# Reflection

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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.