Properties of Fermi Coordinates

Theorem

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

Let $P$ be an embedded $p$-dimensional submanifold.

Let $U$ be a normal neighborhood of $P$ in $M$.

Let $U_0 \subseteq U$ be an open subset.

Let $\tuple{x^1, \ldots, x^p; v^1, \ldots, v^{n-p} }$ be Fermi coordinates on $U_0$.

Let $x^{p + j} = v^j$ for $j = 1, \ldots, n - p$.

Let $\pi : NP \to P$ be the normal bundle of $P$ in $M$.

Let $\tuple {E_1, \ldots, E_{n - p}}$ be a local orthonormal frame for $NP$.

Let $\Gamma^k_{ij}$ be the Christoffel symbol.

Let $\gamma_v : \R \to M$ be a geodesic such that:

$\map {\gamma_v} 0 = q$

$\map {\gamma'_v} 0 = v$

where $\gamma'_v$ denotes the velocity of $\gamma_v$.

Then $\forall q \in P \cap U_0$:

$(1): \quad \map {x^{p + 1} } q = \ldots = \map {x^n} q = 0$

$(2): \quad g_{ij} = g_{ji} = \begin{cases}

0 & : 1 \le i \le p \text{ and } p + 1 \le j \le n\\ \delta_{ij} & : p + 1 \le i, j \le n \end{cases}$

$(3): \quad \forall v = v^1 \valueat {E_1} q + \ldots + v^{n - p} \valueat {E_{n - p} } q \in N_q P : \map {\gamma_v} t = \tuple {\map {x^1} q, \ldots, \map {x^p} q, t v^1, \ldots, t v^{n - p} }$

$(4): \quad \forall i, j \in \N : p + 1 \le i, j \le n : \map {\Gamma^k_{ij} } q = 0$

$(5): \quad \forall i, j, k \in \N : p + 1 \le i, j, k \le n : \partial_i \map {g_{jk} } q = 0$

Proof

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Sources

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