Regularity criteria

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)