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A mapping <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r080/r080510/r0805101.png" /> of an <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r080/r080510/r0805102.png" />-dimensional simply-connected space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r080/r080510/r0805103.png" /> of constant curvature (i.e. of a Euclidean or affine space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r080/r080510/r0805104.png" />, a sphere <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r080/r080510/r0805105.png" /> or a hyperbolic (Lobachevskii) space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r080/r080510/r0805106.png" />) the set of fixed points <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r080/r080510/r0805107.png" /> of which is an <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r080/r080510/r0805108.png" />-dimensional hyperplane. The set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r080/r080510/r0805109.png" /> is called the mirror of the mapping <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r080/r080510/r08051010.png" />; in other words, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r080/r080510/r08051011.png" /> is a reflection in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r080/r080510/r08051012.png" />. Every reflection is uniquely defined by its mirror. The order (period) of a reflection in the group of all motions of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r080/r080510/r08051013.png" /> is equal to 2, i.e. <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r080/r080510/r08051014.png" />.
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The Euclidean or affine space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r080/r080510/r08051015.png" /> can be identified with the vector space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r080/r080510/r08051016.png" /> of its parallel translations. The mapping <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r080/r080510/r08051017.png" /> is then a linear orthogonal transformation of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r080/r080510/r08051018.png" /> with matrix
+
{{TEX|auto}}
 +
{{TEX|done}}
  
<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/r/r080/r080510/r08051019.png" /></td> </tr></table>
+
A mapping  $  \sigma $
 +
of an  $  n $-
 +
dimensional simply-connected space  $  X  ^ {n} $
 +
of constant curvature (i.e. of a Euclidean or affine space  $  E  ^ {n} $,
 +
a sphere  $  S  ^ {n} $
 +
or a hyperbolic (Lobachevskii) space  $  \Lambda  ^ {n} $)
 +
the set of fixed points  $  \Gamma $
 +
of which is an  $  ( n- 1) $-
 +
dimensional hyperplane. The set  $  \Gamma $
 +
is called the mirror of the mapping  $  \sigma $;  
 +
in other words,  $  \sigma $
 +
is a reflection in  $  \Gamma $.
 +
Every reflection is uniquely defined by its mirror. The order (period) of a reflection in the group of all motions of  $  X  ^ {n} $
 +
is equal to 2, i.e. $  \sigma  ^ {2} =  \mathop{\rm Id} _ {X  ^ {n}  } $.
  
in a certain orthonormal basis, and conversely, every orthogonal transformation of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r080/r080510/r08051020.png" /> with this matrix in a certain orthonormal basis is a reflection in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r080/r080510/r08051021.png" />. More generally, a linear transformation <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r080/r080510/r08051022.png" /> of an arbitrary vector space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r080/r080510/r08051023.png" /> over a field <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r080/r080510/r08051024.png" />, of characteristic other than 2, is called a linear reflection if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r080/r080510/r08051025.png" /> and if the rank of the transformation <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r080/r080510/r08051026.png" /> is equal to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r080/r080510/r08051027.png" />. In this case, the subspace <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r080/r080510/r08051028.png" /> of fixed vectors relative to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r080/r080510/r08051029.png" /> has codimension <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r080/r080510/r08051030.png" /> in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r080/r080510/r08051031.png" />, the subspace <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r080/r080510/r08051032.png" /> of eigenvectors with eigenvalue <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r080/r080510/r08051033.png" /> has dimension <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r080/r080510/r08051034.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r080/r080510/r08051035.png" />. If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r080/r080510/r08051036.png" /> is a linear form on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r080/r080510/r08051037.png" /> such that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r080/r080510/r08051038.png" /> when <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r080/r080510/r08051039.png" />, and if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r080/r080510/r08051040.png" /> is an element such that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r080/r080510/r08051041.png" />, then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r080/r080510/r08051042.png" /> is defined by the formula
+
The Euclidean or affine space  $  E  ^ {n} $
 +
can be identified with the vector space $  V  ^ {n} $
 +
of its parallel translations. The mapping  $  \sigma $
 +
is then a linear orthogonal transformation of $  V  ^ {n} $
 +
with matrix
  
<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/r/r080/r080510/r08051043.png" /></td> </tr></table>
+
$$
 +
\left \|
 +
\begin{array}{lllllr}
 +
1  &{}  &{}  &{}  &{}  & 0  \\
 +
{}  &\cdot  &{}  &{}  &{}  &{}  \\
 +
{}  &{}  &\cdot  &{}  &{}  &{}  \\
 +
{}  &{}  &{}  &\cdot  &{}  &{}  \\
 +
{}  &{}  &{}  &{}  & 1  &{}  \\
 +
0  &{}  &{}  &{}  &{}  &- 1  \\
 +
\end{array}
 +
\right \|
 +
$$
  
