My understanding is that certain laws of physics (such as the motion in gravitational field) can be rewritten as a problem of geometry of an "abstract" space.

For example trajectories under gravitational fields can be thought as the geodesics of a space defined by the dimensions (x,y,z,-ict) . No gravity means the space (x,y,z,-ict) is flat and the geodesics are a straight line. Indeed, in absence of any external force we move indefinite along a straight line at constant speed. If we have gravity then the space (x,y,z,-ict) has intrinsic curvature and in free fall motion we simply follow the geodesic of this (intrinsically) curved 4D-space.

Note that this 4D-space(x,y,z,ict) to me is an "abstract" space and not the "real" geometrical space we live in. The "abstract" part came in when the additional dimension "ict" is added.