Semi-geodesic coordinates
geodesic normal coordinates
Coordinates in an
-dimensional Riemannian space, defined by the following characteristic property: the coordinate curves in the direction of
are geodesics for which
is the arc length parameter, and the coordinate surfaces
are orthogonal to these geodesics. In terms of semi-geodesic coordinates, the squared line element is given by
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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
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The total (Gaussian) curvature may be determined from the formula
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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 . In this case all geodesic coordinate curves
intersect at one point (the pole) and
is the angle between the coordinate curves
and
. Any curve
is called a geodesic circle. The squared line element in a neighbourhood of the pole is written as
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in geodesic polar coordinates, where is the total (Gaussian) curvature at the point
,
is the derivative of
with respect to
at
in the direction of the geodesic
, and
is the similarly defined derivative in the direction of the geodesic
.
When geodesic coordinates are defined in a pseudo-Riemannian space, it is often stipulated that the geodesics corresponding to should not be isotropic. In the case the squared line element is written as
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(the plus or minus sign depends on the sign of the square of the integral of the tangent vector to the -curve).
Comments
Results similar to the -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
-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) |
Semi-geodesic coordinates. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Semi-geodesic_coordinates&oldid=48658