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.