The description of a reflection in an arbitrary simply-connected space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r080/r080510/r08051044.png" /> of constant curvature can be reduced to the description of linear reflections in the following way. Every such space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r080/r080510/r08051045.png" /> can be imbedded as a hypersurface in a real <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r080/r080510/r08051046.png" />-dimensional vector space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r080/r080510/r08051047.png" /> in such a way that the motions of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r080/r080510/r08051048.png" /> can be extended to linear transformations of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r080/r080510/r08051049.png" />. Moreover, in a suitable coordinate system in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r080/r080510/r08051050.png" /> the equations of the hypersurface can be written in the following way:
+
in a certain orthonormal basis, and conversely, every orthogonal transformation of $  V  ^ {n} $
 +
with this matrix in a certain orthonormal basis is a reflection in $  E  ^ {n} $.
 +
More generally, a linear transformation  $  \phi $
 +
of an arbitrary vector space $  W $
 +
over a field  $  k $,
 +
of characteristic other than 2, is called a linear reflection if  $  \phi  ^ {2} = \mathop{\rm Id} _ {W} $
 +
and if the rank of the transformation  $  \mathop{\rm Id} - \phi $
 +
is equal to $  1 $.
 +
In this case, the subspace  $  W _ {1} $
 +
of fixed vectors relative to  $  \phi $
 +
has codimension  $  1 $
 +
in $  W $,
 +
the subspace  $  W _ {-} 1 $
 +
of eigenvectors with eigenvalue  $  - 1 $
 +
has dimension  $  1 $
 +
and  $  W = W _ {1} \oplus W _ {-} 1 $.  
 +
If  $  \alpha $
 +
is a linear form on  $  W $
 +
such that  $  \alpha ( W) = 0 $
 +
when  $  w \in W _ {1} $,
 +
and if  $  h \in W _ {-} 1 $
 +
is an element such that $  \alpha ( h) = 2 $,  
 +
then  $  \phi $
 +
is defined by the formula
  
<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/r/r080/r080510/r08051051.png" /></td> </tr></table>
+
$$
 +
\phi w  = w - \alpha ( w) h,\  w \in W.
 +
$$
  
<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/r/r080/r080510/r08051052.png" /></td> </tr></table>
+
The description of a reflection in an arbitrary simply-connected space  $  X  ^ {n} $
 +
of constant curvature can be reduced to the description of linear reflections in the following way. Every such space  $  X  ^ {n} $
 +
can be imbedded as a hypersurface in a real  $  ( n+ 1) $-
 +
dimensional vector space  $  V  ^ {n+} 1 $
 +
in such a way that the motions of  $  X  ^ {n} $
 +
can be extended to linear transformations of  $  V  ^ {n+} 1 $.  
 +
Moreover, in a suitable coordinate system in  $  V  ^ {n+} 1 $
 +
the equations of the hypersurface can be written in the following way:
  
<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/r/r080/r080510/r08051053.png" /></td> </tr></table>
+
$$
 +
x _ {0}  ^ {2} + \dots + x _ {n}  ^ {2}  = 1 \ \
 +
\textrm{ for }  S  ^ {n} ;
 +
$$
  
Every hypersurface in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r080/r080510/r08051054.png" />, given this imbedding, is the intersection with <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r080/r080510/r08051055.png" /> of a certain <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r080/r080510/r08051056.png" />-dimensional subspace in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r080/r080510/r08051057.png" />, and every reflection in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r080/r080510/r08051058.png" /> is induced by a linear reflection in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r080/r080510/r08051059.png" />.
+
$$
 +
x _ {0= 1 \  \textrm{ for }  E  ^ {n} ;
 +
$$
  
If, in the definition of a linear reflection, the requirement that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r080/r080510/r08051060.png" /> is dropped, then the more general concept of a pseudo-reflection is obtained. If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r080/r080510/r08051061.png" /> is the field of complex numbers and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r080/r080510/r08051062.png" /> is a pseudo-reflection of finite order (not necessarily equal to 2), then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r080/r080510/r08051063.png" /> is called a unitary reflection. Every biholomorphic automorphism of finite order of a bounded symmetric domain in a complex space the set of fixed points of which has a complex codimension 1 is also called a unitary reflection.
+
$$
 +
x _ {0}  ^ {2} - \dots - x _ {n}  ^ {2}  =  1 \  \textrm{ and }
 +
\  x _ {0}  >  0 \  \textrm{ for }  \Lambda  ^ {n} .
 +
$$
 +
 
