Given a Riemannian manifold $(M,g)$ with Ricci tensor $ R_{mn} = k g_{mn} $. Suppose the Ricci scalar you get is
$$ R > 0 $$
What can you tell about the manifold $globally$ ? In particular, can you say anything about the topology of this manifold (e.g is this compact?) ?
This question arise in a Physics situation: in 11-dimensional supergravity, one can find solutions to equations with a factorised metric describing $M_4 \times M_7$, where the Riemannian manifold $M_7$ has the geometry described above (Einstein manifold with positive Ricci curvature). These solutions are said to furnish a spontaneous conpactification because $M_7$ is "automatically" compact. But I don't really understand why this is the case.
PS: Useful references where to study these topics in differential geometry? I just know basics (in order to understand General Relativity and String Theory)
Myers's Theorem says that if a manifold has positive lower bound for Ricci curvature, then it must be compact. In particular, if $M$ is Einstein with positive scalar curvature, we have $$Ric=\frac{R}{n}g.$$ Note that Einstein manifold must have constant scalar curvature (which follows from Bianchi identity). Combining all these, we can conclude that $M$ is compact.