# Estimate on the size of flat balls where the Weyl tensor vanishes

Let $(M,g)$ be a Riemannian manifold and suppose that the Weyl tensor of $g$ vanishes at a point $p \in M$. Can one estimate the size of the largest geodesic ball around $p$ that we can make $g$ flat on through conformal deformation in terms of geometric data, i.e., $g$ and the usual curvature invariants? Is this just a function of the injectivity radius?

Also: suppose $g$ is locally conformally Einstein. Can we produce similiar estimates on the largest geodesic ball around a given point that we can make $g$ Einstein on?

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I believe that the answer to the first question is the injectivity radius if the manifold is locally conformally flat. The injectivity radius provides us with an estimate on the size of a chart that contains the point $p$. In this chart the metric has to take the form $h dx^i dx^j$, where $h$ is a smooth, positive function, if the Weyl tensor vanishes everywhere. It follows that we can make the metric flat through conformal deformation on the whole of the interior of the chart.