Local flattening of a Riemannian metric
= Local flattening of a Riemannian metric
{title2=$\widetilde g=(1-\chi)g+\chi\phi^*\delta$}
A <smooth bump function> supported in a chosen coordinate neighborhood and equal to one near a chosen point allows a <Riemannian metric> to be replaced locally by a flat one. The displayed convex combination stays positive definite and glues smoothly to the original metric outside the support. This is a local modification of the metric, not a claim that the original metric is flat.