On Vanishing Riemann curvature tensor

Question

If a manifold $\mathcal{M}$ has a vanishing Riemann curvature tensor, then what

i) does this imply for the manifold? and

ii) What is such a manifold called?

Answer 1

Such Riemannian manifold $(M,g)$ is called "locally flat" or "locally Euclidean". Its metric is locally isometric to the standard metric on $R^n$ where $n$ is the dimension of the manifold. Without further hypothesis, this does not tell you much. Suppose however, that the metric is complete (e.g. the manifold is compact). Then $(M,g)$ is the quotient of the Euclidean $n$-space by a properly discontinuous group of isometries acting freely. Topologically, it follows that such a manifold is finitely covered by the product $T^k\times R^{n-k}$. Take a look at the book

J. Wolf, "Spaces of constant curvature"

for details.