Given a manifold as algebraic variety, say sphere, how do we obtain possible metrics? how do we classify them? If spcaetime manifold is n-sphere, Einstein's vacuum (for now) equation would be some special metric among many other possible metrics? i'm curious what role Einstein's equation plays. It is not determining spacetime manifold itself. It determines some metric as its solution. Then why do we say matter (source term to EInstein equation) curve the spacetime? It changes metric, but what about spacetime itself?