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Diametrically-opposite points on a sphere. Borsuk's antipodal-point theorems apply [[#References|[1]]]: 1) For any continuous mapping of the sphere <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a012/a012720/a0127201.png" /> into the Euclidean space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a012/a012720/a0127202.png" /> there exist antipodes with a common image; 2) Any mapping of the sphere <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a012/a012720/a0127203.png" /> into itself in which the images of antipodes are antipodes is an essential mapping.
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Diametrically-opposite points on a sphere. Borsuk's antipodal-point theorems apply [[#References|[1]]]: 1) For any continuous mapping of the sphere $S^n$ into the Euclidean space $E^n$ there exist antipodes with a common image; 2) Any mapping of the sphere $S^n$ into itself in which the images of antipodes are antipodes is an [[essential mapping]].
  
 
====References====
 
====References====
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  K. Borsuk,  "Drei Sätze über die <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a012/a012720/a0127204.png" />-dimensionale euklidische Sphäre"  ''Fund. Math.'' , '''20'''  (1933)  pp. 177–190</TD></TR></table>
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<table><TR><TD valign="top">[1]</TD> <TD valign="top">  K. Borsuk,  "Drei Sätze über die $n$-dimensionale euklidische Sphäre"  ''Fund. Math.'' , '''20'''  (1933)  pp. 177–190</TD></TR></table>
  
  
  
 
====Comments====
 
====Comments====
The first result mentioned above is known as the Borsuk–Ulam theorem (on antipodes). The following result also goes by the name of Borsuk's antipodal theorem: There is no continuous mapping <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a012/a012720/a0127205.png" /> of the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a012/a012720/a0127206.png" />-ball <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a012/a012720/a0127207.png" /> into the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a012/a012720/a0127208.png" />-sphere <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a012/a012720/a0127209.png" /> such that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a012/a012720/a01272010.png" />, cf. [[#References|[a1]]], p. 131.
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The first result mentioned above is known as the [[Borsuk–Ulam theorem]] (on antipodes). The following result also goes by the name of Borsuk's antipodal theorem: There is no continuous mapping $f$ of the $(n+1)$-ball $B^{n+1}$ into the $n$-sphere $S^n$ such that $f(x)=-f(-x)$, cf. [[#References|[a1]]], p. 131.
  
 
====References====
 
====References====
 
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  V.I. Istrătescu,  "Fixed point theory" , Reidel  (1981)</TD></TR></table>
 
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  V.I. Istrătescu,  "Fixed point theory" , Reidel  (1981)</TD></TR></table>

Latest revision as of 06:23, 28 October 2017

Diametrically-opposite points on a sphere. Borsuk's antipodal-point theorems apply [1]: 1) For any continuous mapping of the sphere $S^n$ into the Euclidean space $E^n$ there exist antipodes with a common image; 2) Any mapping of the sphere $S^n$ into itself in which the images of antipodes are antipodes is an essential mapping.

References

[1] K. Borsuk, "Drei Sätze über die $n$-dimensionale euklidische Sphäre" Fund. Math. , 20 (1933) pp. 177–190


Comments

The first result mentioned above is known as the Borsuk–Ulam theorem (on antipodes). The following result also goes by the name of Borsuk's antipodal theorem: There is no continuous mapping $f$ of the $(n+1)$-ball $B^{n+1}$ into the $n$-sphere $S^n$ such that $f(x)=-f(-x)$, cf. [a1], p. 131.

References

[a1] V.I. Istrătescu, "Fixed point theory" , Reidel (1981)
How to Cite This Entry:
Antipodes. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Antipodes&oldid=13716
This article was adapted from an original article by A.V. Chernavskii (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article