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''geometry of geodesics''
 
''geometry of geodesics''
  
The geometry of a metric space (a <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g044/g044100/g0441002.png" />-space) characterized by the fact that extensions of geodesic lines (cf. [[Geodesic line|Geodesic line]]), defined as locally shortest lines, are unique.
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The geometry of a metric space (a $G$-space) characterized by the fact that extensions of geodesic lines (cf. [[Geodesic line|Geodesic line]]), defined as locally shortest lines, are unique.
  
A <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g044/g044100/g0441003.png" />-space is defined by the following system of axioms:
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A $G$-space is defined by the following system of axioms:
  
1) <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g044/g044100/g0441004.png" /> is a metric space; <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g044/g044100/g0441005.png" /> is the distance in the space;
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1) $G$ is a metric space; $\rho(x,y)$ is the distance in the space;
  
2) <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g044/g044100/g0441006.png" /> is finitely compact, i.e. bounded infinite sets in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g044/g044100/g0441007.png" /> have limit points;
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2) $G$ is finitely compact, i.e. bounded infinite sets in $G$ have limit points;
  
3) <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g044/g044100/g0441008.png" /> is convex in the sense of Menger, i.e. for two points <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g044/g044100/g0441009.png" /> there exists a third point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g044/g044100/g04410010.png" /> distinct from <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g044/g044100/g04410011.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g044/g044100/g04410012.png" /> and such that
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3) $G$ is convex in the sense of Menger, i.e. for two points $x\neq y$ there exists a third point $z$ distinct from $x$ and $y$ and such that
  
<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/g/g044/g044100/g04410013.png" /></td> </tr></table>
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$$\rho(x,z)+\rho(z,y)=\rho(x,y);$$
  
4) for each point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g044/g044100/g04410014.png" /> there exists an <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g044/g044100/g04410015.png" /> such that in the ball <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g044/g044100/g04410016.png" /> there exists, for the points <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g044/g044100/g04410017.png" />, a third point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g044/g044100/g04410018.png" /> distinct from <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g044/g044100/g04410019.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g044/g044100/g04410020.png" /> with <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g044/g044100/g04410021.png" /> (the axiom of local extension);
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4) for each point $a$ there exists an $r>0$ such that in the ball $\rho(a,x)<r$ there exists, for the points $x\neq y$, a third point $z$ distinct from $x$ and $y$ with $\rho(x,y)+\rho(y,z)=\rho(x,z)$ (the axiom of local extension);
  
5) if, in axiom 4), two points <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g044/g044100/g04410022.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g044/g044100/g04410023.png" /> such that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g044/g044100/g04410024.png" /> have been found, then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g044/g044100/g04410025.png" /> (the axiom of unique extension).
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5) if, in axiom 4), two points $z_1$ and $z_2$ such that $\rho(y,z_1)=\rho(y,z_2)$ have been found, then $z_1=z_2$ (the axiom of unique extension).
  
The class of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g044/g044100/g04410026.png" />-spaces includes, in particular, Riemannian spaces and Finsler spaces (cf. [[Finsler space|Finsler space]]; [[Riemannian space|Riemannian space]]).
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The class of $G$-spaces includes, in particular, Riemannian spaces and Finsler spaces (cf. [[Finsler space|Finsler space]]; [[Riemannian space|Riemannian space]]).
  
<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g044/g044100/g04410027.png" />-spaces in which extension of a geodesic is possible in general and any segment of the geodesic remains a shortest line are called straight spaces. They include, for example, Euclidean, Minkowski and Lobachevskii spaces, and all simply-connected Riemannian spaces of non-positive curvature. In straight spaces and in some <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g044/g044100/g04410028.png" />-spaces of special type (elliptic), a geodesic is determined by two points.
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$G$-spaces in which extension of a geodesic is possible in general and any segment of the geodesic remains a shortest line are called straight spaces. They include, for example, Euclidean, Minkowski and Lobachevskii spaces, and all simply-connected Riemannian spaces of non-positive curvature. In straight spaces and in some $G$-spaces of special type (elliptic), a geodesic is determined by two points.
  
In general <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g044/g044100/g04410029.png" />-spaces, unlike in Minkowski spaces, a sphere is not always convex. Perpendicularity, defined in terms of the shortest line to the geodesic, is not necessarily symmetric, unlike in Euclidean spaces. Criteria have been formulated in terms of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g044/g044100/g04410030.png" />-spaces to distinguish Euclidean spaces, spheric spaces, and Minkowski spaces.
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In general $G$-spaces, unlike in Minkowski spaces, a sphere is not always convex. Perpendicularity, defined in terms of the shortest line to the geodesic, is not necessarily symmetric, unlike in Euclidean spaces. Criteria have been formulated in terms of $G$-spaces to distinguish Euclidean spaces, spheric spaces, and Minkowski spaces.
  
