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A mapping <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052810/i0528101.png" /> from a metric space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052810/i0528102.png" /> into a metric space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052810/i0528103.png" /> preserving distances between points: If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052810/i0528104.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052810/i0528105.png" />, then
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A mapping $f$ from a metric space $A$ into a metric space $B$ preserving distances between points: If $x, y \in A$ and $f \left({x}\right), f \left({y}\right) \in B$, then
  
<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/i/i052/i052810/i0528106.png" /></td> </tr></table>
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$$ \rho_A \! \left({x, y}\right) = \rho_B \! \left({f \left({x}\right), f \left({y}\right)}\right). $$
  
An isometric mapping is an injective mapping of a special type, indeed it is an [[Immersion|immersion]]. If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052810/i0528107.png" />, that is, if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052810/i0528108.png" /> is a bijection, then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052810/i0528109.png" /> is said to be an isometry from <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052810/i05281010.png" /> onto <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052810/i05281011.png" />, and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052810/i05281012.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052810/i05281013.png" /> are said to be in isometric correspondence, or to be isometric to each other. Isometric spaces are homeomorphic. If in addition <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052810/i05281014.png" /> is the same as <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052810/i05281015.png" />, then the isometric mapping is said to be an isometric transformation, or a motion, of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052810/i05281016.png" />.
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An isometric mapping is an injective mapping of a special type, indeed it is an [[Immersion|immersion]]. If $f \left({A}\right) = B$, that is, if $f$ is a bijection, then $f$ is said to be an isometry from $A$ onto $B$, and $A$ and $B$ are said to be in isometric correspondence, or to be isometric to each other. Isometric spaces are homeomorphic. If in addition $B$ is the same as $A$, then the isometric mapping is said to be an isometric transformation, or a motion, of $A$.
  
If the metric spaces <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052810/i05281017.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052810/i05281018.png" /> are subsets of some topological space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052810/i05281019.png" /> and if there exists a [[Deformation|deformation]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052810/i05281020.png" /> such that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052810/i05281021.png" /> is an isometric mapping from <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052810/i05281022.png" /> onto <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052810/i05281023.png" /> for each <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052810/i05281024.png" />, then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052810/i05281025.png" /> is called an isometric deformation of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052810/i05281026.png" /> into <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052810/i05281027.png" />.
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If the metric spaces $A_0$ and $A_1$ are subsets of some topological space $B$ and if there exists a [[Deformation|deformation]] $F_t : A \to B$ such that $F_t$ is an isometric mapping from $A$ onto $A_t$ for each $t$, then $\left\{{A_t}\right\}$ is called an isometric deformation of $A_0$ into $A_1$.
  
 
An isometry of real Banach spaces is an affine mapping. Such a linear isometry is realized by (and called) an [[Isometric operator|isometric operator]].
 
An isometry of real Banach spaces is an affine mapping. Such a linear isometry is realized by (and called) an [[Isometric operator|isometric operator]].
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====References====
 
====References====
 
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  S. Mazur,  S. Ulam,  "Sur les transformations isométriques d'espaces vectoriels"  ''C.R. Acad. Sci. Paris'' , '''194'''  (1932)  pp. 946–948</TD></TR></table>
 
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  S. Mazur,  S. Ulam,  "Sur les transformations isométriques d'espaces vectoriels"  ''C.R. Acad. Sci. Paris'' , '''194'''  (1932)  pp. 946–948</TD></TR></table>
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Latest revision as of 06:49, 14 January 2017

A mapping $f$ from a metric space $A$ into a metric space $B$ preserving distances between points: If $x, y \in A$ and $f \left({x}\right), f \left({y}\right) \in B$, then

$$ \rho_A \! \left({x, y}\right) = \rho_B \! \left({f \left({x}\right), f \left({y}\right)}\right). $$

An isometric mapping is an injective mapping of a special type, indeed it is an immersion. If $f \left({A}\right) = B$, that is, if $f$ is a bijection, then $f$ is said to be an isometry from $A$ onto $B$, and $A$ and $B$ are said to be in isometric correspondence, or to be isometric to each other. Isometric spaces are homeomorphic. If in addition $B$ is the same as $A$, then the isometric mapping is said to be an isometric transformation, or a motion, of $A$.

If the metric spaces $A_0$ and $A_1$ are subsets of some topological space $B$ and if there exists a deformation $F_t : A \to B$ such that $F_t$ is an isometric mapping from $A$ onto $A_t$ for each $t$, then $\left\{{A_t}\right\}$ is called an isometric deformation of $A_0$ into $A_1$.

An isometry of real Banach spaces is an affine mapping. Such a linear isometry is realized by (and called) an isometric operator.


Comments

The fact that isometries of real Banach spaces are affine is due to S. Ulam and S. Mazur [a1].

References

[a1] S. Mazur, S. Ulam, "Sur les transformations isométriques d'espaces vectoriels" C.R. Acad. Sci. Paris , 194 (1932) pp. 946–948
How to Cite This Entry:
Isometric mapping. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Isometric_mapping&oldid=15201
This article was adapted from an original article by M.I. Voitsekhovskii (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article