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In a [[Manifold|manifold]]  $  M $
with [[Riemannian metric|Riemannian metric]]  $  g $, 
a natural object characterizing the geometry is the distance  $  d ( P,Q ) $
between any two points  $  P $, 
$  Q $
in  $  M $, 
defined as the infimum of the length of all curves connecting the points. This does not itself generalize to pseudo-Riemannian manifolds (cf. [[Pseudo-Riemannian space|Pseudo-Riemannian space]]; [[Pseudo-Riemannian geometry|Pseudo-Riemannian geometry]]). In that case, however, for any  $  P $
there is an open neighbourhood  $  U _ {P} $
such that for  $  Q \in U _ {P} $
the two points are joined by a unique geodesic  $  \gamma _ {PQ }  $
in  $  U _ {P} $. 
If  $  u $
is an [[Affine parameter|affine parameter]] with  $  \gamma _ {PQ }  ( u _ {1} ) = P $, 
$  \gamma _ {PQ }  ( u _ {2} ) = Q $, 
and  $  X = d \gamma _ {PQ }  /du $
is the corresponding tangent vector, then the quantity

$$ 
\Omega ( P,Q ) = {
\frac{1}{2}
 } ( u _ {2} - u _ {1} )  ^ {2} g ( X,X )
$$

is independent of the choice of affine parameter and is well defined for both Riemannian and pseudo-Riemannian metrics  $  g $. 
In the Riemannian case it reduces to  $  d ( P,Q )  ^ {2} /2 $, 
while in the case of general relativity (pseudo-Riemannian metric of Lorentz signature) it evaluates to zero if  $  P $, 
$  Q $
are null-separated and to plus or minus half the square of the space- or time-separation of the points otherwise. This function was introduced by H.S. Ruse [[#References|[a3]]] and popularized by J.L. Synge [[#References|[a4]]] as the world function for general relativity.

If  $  g $
is differentiable of class  $  C  ^ {k} $(
$  k \geq  2 $), 
then  $  \Omega $
is of class  $  C ^ {k - 2 } $
on the manifold  $  D $
of pairs  $  \{ {( P,Q ) } : {Q \in U _ {P} } \} $. 
On the diagonal  $  \Delta = \{ {( P,P ) } : {P \in M } \} $
the first few partial derivatives with respect to the coordinates of the first argument are given, in index notation, by

$$ 
\Omega _ {,i }  = 0,  \Omega _ {,ij }  = g _ {ij } 
$$

$$ 
\Omega _ {,ijk }  = 0,  \Omega _ {,ijkl }  = - {
\frac{1}{3}
 } ( R _ {ikjm }  + R _ {imjk }  ) ,
$$

where  $  R $
is the [[Riemann tensor|Riemann tensor]] of  $  g $(
with sign convention  $  R  ^ {i} _ {jkl }  = \partial  _ {k} \Gamma  ^ {i} _ {jl }  - \dots $).

Since the world function is physically interpretable in terms of the squares of lengths and times, and is linked by the above formulas to the [[Curvature|curvature]], it could be used by Synge [[#References|[a4]]] as a systematic tool in deriving the basic formulas of [[Differential geometry|differential geometry]] together with their physical interpretation. In this spirit it was used by C.J.S. Clarke and F. de Felice [[#References|[a1]]] to express the curvature corrections to radar-ranging measurements; further formulas of this kind have been presented in [[#References|[a2]]].

====References====
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  C.J.S. Clarke,   F. de Felice,   "Relativity on curved manifolds" , Cambridge Univ. Press  (1990)</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top">  J.M. Gambi,   P. Romero,   A. Sanmiguel,   F. Vicente,   "Fermi coordinate transformation under base-line change in relativistic celestial mechanics"  ''Int. J. Theor. Phys.'' , '''30'''  (1991)  pp. 1097–1116</TD></TR><TR><TD valign="top">[a3]</TD> <TD valign="top">  H.S. Ruse,   "Taylor's theorem in the tensor calculus"  ''Proc. London Math. Soc.'' , '''32'''  (1931)  pp. 87</TD></TR><TR><TD valign="top">[a4]</TD> <TD valign="top">  J.L. Synge,   "Relativity: the general theory" , North-Holland  (1960)</TD></TR></table>