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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Sources
- 2018: John M. Lee: Introduction to Riemannian Manifolds (2nd ed.) ... (previous) ... (next): $\S 5$: The Levi-Civita Connection. Normal Neighborhoods and Normal Coordinates