Relation between Normal Neighborhood Charts on Riemannian Manifold

Theorem
Let $\struct {M, g}$ be an $n$-dimensional Riemannian manifold.

Let $\tuple {x^i}$ and $\tuple {\tilde x^j}$ be normal neighborhood charts on $M$.

Let $\map O {n, \R}$ be the orthogonal group.

Let $A^j_i \in \map O {n, \R}$ be a constant matrix.

Then:


 * $\forall \tuple {x^i} : \forall \tuple {\tilde x^i} : \exists A^j_i \in \map O {n, \R} : \tilde x^j = A^j_i x^i$