Singularities of flat manifolds

Question

Can a flat manifold have infinitely many singularities? Let me explain: I am working on riemannian geometry applied to thermodynamics. I am analyzing closed simple systems whose state of equilibrium states is a flat riemannian manifold. I solve the differential equation for the scalar curvature equal to zero and I assume that one of the components of the diagonalized metric is constant. The result is that the chosen coordinate system might have infinitely many singularities. Is this actually possible? Is there any topological/geometrical result that forbids this result I'm getting?

Answer 1

Suppose that $(M,g)$ is a flat, complete, smooth Riemannian manifold. If $N$ is a smooth manifold and there is a diffeomorphism $f: N \to M$, then $(N, f^*g)$ is also a flat Riemannian manifold. Thus, the property of being flat gives no control over "how bad" the metric looks in local coordinates. For instance, choose a sequence of $C^{\infty}$ diffeomorphisms $f_i: M \to M$ so that the limit $f_{\infty}$ is a diffeomorphism except on an infinite singular set. Then you have a sequence of smooth, flat metrics $f^*g_i$ that develop arbitrarily many singularities in rather arbitrary ways. The limiting "metric" is then a flat metric with arbitrary coordinate singularities.