The "within our current understanding of physics" makes this hard. Our current understanding is that the curvature of spacetime is intricately linked to the distribution of matter and energy in spacetime.
Now there is the well known Alcubierre drive which allows for ftl travel (in a global frame, nothing travels faster than light locally). However the Alcubierre drive requires exotic material (with negative mass) and extreme consumption of energy (the power of several billion stars) to work. Our current understanding does not permit for a functioning Alcubierre drive to be built.
There are similar, but low tech ways of bending spacetime. For example, if you fling a planet out into space, it can drag a spaceship behind it gravitationally. That's probably not what you were thinking, yet it does involve bending spacetime to translate motion towards a centre of gravity into directional movement.
Essentially the problem is this, to bend spacetime you need to move large amounts of mass around, and if you can move large masses around, why don't you just move the spaceship? There is a very exception to this. If the large mass is already moving, you can steal a little of its momentum, and gain a lot of speed. This is called a gravitational slingshot, and is very much "real physics".
Taken to extreme is the possibility of gaining velocity by dipping into the Ergosphere of a rotating black hole. This could allow for an object to gain a very large amount of energy from the black hole (up to 20% of the object's mass-energy). Again, this isn't a drive that you can take with you, but a way of boosting your speed. Flying your spacecraft around a black hole has a number of health and safety issues.