Definition:Fermi Coordinates

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

Let $P \subseteq M$ an embedded $p$-dimensional submanifold.

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

Let $V \subseteq NP$ be an open subset.

Let $E$ be the normal exponential map.

Let $U \subseteq M$ be the normal neighborhood such that $U = \map E V$.

Let $\struct {W_0, \psi}$ be a smooth coordinate chart for $P$.

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

Let $\hat W_0 = \map \psi {W_0} \subseteq \R^p$.

Let $B : \hat W_0 \times \R^{n - p} \to \valueat {NP} {W_0}$ be a diffeomorphism such that:


 * $\map B {x^1, \ldots, x^p, v^1, \ldots, v^{n - p} } = \tuple {q, v^1 \valueat {E_1} q + \ldots + v^{n - p} \valueat {E_{n - p} } q }$

where:


 * $q = \map {\psi^{-1} } {x^1, \ldots, x^p}$

Let:


 * $V_0 = V \cap \valueat {NP} {W_0} \subseteq NP$

Let:


 * $U_0 = \map E {V_0} \subseteq M$

Let $\phi : U_0 \to \R^n$ be a smooth coordinate map such that:


 * $\phi = B^{-1} \circ \paren {\valueat E {V_0} }^{-1} : \map E {q, v^1 \valueat {E_1} q + \ldots + v^{n - p} \valueat {E_{n - p} } q} \mapsto \tuple {\map {x^1} q, \ldots, \map {x^p} q, v^1, \ldots, v^{n - p} }$

Then the coordinates defined according to the coordinate map above are called Fermi coordinates.