Curvature and Christoffel symbols

Question

Let (M,g) be a Riemannian manifold and $\nabla$ its Levi-Civita connection. Suppose that the curvature $R$ is zero. Is $\nabla$ locally trivial?, i.e., do the Christoffel symbols equal to zero?

Thank you very much!

Answer 1

This is a strange use of the term "locally trivial": As phrased it suggests an (invariant) property of $\nabla$, but the Christoffel symbols of a connection are relative to (depend on) a choice of coordinates, and they can be identically zero in one chart and not identically zero in another.

If the Riemannian curvature of $(M, g)$ is zero everywhere, then $g$ is locally flat; equivalently, around every point there are coordinates $(x^a)$ so that, in that coordinate chart, $g$ is given by $\sum (dx^a)^2$. Since all of the coefficients $g_{ab} = \delta_{ab}$ are constant, the Christoffel symbols of the Levi-Civita connection $\nabla$ of $g$ with respect to these coordinates are all zero.

On the other hand, computing from the definition shows that the Christoffel symbols for Levi-Civita connection of the standard (flat) metric on $\Bbb R^2$ with respect to polar coordinates are not all zero.