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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 :)