Semi-geodesic coordinates

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 [a2]. For the introduction of semi-geodesic coordinates (in a sufficiently small neighbourhood of an arbitrary point) in a Riemannian space see [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

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