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A connected set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g044/g044160/g0441601.png" /> of points on a surface <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g044/g044160/g0441602.png" /> such that for each point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g044/g044160/g0441603.png" /> there exists a disc <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g044/g044160/g0441604.png" /> with centre at <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g044/g044160/g0441605.png" /> such that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g044/g044160/g0441606.png" /> has one of the following forms: 1) <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g044/g044160/g0441607.png" />; 2) <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g044/g044160/g0441608.png" /> is a semi-disc of the disc; 3) <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g044/g044160/g0441609.png" /> is a sector of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g044/g044160/g04416010.png" /> other than a semi-disc; or 4) <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g044/g044160/g04416011.png" /> consists of a finite number of sectors <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g044/g044160/g04416012.png" /> of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g044/g044160/g04416013.png" /> with no common points except <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g044/g044160/g04416014.png" />.
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A connected set $G$ of points on a surface $F$ such that for each point $x$ there exists a disc $K(x)$ with centre at $x$ such that $K_G=G\cap K(x)$ has one of the following forms: 1) $K_G(x)=K(x)$; 2) $K_G(x)$ is a semi-disc of the disc; 3) $K_G(x)$ is a sector of $K(x)$ other than a semi-disc; or 4) $K_G(x)$ consists of a finite number of sectors $u_i(x)$ of $K(x)$ with no common points except $x$.
  
A point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g044/g044160/g04416015.png" /> is called a regular interior point in the first case, a regular boundary point in the second, an angular point in the third, and a nodal point in the fourth case. A geodesic region that is compact in itself and has no nodal points is called a normal region. A normal region is either a closed surface or a surface with boundary consisting of a finite number of pairwise non-intersecting Jordan polygons.
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A point $x$ is called a regular interior point in the first case, a regular boundary point in the second, an angular point in the third, and a nodal point in the fourth case. A geodesic region that is compact in itself and has no nodal points is called a normal region. A normal region is either a closed surface or a surface with boundary consisting of a finite number of pairwise non-intersecting Jordan polygons.
  
A geodesic region may be considered as a metric space by introducing the so-called <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g044/g044160/g04416017.png" />-distance <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g044/g044160/g04416018.png" /> between two points <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g044/g044160/g04416019.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g044/g044160/g04416020.png" /> (the greatest lower bound of the lengths of all rectifiable curves connecting <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g044/g044160/g04416021.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g044/g044160/g04416022.png" /> and completely contained in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g044/g044160/g04416023.png" />). A rectifiable arc in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g044/g044160/g04416024.png" /> with ends <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g044/g044160/g04416025.png" /> is called a <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g044/g044160/g04416027.png" />-segment if it is the shortest connection between <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g044/g044160/g04416028.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g044/g044160/g04416029.png" /> in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g044/g044160/g04416030.png" />. Single points are considered to be <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g044/g044160/g04416031.png" />-segments of length zero. For all points of a <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g044/g044160/g04416032.png" />-segment the equation <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g044/g044160/g04416033.png" /> is valid. A <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g044/g044160/g04416035.png" />-ray is a ray inside a geodesic region each partial arc of which is a <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g044/g044160/g04416036.png" />-segment. A <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g044/g044160/g04416038.png" />-line consists of two rays with no points in common other than the origin, such that each arc contained in the line is a <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g044/g044160/g04416039.png" />-segment.
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A geodesic region may be considered as a metric space by introducing the so-called $G$-distance $\rho_G$ between two points $a$ and $b$ (the greatest lower bound of the lengths of all rectifiable curves connecting $a$ and $b$ and completely contained in $G$). A rectifiable arc in $G$ with ends $a,b$ is called a $G$-segment if it is the shortest connection between $a$ and $b$ in $G$. Single points are considered to be $G$-segments of length zero. For all points of a $G$-segment the equation $\rho_G(a,x)+\rho_G(x,b)=\rho_G(a,b)$ is valid. A $G$-ray is a ray inside a geodesic region each partial arc of which is a $G$-segment. A $G$-line consists of two rays with no points in common other than the origin, such that each arc contained in the line is a $G$-segment.
  
