Is it possible to distinguish rest and movement in hyperbolic universe? Imagine a large body (for example, a planet) in 3D hyperbolic space. Now imagine the planet starts moving in a straight line at constant speed.
In Euclidean space, all points would move along parallel lines. In hyperbolic space, however, most points will be moving along equidistant curves, or hypercycles.
But points moving along a curve should experience some sort of force. If an observer is on the surface of the planet, he should be able to measure larger or smaller force depending on how close he is to the axis of movement.
This leads to conclusion that, unlike Euclidean space, hyperbolic space would allow an observer to determine whether the planet is in rest or whether it is moving without referring to any other body -- absolute rest/movement, in effect.
Is this line of reasoning correct?
 A: Contrary to my initial gut feeling, this sounds highly plausible. If you assume that any body which is not moving (with constant speed) along a geodesic experiences some force, then you can simply take four test bodies, forming a regular hyperbolic tetrahedron, and consider translating that through space. As you said, a hyperbolic translation only moves points on a specific geodesic along that same geodesic, while all other points are moved along curves equidistant to that geodesic. So at most two corners of the tetrahedron can remain without force, at least two others will take a curved trajectory and hence be subject to some force. So I'd say you are right.
Whether the force is sufficiently large to be measured in practice depends of course on the scale of things, so we still don't know whether the universe is flat or not.
A: To make it a discussion and hoping for insights:
NO it is not possible to distinguish rest and movement in hyperbolic universe ( I follow my gut feeling) 
(sorry this is a bit rambling but i found the question intruiging)
I would suggest the following two counter examples:
Counter example 1: "moving earth"
The earth is moving at quite a high speed (around the sun) but standing on the earth can i measure intrinsicly (thus not calculate with known positions of the sun and earth) in which direction i am moving?
Counter example 2: "spherical triangle"
the idea behind the question seems to be that somehow you can measure some kind of distortion (because points on the surface don't move in straight lines) 
the same would apply to a moving triangle on a sphere (on a sphere the geodesics are also not equidistant), still triangles can move around on spheres without distortion. so also no measurement of distortions can be made.
General remarks:
it all seems to be going back to what in the old days (1890-1910) was called "the axiom of free (undistorted) movement". I think in short it was that for free movement the containing space must have a constant curvature. 
so figures / objects don't change form/angles while moving.
a hyperbolic universe has an constant negative curvature.
an example of a structure that does not have a constant curvature is a (normal) torus, triangles on a torus that have the same side lengths can have different angles, compare a triangle on the inside (negative curvature) with one on the outside (positive curvature) of a torus, for this reason 
triangles cannot be moved in every directions without distortion.
Discussion
I make this post a community wiki, let people who agree with my points add  their remarks here, and I hope MvG will do the same so people suportingbhim can add to his points.  (ps don't remove comments of others , but do improve them)
