Fermi coordinates are local coordinates adapted to an embedded submanifold 
of a Riemannian manifold 
. Choose local coordinates 
on 
and a local orthonormal frame 
of its normal bundle.
Nearby points are represented by
 |
where 
has coordinates 
and 
is the exponential map.
The resulting coordinates are 
.
Along 
,
the tangential and normal coordinate directions are orthogonal and the normal block
of the metric tensor is the identity
matrix. When 
is a geodesic, these are called Fermi normal coordinates;
the metric tensor has Euclidean components and vanishing
first derivatives along the geodesic. Fermi coordinates
generalize normal coordinates, which are adapted
to a single point.