A geodesic region has a total curvature if and only if for any sequence of normal regions exhausting the geodesic region the total curvatures tend to a common value. If the Gaussian curvature of the domain is nowhere negative or if it is nowhere positive, then the domain has a total curvature. If the domain does not have a total curvature, then it is always possible to find an exhausting sequence of normal regions with total curvatures tending to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g044/g044160/g04416040.png" />. If the boundary of a complete geodesic region, homeomorphic to a closed half-plane, has only a finite number of angular points and if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g044/g044160/g04416041.png" /> are the respective angles measured in the geodesic region, then the inequality
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A geodesic region has a total curvature if and only if for any sequence of normal regions exhausting the geodesic region the total curvatures tend to a common value. If the Gaussian curvature of the domain is nowhere negative or if it is nowhere positive, then the domain has a total curvature. If the domain does not have a total curvature, then it is always possible to find an exhausting sequence of normal regions with total curvatures tending to $\pm\infty$. If the boundary of a complete geodesic region, homeomorphic to a closed half-plane, has only a finite number of angular points and if $\omega_1,\dots,\omega_n$ are the respective angles measured in the geodesic region, then the inequality
  
<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/g044160/g04416042.png" /></td> </tr></table>
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$$C(G)\leq\pi-\sum_{i=1}^n(\pi-\omega_i)$$
  
is valid for the total curvature <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g044/g044160/g04416043.png" />.
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is valid for the total curvature $C(G)$.
  
 
====References====
 
====References====
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  S.E. Cohn-Vossen,  "Kürzeste Wege und Totalkrümmung auf Flächen"  ''Compos. Math.'' , '''2'''  (1935)  pp. 69–133</TD></TR></table>
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<table>
 
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<TR><TD valign="top">[1]</TD> <TD valign="top">  S.E. Cohn-Vossen,  "Kürzeste Wege und Totalkrümmung auf Flächen"  ''Compos. Math.'' , '''2'''  (1935)  pp. 69–133</TD></TR>
 
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<TR><TD valign="top">[a1]</TD> <TD valign="top">  J. Cheeger,  D.G. Ebin,  "Comparison theorems in Riemannian geometry" , North-Holland  (1975)</TD></TR>
 
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</table>
====Comments====
 
 
 
 
 
====References====
 
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  J. Cheeger,  D.G. Ebin,  "Comparison theorems in Riemannian geometry" , North-Holland  (1975)</TD></TR></table>
 

Latest revision as of 11:10, 16 April 2023

A connected set $G$ of points on a surface $F$ such that for each point $x$ there exists a disc $K(x)$ with centre at $x$ such that $K_G=G\cap K(x)$ has one of the following forms: 1) $K_G(x)=K(x)$; 2) $K_G(x)$ is a semi-disc of the disc; 3) $K_G(x)$ is a sector of $K(x)$ other than a semi-disc; or 4) $K_G(x)$ consists of a finite number of sectors $u_i(x)$ of $K(x)$ with no common points except $x$.

A point $x$ is called a regular interior point in the first case, a regular boundary point in the second, an angular point in the third, and a nodal point in the fourth case. A geodesic region that is compact in itself and has no nodal points is called a normal region. A normal region is either a closed surface or a surface with boundary consisting of a finite number of pairwise non-intersecting Jordan polygons.

A geodesic region may be considered as a metric space by introducing the so-called $G$-distance $\rho_G$ between two points $a$ and $b$ (the greatest lower bound of the lengths of all rectifiable curves connecting $a$ and $b$ and completely contained in $G$). A rectifiable arc in $G$ with ends $a,b$ is called a $G$-segment if it is the shortest connection between $a$ and $b$ in $G$. Single points are considered to be $G$-segments of length zero. For all points of a $G$-segment the equation $\rho_G(a,x)+\rho_G(x,b)=\rho_G(a,b)$ is valid. A $G$-ray is a ray inside a geodesic region each partial arc of which is a $G$-segment. A $G$-line consists of two rays with no points in common other than the origin, such that each arc contained in the line is a $G$-segment.

A geodesic region has a total curvature if and only if for any sequence of normal regions exhausting the geodesic region the total curvatures tend to a common value. If the Gaussian curvature of the domain is nowhere negative or if it is nowhere positive, then the domain has a total curvature. If the domain does not have a total curvature, then it is always possible to find an exhausting sequence of normal regions with total curvatures tending to $\pm\infty$. If the boundary of a complete geodesic region, homeomorphic to a closed half-plane, has only a finite number of angular points and if $\omega_1,\dots,\omega_n$ are the respective angles measured in the geodesic region, then the inequality

$$C(G)\leq\pi-\sum_{i=1}^n(\pi-\omega_i)$$

is valid for the total curvature $C(G)$.

References

[1] S.E. Cohn-Vossen, "Kürzeste Wege und Totalkrümmung auf Flächen" Compos. Math. , 2 (1935) pp. 69–133
[a1] J. Cheeger, D.G. Ebin, "Comparison theorems in Riemannian geometry" , North-Holland (1975)
How to Cite This Entry:
Geodesic region. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Geodesic_region&oldid=12875
This article was adapted from an original article by Yu.S. Slobodyan (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article