Roughly speaking, in addition to Einstein equations, the (spatial) FLRW metric is constructed by assuming that, at fixed time, in the Riemannian manifold defining space:

(a) metric properties are invariant under 3-roatations,

(b) metric properties are the same at every spatial point.

Hypothesis (b) is the spatial homogeneity notion that is assumed. There is no supposition about how to move from a point in the space (at given time) to another point in the space (at same time).

It turns out that one moves through the space using a maximal group of continuous isometries of a $3D$ Riemannian manifold with constant curvature.

Up to geometric identitifcations under discrete subgroups of isometries, only three well known possibilities exist. One is, an Euclidean (affine) space, where movements are described in terms of vectors. The remaining two cases are more subtle, but permitted by the assumed hypotheses.

Roughly speaking, in addition to Einstein equations, the (spatial) FLRW metric is constructed by assuming that, at fixed time, in the Riemannian manifold defining space:

(a) metric properties are invariant under 3-rotations,

(b) metric properties are the same at every spatial point.

Hypothesis (b) is the spatial homogeneity notion that is assumed. There is no supposition about how to move from a point in the space (at given time) to another point in the space (at same time).

It turns out that one moves through the space using a maximal group of continuous isometries of a $3D$ Riemannian manifold with constant curvature.

Up to geometric identitifcations under discrete subgroups of isometries, only three well known possibilities exist. One is, an Euclidean (affine) space, where movements are described in terms of vectors. The remaining two cases are more subtle, but permitted by the assumed hypotheses.