Local flattening of a Riemannian metric ($\tilde{g}=(1-\chi )g+\chi {\varphi }^{\ast }\delta$)

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.