Will curvature of space time break a spaceship in half , if its top half is in a curved region and bottom half is not? My question is simple. If we have a spaceship, whose top half is in a region of highly curved space, while the bottom half is in a region of flat space, will the top half bend in a way, which will break the ship in two?
Or will it not break, because the ship is not bending, but rather space itself is bending?
I am not concerned about how realistic it is to have such a sharp difference in curvature between two nearby regions of space. My question is IF there was such a situation, then what would happen?
 A: The ship will break due to tidal forces. If the gravity strength is high enough then the ship will break in two where the ship crosses the surface of gravity-no-gravity. It can be of course that the difference in force (tidal force) between just inside and just outside the surface to make the ship break is already present somewhere inside the surface. In that case, the ship will break in the gravity region.
Note though that the kind of gravity field you envision isn't present in reality.
A: Here is what happens when there is a sharp shift in the curvature of spacetime.
First of all, those big shifts occur in the immediate vicinity of a very dense object like a neutron star or a small black hole. In either case, the curvature grows more severe the closer you (in your spacesuit) get to the object, and the difference in curvature between the end of you closest to the object- your feet, say- and the part of you furthest away (your head) is so great that your body gets pulled apart into a thin string and you die in a process called spaghettification.
So if your spaceship made a close pass to such an object, it wouldn't get broken in half- it (and its contents) would get squished into spaceship spaghetti.
Note that there isn't any process by which a portion of severely curved space can exist right next to completely flat space; the curvature difference I describe above may be extremely steep but nonetheless it is smooth rather than discontinuous.
A: We have experimental evidence of what happens to matter when the curvature of space is high. Matter "melts".
So depending how high the curvature is  , it will melt. I do not think nature creates  curvatures of space time that are discontinuous in the way that a spaceship could be half in half out.
A: 
highly curved space

In simple terms this means you have a intense gravitational field.

flat space

In equally simple terms this means very little gravitational field.
There will be a large stress between the two parts of the object if one end has a low field and one a high one.  Exactly what will happen will depend on the details of those fields and the materials and structures involved.  The object could compress, suffer some form of structural failure or it could survive.  There's no way way to know exactly what would happen in vaguely defined circumstances.

Or will it not break, because ship is not bending, but rather the space itself is bending ?

The "bending of space" is best thought of (in this context) as a stress on the materials.  We detect such stresses in the LIGO detectors which measure the changing stresses caused by gravitational waves.  The stresses in spacetime cause real physical changes.
