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A kinematic mapping for projective and affine motions and some applications. (English) Zbl 0956.53012

Dillen, F. (ed.) et al., Geometry and topology of submanifolds, VIII. Proceedings of the international meeting on geometry of submanifolds, Brussels, Belgium, July 13-14, 1995 and Nordfjordeid, Norway, July 18-August 7, 1995. Singapore: World Scientific. 292-301 (1996).
Introduction: The kinematic mapping \(\Gamma\) for projective motions defined in [the author, Abh. Math. Semin. Univ. Hamb. 63, 177-196 (1993; Zbl 0791.53014)] is a generalization of the mapping which was defined for the group of Euclidean and equiform transformations in the plane by W. Wunderlich [Z. Angew. Math. Mech. 66, 421-428 (1986; Zbl 0596.70001)] and O. Bottema and G. R. Veldkamp [Tagungsber. Oberwolfach 17 (1982)], respectively. The mapping \(\Gamma\) assigns a point in an \(m\)-dimensional projective space \(P^m\), called kinematic image space or matrix space, to a collineation from an \(n\)-dimensional projective space \(P^n\) to an \(n\)-dimensional projective space \(\overline{P^n}\). The main theorem says that the paths of points are images of the kinematic image of the motion under different linear mappings, so that these mappings depend on the point in the moving system only.
The set of images of singular collineations with rank \(r\) at most is called rank-\(r\)-manifold. Since these manifolds are important for the following considerations they will be discussed in detail. A characterization of all maximal subspaces on the rank-\(n\)-manifold for dimensions \(n= 2,3\) is presented here the first time.
The kinematic mapping \(\Gamma\) and its special cases for subgroups such as affine or Euclidean motions have turned out to be a good tool to characterize Darboux motions and convex motions.
For the entire collection see [Zbl 0901.00043].

MSC:

53A17 Differential geometric aspects in kinematics
53A20 Projective differential geometry
53A15 Affine differential geometry
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