In Miranda Jr, Pallara, Paronetto and Preunkert's Heat semigroup and functions of bounded variation on Riemannian manifolds (p.106, property (4)), the authors state that for an $n\ge 2$-dimensional Riemannian manifold $M$ without boundary, one can construct a collection of coordinate charts $(U_i^\eta,\psi_i)$ such that $T^*(U_i^\eta)\simeq U_i^\eta \times \mathbb{R}^n$, and $d\psi_i$ is an isometry between $T^*_x(M)$ and $\mathbb{R}^n$ for every $x \in U_i^\eta$.
Why is this possible when $M$ is not flat? The coordinate system $(U_i^\eta, \psi_i)$ seems to have an orthonormal coordinate frame, which is only possible if $M$ has a flat metric on $M$ (by e.g. Theorem 13.14 in Lee's Introduction to Smooth Manifolds, or this question).
Miranda, M. jun.; Pallara, D.; Paronetto, F.; Preunkert, M., Heat semigroup and functions of bounded variation on Riemannian manifolds, J. Reine Angew. Math. 613, 99-119 (2007). ZBL1141.58014.
The existence of a local isometry implies flatness, since the curvature tensor is a local invariant. So this condition implies $(M, g)$ is flat on the neighborhood of interest.
One can arrange for coordinate charts to be an isometry at one point and have all first derivatives of the metric vanish at that point - these are called exponential coordinates - but no better is possible in general. It is likely the case (but one should check) that only this or something similar which can be arranged for using previously assumed hypotheses is actually used in the proofs involving this condition.