what i meant in my question was, even if the metric changes due to matter, it doesn't change spacetime manifold itself. n-sphere in out example. presence of matter on the sphere could deform metric field on it, but not sphere itself. (changes only up to diffeomorphism)
i don't know what...
actually definition above is not completely true. i believe that definition is given in goldstein's book.
but in that logic, you can not distinguish one dynamical system from another. (Both have canonical looking equation) So if you write down two equations and start claiming both of them are...
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...