It is customary to specify the geometry of a Riemannian
N
-space by writing down a quadratic line-element, the coefficients being ½
N
(
N
+ 1) functions of the coordinates. But since there is an
N
-fold arbitrariness in the choice of coordinates, there is an
N
-fold arbitrariness in the metric tensor, and one expresses this by saying that the metric tensor satisfies
N
coordinate conditions, so that there are essentially only ½
N
(
N
- 1) components. If the coordinate system is made definite by constructing it according to some geometrical plan, the coordinate conditions may be made explict; their form is well known for Riemannian coordinates (based on geodesics drawn out from a point) and for Gaussian coordinates (based on geodesics drawn orthogonal to an (
N
- 1)-space), and in some other cases. Our purpose is to present in a single argument the coordinate conditions for coordinates based on geodesics drawn orthogonal to a subspace of
M
dimensions (
M
= 0, 1, ...,
N
- 1). These conditions are very simple in form. They are used to express the metric tensor in terms of integrals of the linear part
L
ijkm
of the covariant Riemann tensor. If in these integrals
L
ijkm
is replaced by any other set of functions
E
ijkm
having the same symmetries as
L
ijkm
, then
L
ijkm
and
E
ijkm
differ only by terms evaluated on the subspace. All the results are applicable to the space-time of general relativity if one puts
N
= 4.
Generalized embedding variables for geometrodynamics and space‐time diffeomorphisms: Ultralocal coordinate conditions
