Can they exist ?? Like Magnetic Field lines and Electric Field lines
I think you might be confused by what "exists" means in this context. http://en.wikipedia.org/wiki/Field_line" [Broken] are often physically significant, they are still just mathematical tools. In that sense, none of them "exist". But the gravitational field isn't any different, in terms of 'having' field lines, from the EM field, or any vector field, really.
Like an Electric field connects one point to the charge to the other..in that context , would a Grav. field line connect any two centre's of gravity?
Also it will always act radially inwards for a particle , so is the geometry similar to other Field lines?