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Curves in pseudo-Galilean geometry. (English) Zbl 0932.53005

A pseudo-Galilean space \(G_3^1\) is a three-dimensional affine space \(A_3\) having as an absolute an ordered triple \(\{ \omega ,f, I\}\) where \(\omega\) is the ideal plane, \(f\) a line in \(\omega\) and \(I\) is a fixed hyperbolic involution of the points of \(f\). On \(G_3^1\), a six-parameter group \(\overline{B}_6\) acts as the motion group. The author develops the theory of spacelike and timelike curves in \(G_3^1\) by constructing a pseudo-Galilean Frenet frame.
As the main theorem, the author shows the fundamental theorem for pseudo-Galilean curves, which differs crucially from the analogous theorems in Euclidean, isotropic or Galilean space since uniqueness is not fulfilled. The second theorem is concerned with the admissible spacelike (timelike) curves with the same natural equations.

MSC:

53A04 Curves in Euclidean and related spaces
53A40 Other special differential geometries
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