Equivariant mappings from vector product into \(G\)-space of vectors and \(\varepsilon \)-vectors with \(G=O(n,1,\mathbb {R})\). (English) Zbl 1108.53009

Glanc, Barbara; Misiak, Aleksander; Stȩpień, Zofia

Math. Bohem. 130, No. 3, 265-275 (2005).

Summary: In this note all vectors and \(\varepsilon \)-vectors of a system of \(m\leq n\) linearly independent contravariant vectors in the \(n\)-dimensional pseudo-Euclidean geometry of index one are determined. The problem is resolved by finding the general solution of the functional equation \(F( A{\underset {1} u}, A{\underset {2} u},\dots ,A{\underset {m} u}) =( \text{det} A)^{\lambda }\cdot A\cdot F( {\underset {1} u},{\underset {2} u},\dots , {\underset {m} u})\) with \(\lambda =0\) and \(\lambda =1\), for an arbitrary pseudo-orthogonal matrix \(A\) of index one and given vectors \( {\underset {1} u},{\underset {2} u},\dots ,{\underset {m} u}.\)

MSC:

53A55

Differential invariants (local theory), geometric objects

Keywords:

equivariant map; pseudo-Euclidean geometry; functional equation