Relation between Normal Neighborhood Charts on Riemannian Manifold

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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:

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


Proof




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