The theory of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g044/g044100/g04410031.png" />-spaces showed that many results of differential geometry are not connected with conditions of differentiability. This theory extended the studies on Finsler spaces; made it possible to study those metrizations of affine and projective spaces which convert straight lines to geodesic lines; and to study the freedom of choice of geodesic nets under metrization. A number of hitherto unsolved problems are connected with the possible topological structure of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g044/g044100/g04410032.png" />-spaces [[#References|[1]]].
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The theory of $G$-spaces showed that many results of differential geometry are not connected with conditions of differentiability. This theory extended the studies on Finsler spaces; made it possible to study those metrizations of affine and projective spaces which convert straight lines to geodesic lines; and to study the freedom of choice of geodesic nets under metrization. A number of hitherto unsolved problems are connected with the possible topological structure of $G$-spaces [[#References|[1]]].
  
 
====References====
 
====References====
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====Comments====
 
====Comments====
For further generalizations and results see [[#References|[a1]]]. The phrase  "G-space" , in analogy with <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g044/g044100/g04410033.png" />-set, is also used to denote the totally different notion of a (topological) space with an action of a (topological) group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g044/g044100/g04410034.png" /> on it.
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For further generalizations and results see [[#References|[a1]]]. The phrase  "G-space" , in analogy with $G$-set, is also used to denote the totally different notion of a (topological) space with an action of a (topological) group $G$ on it.
  
 
====References====
 
====References====
 
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  M. Gromov,  "Structures métriques pour les variétés Riemanniennes" , F. Nathan  (1981)  (Translated from Russian)</TD></TR></table>
 
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  M. Gromov,  "Structures métriques pour les variétés Riemanniennes" , F. Nathan  (1981)  (Translated from Russian)</TD></TR></table>

Latest revision as of 16:23, 1 May 2014

geometry of geodesics

The geometry of a metric space (a $G$-space) characterized by the fact that extensions of geodesic lines (cf. Geodesic line), defined as locally shortest lines, are unique.

A $G$-space is defined by the following system of axioms:

1) $G$ is a metric space; $\rho(x,y)$ is the distance in the space;

2) $G$ is finitely compact, i.e. bounded infinite sets in $G$ have limit points;

3) $G$ is convex in the sense of Menger, i.e. for two points $x\neq y$ there exists a third point $z$ distinct from $x$ and $y$ and such that

$$\rho(x,z)+\rho(z,y)=\rho(x,y);$$

4) for each point $a$ there exists an $r>0$ such that in the ball $\rho(a,x)<r$ there exists, for the points $x\neq y$, a third point $z$ distinct from $x$ and $y$ with $\rho(x,y)+\rho(y,z)=\rho(x,z)$ (the axiom of local extension);

5) if, in axiom 4), two points $z_1$ and $z_2$ such that $\rho(y,z_1)=\rho(y,z_2)$ have been found, then $z_1=z_2$ (the axiom of unique extension).

The class of $G$-spaces includes, in particular, Riemannian spaces and Finsler spaces (cf. Finsler space; Riemannian space).

$G$-spaces in which extension of a geodesic is possible in general and any segment of the geodesic remains a shortest line are called straight spaces. They include, for example, Euclidean, Minkowski and Lobachevskii spaces, and all simply-connected Riemannian spaces of non-positive curvature. In straight spaces and in some $G$-spaces of special type (elliptic), a geodesic is determined by two points.

In general $G$-spaces, unlike in Minkowski spaces, a sphere is not always convex. Perpendicularity, defined in terms of the shortest line to the geodesic, is not necessarily symmetric, unlike in Euclidean spaces. Criteria have been formulated in terms of $G$-spaces to distinguish Euclidean spaces, spheric spaces, and Minkowski spaces.

The theory of $G$-spaces showed that many results of differential geometry are not connected with conditions of differentiability. This theory extended the studies on Finsler spaces; made it possible to study those metrizations of affine and projective spaces which convert straight lines to geodesic lines; and to study the freedom of choice of geodesic nets under metrization. A number of hitherto unsolved problems are connected with the possible topological structure of $G$-spaces [1].

References

[1] H. Busemann, "The geometry of geodesics" , Acad. Press (1955)


Comments

For further generalizations and results see [a1]. The phrase "G-space" , in analogy with $G$-set, is also used to denote the totally different notion of a (topological) space with an action of a (topological) group $G$ on it.

References

[a1] M. Gromov, "Structures métriques pour les variétés Riemanniennes" , F. Nathan (1981) (Translated from Russian)
How to Cite This Entry:
Geodesic geometry. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Geodesic_geometry&oldid=11711
This article was adapted from an original article by V.A. Zalgaller (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article