 +
Every hypersurface in  $  X  ^ {n} $,
 +
given this imbedding, is the intersection with  $  X  ^ {n} $
 +
of a certain  $  n $-
 +
dimensional subspace in  $  V  ^ {n+} 1 $,
 +
and every reflection in  $  X  ^ {n} $
 +
is induced by a linear reflection in  $  V  ^ {n+} 1 $.
 +
 
 +
If, in the definition of a linear reflection, the requirement that $  \phi  ^ {2} = \mathop{\rm Id} _ {W} $
 +
is dropped, then the more general concept of a pseudo-reflection is obtained. If $  k $
 +
is the field of complex numbers and $  \phi $
 +
is a pseudo-reflection of finite order (not necessarily equal to 2), then $  \phi $
 +
is called a unitary reflection. Every biholomorphic automorphism of finite order of a bounded symmetric domain in a complex space the set of fixed points of which has a complex codimension 1 is also called a unitary reflection.
  
 
See also [[Reflection group|Reflection group]].
 
See also [[Reflection group|Reflection group]].
Line 25: Line 114:
 
====References====
 
====References====
 
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  N. Bourbaki,  "Groupes et algèbres de Lie" , ''Eléments de mathématiques'' , Hermann  (1968)  pp. Chapts. 4–6</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  E.B. Vinberg,  "Discrete linear groups generated by reflections"  ''Math. USSR Izv.'' , '''35''' :  5  (1971)  pp. 1083–1119  ''Izv. Akad. Nauk SSSR Ser. Mat.'' , '''35''' :  5  (1971)  pp. 1072–1112</TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top">  E. Gottschling,  "Reflections in bounded symmetric domains"  ''Comm. Pure Appl. Math.'' , '''22'''  (1969)  pp. 693–714</TD></TR><TR><TD valign="top">[4]</TD> <TD valign="top">  B.A. Rozenfel'd,  "Non-Euclidean spaces" , Moscow  (1969)  (In Russian)</TD></TR></table>
 
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  N. Bourbaki,  "Groupes et algèbres de Lie" , ''Eléments de mathématiques'' , Hermann  (1968)  pp. Chapts. 4–6</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  E.B. Vinberg,  "Discrete linear groups generated by reflections"  ''Math. USSR Izv.'' , '''35''' :  5  (1971)  pp. 1083–1119  ''Izv. Akad. Nauk SSSR Ser. Mat.'' , '''35''' :  5  (1971)  pp. 1072–1112</TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top">  E. Gottschling,  "Reflections in bounded symmetric domains"  ''Comm. Pure Appl. Math.'' , '''22'''  (1969)  pp. 693–714</TD></TR><TR><TD valign="top">[4]</TD> <TD valign="top">  B.A. Rozenfel'd,  "Non-Euclidean spaces" , Moscow  (1969)  (In Russian)</TD></TR></table>
 
 
  
 
====Comments====
 
====Comments====

Latest revision as of 14:55, 7 June 2020


A mapping $ \sigma $ of an $ n $- dimensional simply-connected space $ X ^ {n} $ of constant curvature (i.e. of a Euclidean or affine space $ E ^ {n} $, a sphere $ S ^ {n} $ or a hyperbolic (Lobachevskii) space $ \Lambda ^ {n} $) the set of fixed points $ \Gamma $ of which is an $ ( n- 1) $- dimensional hyperplane. The set $ \Gamma $ is called the mirror of the mapping $ \sigma $; in other words, $ \sigma $ is a reflection in $ \Gamma $. Every reflection is uniquely defined by its mirror. The order (period) of a reflection in the group of all motions of $ X ^ {n} $ is equal to 2, i.e. $ \sigma ^ {2} = \mathop{\rm Id} _ {X ^ {n} } $.

