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

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
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