Components of Metric at Origin of Normal Neighborhood of Pseudo-Riemannian Neighborhood

Theorem

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

Let $U_p$ be the normal neighborhood for $p \in M$.

Let $\struct {U_p, \tuple {x^i}}$ be a normal coordinate chart.

Suppose $g_{ij}$ are the components of metric $g$ in coordinates $\tuple {x^i}$ at $p \in M$.

Then:

$g_{ij} = \pm_{ij} \delta_{ij}$

where $\delta_{ij}$ denotes the Kronecker delta, and $\pm_{ij}$ is a sign that depends on $i$ and $j$.

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