Difference between revisions of "Antipodes"
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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 | + | {{TEX|done}} |
| + | 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 | + | <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 | + | 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> | ||
Revision as of 19:22, 17 August 2014
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=32986
Antipodes. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Antipodes&oldid=32986
This article was adapted from an original article by A.V. Chernavskii (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article