When people talk about how mass is curving the space-time and the geometry of it pretty much accounts for gravity, does the curvature actually means what it is usually represented as - a curve in a higher dimension (from plane to a 3D space)? In our 3D space case it would be a curve in 4th spacial dimension?

If that is the case and it really has such a meaning, doesn't it mean that the curvature can be both "up" and "down" from a 2D plane (not sure if two or more directions for 3D ..) and gravity from both would behave exactly the same?

Curvature to the "up", instead of the usual "down", would NOT have a "negative mass" (well, depends what we mean by that, doesn't it?) or "anti-gravity" properties or something like that, BUT - what would happen if those two came together? They would cancel out, wouldn't they? Would holding a chunk of this oppositely curved mass in my spaceship cancel out some of the mass and make it lighter?

Can anti-matter be thought as simply a space-time curvature in the opposite direction?