The Euclidean or affine space $ E ^ {n} $ can be identified with the vector space $ V ^ {n} $ of its parallel translations. The mapping $ \sigma $ is then a linear orthogonal transformation of $ V ^ {n} $ with matrix

$$ \left \| \begin{array}{lllllr} 1 &{} &{} &{} &{} & 0 \\ {} &\cdot &{} &{} &{} &{} \\ {} &{} &\cdot &{} &{} &{} \\ {} &{} &{} &\cdot &{} &{} \\ {} &{} &{} &{} & 1 &{} \\ 0 &{} &{} &{} &{} &- 1 \\ \end{array} \right \| $$

in a certain orthonormal basis, and conversely, every orthogonal transformation of $ V ^ {n} $ with this matrix in a certain orthonormal basis is a reflection in $ E ^ {n} $. More generally, a linear transformation $ \phi $ of an arbitrary vector space $ W $ over a field $ k $, of characteristic other than 2, is called a linear reflection if $ \phi ^ {2} = \mathop{\rm Id} _ {W} $ and if the rank of the transformation $ \mathop{\rm Id} - \phi $ is equal to $ 1 $. In this case, the subspace $ W _ {1} $ of fixed vectors relative to $ \phi $ has codimension $ 1 $ in $ W $, the subspace $ W _ {-} 1 $ of eigenvectors with eigenvalue $ - 1 $ has dimension $ 1 $ and $ W = W _ {1} \oplus W _ {-} 1 $. If $ \alpha $ is a linear form on $ W $ such that $ \alpha ( W) = 0 $ when $ w \in W _ {1} $, and if $ h \in W _ {-} 1 $ is an element such that $ \alpha ( h) = 2 $, then $ \phi $ is defined by the formula

$$ \phi w = w - \alpha ( w) h,\ w \in W. $$

The description of a reflection in an arbitrary simply-connected space $ X ^ {n} $ of constant curvature can be reduced to the description of linear reflections in the following way. Every such space $ X ^ {n} $ can be imbedded as a hypersurface in a real $ ( n+ 1) $- dimensional vector space $ V ^ {n+} 1 $ in such a way that the motions of $ X ^ {n} $ can be extended to linear transformations of $ V ^ {n+} 1 $. Moreover, in a suitable coordinate system in $ V ^ {n+} 1 $ the equations of the hypersurface can be written in the following way:

$$ x _ {0} ^ {2} + \dots + x _ {n} ^ {2} = 1 \ \ \textrm{ for } S ^ {n} ; $$

$$ x _ {0} = 1 \ \textrm{ for } E ^ {n} ; $$

$$ x _ {0} ^ {2} - \dots - x _ {n} ^ {2} = 1 \ \textrm{ and } \ x _ {0} > 0 \ \textrm{ for } \Lambda ^ {n} . $$

Every hypersurface in $ X ^ {n} $, given this imbedding, is the intersection with $ X ^ {n} $ of a certain $ n $- dimensional subspace in $ V ^ {n+} 1 $, and every reflection in $ X ^ {n} $ is induced by a linear reflection in $ V ^ {n+} 1 $.

If, in the definition of a linear reflection, the requirement that $ \phi ^ {2} = \mathop{\rm Id} _ {W} $ is dropped, then the more general concept of a pseudo-reflection is obtained. If $ k $ is the field of complex numbers and $ \phi $ is a pseudo-reflection of finite order (not necessarily equal to 2), then $ \phi $ is called a unitary reflection. Every biholomorphic automorphism of finite order of a bounded symmetric domain in a complex space the set of fixed points of which has a complex codimension 1 is also called a unitary reflection.

See also Reflection group.

References

[1] N. Bourbaki, "Groupes et algèbres de Lie" , Eléments de mathématiques , Hermann (1968) pp. Chapts. 4–6
[2] E.B. Vinberg, "Discrete linear groups generated by reflections" Math. USSR Izv. , 35 : 5 (1971) pp. 1083–1119 Izv. Akad. Nauk SSSR Ser. Mat. , 35 : 5 (1971) pp. 1072–1112
[3] E. Gottschling, "Reflections in bounded symmetric domains" Comm. Pure Appl. Math. , 22 (1969) pp. 693–714
[4] B.A. Rozenfel'd, "Non-Euclidean spaces" , Moscow (1969) (In Russian)

Comments

The spelling reflexion also occurs in the literature.

A basic fact is that the reflections generate the orthogonal group; see [a2], Sects. 8.12.12, 13.7.12.

References

[a1] B.A. [B.A. Rozenfel'd] Rosenfel'd, "A history of non-euclidean geometry" , Springer (1988) (Translated from Russian)
[a2] M. Berger, "Geometry" , 1–2 , Springer (1987) (Translated from French)
[a3] H.S.M. Coxeter, "Introduction to geometry" , Wiley (1963)
[a4] M. Greenberg, "Euclidean and non-euclidean geometry" , Freeman (1980)
[a5] B. Artmann, "Lineare Algebra" , Birkhäuser (1986)
[a6] P.R. Halmos, "Finite-dimensional vector spaces" , v. Nostrand (1958)
How to Cite This Entry:
Reflection. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Reflection&oldid=15842
This article was adapted from an original article by V.L. Popov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article