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''geodesic normal coordinates''

Coordinates  $  x  ^ {1} \dots x  ^ {n} $
in an  $  n $-
dimensional Riemannian space, defined by the following characteristic property: the coordinate curves in the direction of  $  x  ^ {1} $
are geodesics for which  $  x  ^ {1} $
is the arc length parameter, and the coordinate surfaces  $  x  ^ {1} = \textrm{ const } $
are orthogonal to these geodesics. In terms of semi-geodesic coordinates, the squared line element is given by

$$ 
d s  ^ {2}  =  ( d x  ^ {1} )  ^ {2} + \sum _ {i , j = 2 } ^ { n } 
g _ {ij}  d x  ^ {i}  d x  ^ {j} .
$$

Semi-geodesic coordinates can be introduced in a sufficiently small neighbourhood of any point of an arbitrary Riemannian space. In many types of two-dimensional Riemannian spaces (such as regular surfaces of strictly negative curvature), semi-geodesic coordinates can be introduced in the large.

In the two-dimensional case, the squared line element is usually written as

$$ 
d s  ^ {2}  =  d u  ^ {2} + B  ^ {2} ( u , v )  d v  ^ {2} .
$$

The total (Gaussian) curvature may be determined from the formula

$$ 
K  =  - 
\frac{1}{B}
 
\frac{\partial   ^ {2} B }{\partial  u  ^ {2} }
 .
$$

In the theory of two-dimensional Riemannian manifolds with curvature of fixed sign, an important role is assigned to a special type of semi-geodesic coordinates — the geodesic polar coordinates  $  ( r , \phi ) $. 
In this case all geodesic coordinate curves  $  \phi = \textrm{ const } $
intersect at one point (the pole) and  $  \phi $
is the angle between the coordinate curves  $  v = 0 $
and  $  \phi = \textrm{ const } $. 
Any curve  $  r = \textrm{ const } $
is called a geodesic circle. The squared line element in a neighbourhood of the pole is written as

$$ 
d s  ^ {2}  =  d r  ^ {2} + r  ^ {2} \left \{ 1 - 
\frac{K _ {0} }{3}
 r  ^ {2\right} . -
$$

$$ 
- \left .

\frac{1}{6}
 ( K _ {1}  \cos  \phi + K _ {2}  \sin \ 
\phi ) r  ^ {3} + o ( r  ^ {3} ) \right \}  d \phi  ^ {2}
$$

in geodesic polar coordinates, where  $  K _ {0} $
is the total (Gaussian) curvature at the point  $  P $, 
$  K _ {1} $
is the derivative of  $  K $
with respect to  $  r $
at  $  P $
in the direction of the geodesic  $  \phi = 0 $, 
and  $  K _ {2} $
is the similarly defined derivative in the direction of the geodesic  $  \phi = \pi / 2 $.

When geodesic coordinates are defined in a pseudo-Riemannian space, it is often stipulated that the geodesics corresponding to  $  x  ^ {1} $
should not be isotropic. In the case the squared line element is written as

$$ 
d s  ^ {2}  =  \pm  ( d x  ^ {1} )  ^ {2} +
\sum _ {i , j = 2 } ^ { n }  g _ {ij}  d x  ^ {i}  d x  ^ {j}
$$

(the plus or minus sign depends on the sign of the square of the integral of the tangent vector to the  $  x  ^ {1} $-
curve).

====Comments====
Results similar to the  $  2 $-
dimensional case hold in arbitrary dimensions [[#References|[a2]]]. For the introduction of semi-geodesic coordinates (in a sufficiently small neighbourhood of an arbitrary point) in a Riemannian space see [[#References|[a1]]]. (It is done as follows: take a small piece of the hypersurface at the point and take for  $  x  ^ {1} $-
coordinates sufficiently short normal geodesics to this hypersurface.)

====References====
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  W. Klingenberg,   "A course in differential geometry" , Springer  (1983)  (Translated from German)</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top">  W. Klingenberg,   "Riemannian geometry" , de Gruyter  (1982)  (Translated from German)</TD></TR><TR><TD valign="top">[a3]</TD> <TD valign="top">  B. O'Neill,   "Semi-Riemannian geometry (with applications to relativity)" , Acad. Press  (1983)</TD></TR><TR><TD valign="top">[a4]</TD> <TD valign="top">  S. Kobayashi,   K. Nomizu,   "Foundations of differential geometry" , '''1–2''' , Interscience  (1963–1969)</TD></TR></table>