On mappings preserving pseudo-Euclidean volume. (Russian) Zbl 0565.51008

Astrakov, S. N.

Sib. Mat. Zh. 26, No. 1(149), 4-11 (1985).

In the pseudo-Euclidean space \({}^ kE_ n(2<n\), \(1\leq k\leq n-1)\), for each m-simplex \(\bar A=A_ 0A_ 1...A_ m\) the pseudo-Euclidean m- volume \(V_ m(\bar A)\) is defined as a complex number \(\rho\) such as \((m!\rho)^ 2\) is the Gramm’s determinant of the vectors \(A_ 0A_ i.\)
The following Theorem is geometrically proved: For each natural number \(m(1<m\leq n)\) and each complex number \(\rho\), if for a map \(f: {}^ kE_ n\to^ kE_ n\) the condition \(V_ m(\bar A)=\rho \Leftrightarrow V_ m(f(\bar A))=\rho\) holds, then f is an affine transformation.
This very natural Theorem seems to have important theoretical consequences.

Reviewer: D.Brânzei

MSC:

51M10	Hyperbolic and elliptic geometries (general) and generalizations
51M25	Length, area and volume in real or complex geometry

Keywords:

m-simplex; pseudo-Euclidean m-volume; affine transformation