spaghetti3451 said:
Look at the sentence below the caption of figure 4 on page 8 of the notes.
Since this is an advanced source, it is assuming that readers already understand in detail the procedure that is being described here; so it doesn't have to explain that procedure in detail and with rigor. You should not be taking this wording to be part of such a detailed rigorous description.
Also, this section is talking about QFT on a Euclidean space, not on a Minkowski spacetime. So the "isometry" it is talking about is not the kind you will see in ordinary Minkowski spacetime. (Note that the word "spacelike", which you used in the OP, does not appear in the sentence you refer to in the paper.)
To illustrate how the foliation concept works in an ordinary (locally Lorentzian) spacetime, consider inertial coordinates on Minkowski spacetime. Each surface of constant coordinate time ##t## is a spacelike hypersurface, and the set of all such hypersurfaces foliates the entire spacetime. Furthermore, the coordinate basis vector ##\partial_t## in these coordinates generates an isometry; heuristically, "isometry" means that each of the spacelike hypersurfaces in the foliation has the same geometry (in this case, each one is just flat Euclidean 3-space). We say that ##\partial_t## "generates" this isometry because integral curves of this vector fields, which are just the worldlines of inertial observers at rest in these coordinates, form a timelike congruence (a family of worldlines that never intersect each other and which cover the entire spacetime), and an observer moving along any of these worldlines sees an unchanging spacetime geometry in his vicinity.
For another example, consider Schwarzschild spacetime outside the event horizon. Here the spacelike hypersurfaces that foliate the spacetime (more precisely, this region of the spacetime) are the surfaces of constant Schwarzschild coordinate time ##t##; and the timelike vector field that generates the isometry is again the coordinate basis vector ##\partial_t##. Again, each spacelike hypersurface has the same geometry (this time it's the Flamm paraboloid), and observers moving along integral curves of the isometry (call these "static" observers) see an unchanging spacetime geometry in their vicinity. The difference in this case, vs. the Minkowski case above, is that the spacetime geometry is different in the vicinity of different static observers (more precisely, static observers at different radius).
So the isometry doesn't "relate" the different spacelike hypersurfaces; it tells us which observers (which worldlines) see unchanging spacetime geometry in their vicinity. The spacelike hypersurfaces all have the same geometry; there's nothing to "relate".
