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## Main Question or Discussion Point

So, I was trying to do a derivation of my own for the FLRW metric, since I couldn't understand the one Wald had. The spatial slice [itex]M[/itex] is a connected Riemannian manifold which is everywhere isotropic. That is, in every point [itex]p\in M[/itex] and unit vectors in [itex]v_1,v_2\in T_p\left(M\right)[/itex] there is an isometry [itex]\phi:M\rightarrow M[/itex] with [itex]\phi_*v_1 = v_2[/itex]. By isotropy, the sectional curvature is a pointwise function and by Schur's Lemma, [itex]M[/itex] it's constant.

My question is the following. Under those assumptions, can it be proved that [itex]M[/itex] is simply connected and complete? The FLRW metric suggests that [itex]M[/itex] is:

(a) a sphere if the curvature is constant.

(b) a hyberboloid if the curvature is negative,

(c) flat if the curvature is zero.

all if which are connected, simply connected and complete manifolds.

P.S. This may be the wrong section for this thread. If so I apologize :)

My question is the following. Under those assumptions, can it be proved that [itex]M[/itex] is simply connected and complete? The FLRW metric suggests that [itex]M[/itex] is:

(a) a sphere if the curvature is constant.

(b) a hyberboloid if the curvature is negative,

(c) flat if the curvature is zero.

all if which are connected, simply connected and complete manifolds.

P.S. This may be the wrong section for this thread. If so I apologize :)