Reflection
A mapping
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}{cccc} 1 &{} &{} & 0 \\ {} &\ddots &{} &{} \\ {} &{} & 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) |
Reflection. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Reflection&oldid=52262