Geodesic preservation by a local isometry

A local isometry intertwines the Levi--Civita covariant derivatives. It therefore sends every affinely parametrized geodesic to an affinely parametrized geodesic.

A nonunit Euclidean dilation sends straight-line geodesics to straight-line geodesics but scales their tangent inner products. Preservation of geodesics alone therefore does not imply local isometry.