the expansion of the universe does not accelerate per volume unit, but that the expansion of the Universe accelerates because there is more and more space that can expand, strengthening the big chill theory and weakening the big rip theory.
Ideas and possibilities
The Meisner effect
Multiple scientific theories, including String Theory, independently predict that gravity and electromagnetism unify in higher dimensions. If an electromagnetic field can be made to leave normal spacetime, it would "trap" gravitons (and antigravitons) into normal spacetime. The Meisner effect is that the magnetic field of supraconductors repell all other magnetic fields, regardless of polarity. If the Meisner fields could be made to repel each other so strongly that much of the fields are forced out of normal spacetime, that would be a solution. Altough Podkletnovs experiment is usually dismissed as pseudoscience, ESA has actually repeated the results under controlled circumstances, multiple times, and also ruled out error sources (Towards a new test of general relativity), but it does conclude that it requires niobium supraconductors cooled by liquid helium (no, the full abstract, Experimental Detection of the Gravitomagnetic London Moment, states that lead works as well, which is much cheaper, altough it still requires liquid helium). Someone should build a device with several such supraconductors spinning very close to each other! Due to its experimentally supported status, this idea shall be rated high importance and be treated as such in the promotional material, including being submitted at www.100yss.org. Another way of making the effect even stronger is to actually conduct electricity through the supraconductors, and not rely on the weak induction the spinning supraconductors gets.
The Casimir effect and potential use for graphene
Multiple scientific theories, including String Theory, independently predict that gravity and electromagnetism unify in higher dimensions. If an electromagnetic field can be made to leave normal spacetime, it would "trap" gravitons (and antigravitons) into normal spacetime. Electromagnetic fields could be made to leave normal spacetime by being "consumed" by a vacuum energy deficiency. And that vacuum energy deficiency can be created by the Casimir effect. Graphene would be very useful for making the thin plates. The space colonization workforce should test if graphene can generate a Casimir effect! It is much stronger than gold leaf and even thinner, making it a much better candidate for generating practically useful amounts of Casimir effect. If you have done a experiment that either proves or disproves that graphene can generate a Casimir effect, please write the results here.
One possibility to create the necessary graphene "sandwich" would be to expose carbon in zero gravity to a interference pattern of electromagnetic waves with the same wavelength as the diameter of a carbon atom, which means x-rays of a somewhat less energetic wavelength than in an ordinary x-ray scan. The constructive and destructive interference would then, due to the weightlessness, organize the carbon atoms into graphene layers spaced the width of one carbon atom apart, a perfect Casimir sandwich.
Rotational effects
It is hypothesized that rapidly spinning objects can transfer their centrifugal effects to spacetime itself through frame dragging. That have been observed in black holes. It would be practically impossible to generate such heavy objects, but even not-so-heavy rapidly spinning objects can still tear spacetime if they are extremely close to each other.