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''geodesic normal coordinates''
 
''geodesic normal coordinates''
  
Coordinates <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084100/s0841001.png" /> in an <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084100/s0841002.png" />-dimensional Riemannian space, defined by the following characteristic property: the coordinate curves in the direction of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084100/s0841003.png" /> are geodesics for which <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084100/s0841004.png" /> is the arc length parameter, and the coordinate surfaces <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084100/s0841005.png" /> are orthogonal to these geodesics. In terms of semi-geodesic coordinates, the squared line element is given by
+
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
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084100/s0841006.png" /></td> </tr></table>
+
$$
 +
d s  ^ {2}  = ( d x  ^ {1} )  ^ {2} + \sum _ {i , j = 2 } ^ { n }
 +
g _ {ij}  d x  ^ {i}  d x  ^ {j} .
 +
$$
  
 
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.
 
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.
Line 9: Line 29:
 
In the two-dimensional case, the squared line element is usually written as
 
In the two-dimensional case, the squared line element is usually written as
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084100/s0841007.png" /></td> </tr></table>
+
$$
 +
d s  ^ {2}  = d u  ^ {2} + B  ^ {2} ( u , v )  d v  ^ {2} .
 +
$$
  
 
The total (Gaussian) curvature may be determined from the formula
 
The total (Gaussian) curvature may be determined from the formula
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084100/s0841008.png" /></td> </tr></table>
+
$$
 
+
= -  
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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084100/s0841009.png" />. In this case all geodesic coordinate curves <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084100/s08410010.png" /> intersect at one point (the pole) and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084100/s08410011.png" /> is the angle between the coordinate curves <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084100/s08410012.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084100/s08410013.png" />. Any curve <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084100/s08410014.png" /> is called a geodesic circle. The squared line element in a neighbourhood of the pole is written as
+
\frac{1}{B}
 +
 +
\frac{\partial  ^ {2} B }{\partial  u  ^ {2} }
 +
.
 +
$$
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084100/s08410015.png" /></td> </tr></table>
+
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
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084100/s08410016.png" /></td> </tr></table>
+
$$
 +
d s  ^ {2}  = d r  ^ {2} + r  ^ {2} \left \{ 1 -  
 +
\frac{K _ {0} }{3}
 +
r  ^ {2\right} . -
 +
$$
  
in geodesic polar coordinates, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084100/s08410017.png" /> is the total (Gaussian) curvature at the point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084100/s08410018.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084100/s08410019.png" /> is the derivative of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084100/s08410020.png" /> with respect to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084100/s08410021.png" /> at <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084100/s08410022.png" /> in the direction of the geodesic <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084100/s08410023.png" />, and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084100/s08410024.png" /> is the similarly defined derivative in the direction of the geodesic <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084100/s08410025.png" />.
+
$$
 +
- \left .
  
When geodesic coordinates are defined in a pseudo-Riemannian space, it is often stipulated that the geodesics corresponding to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084100/s08410026.png" /> should not be isotropic. In the case the squared line element is written as
+
\frac{1}{6}
 +
( K _ {1}  \cos  \phi + K _ {2}  \sin \
 +
\phi ) r  ^ {3} + o ( r  ^ {3} ) \right \}  d \phi  ^ {2}
 +
$$
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084100/s08410027.png" /></td> </tr></table>
+
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 $.
  
(the plus or minus sign depends on the sign of the square of the integral of the tangent vector to the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084100/s08410028.png" />-curve).
+
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====
 
====Comments====
Results similar to the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084100/s08410029.png" />-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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084100/s08410030.png" />-coordinates sufficiently short normal geodesics to this hypersurface.)
+
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====
 
====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>
 
<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>

Latest revision as of 08:13, 6 June 2020


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)
How to Cite This Entry:
Semi-geodesic coordinates. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Semi-geodesic_coordinates&oldid=48658
This article was adapted from an original article by D.D. Sokolov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article