I am taking general relativity and we recently introduced connections and curvature. In the definition of the curvature tensor, the lecture notes said:
"A manifold $M$ with a given affine connection is curved if $R(X,Y)Z$ is not identically zero. If $R(X, Y)Z = 0 $ for all vector fields $X, Y, Z$, then the connection is flat." [emphasis mine]
When I asked, my professor said that it is the manifold that is flat and not the connection, but I have seen other references also say that it is the connection that is flat. What is it that is flat? Is it the connection or the manifold, or is it both (together)?
How different can one manifold with two different connections be, can you choose a connection on a given manifold as to make the curvature tensor identically zero? Intuitively it seems a manifold that is curved should always be curved in some sense, but on the other hand it seems that you have a very large freedom in defining the connection. For example if you have a space with two different metrics, the geometry can be very different.