Generalized Planar Curves and Quaternionic Geometry

Motivated by the analogies between the projective and the almost quaternionic geometries, we first study the generalized planar curves and mappings. We follow, recover, and extend the classical approach, see e.g. [9, 10]. Then we exploit the impact of the general results in the almost quaternionic geometry. In particular we show, that the natural class of H–planar curves coincides with the class of all geodesics of the so called Weyl connections and preserving this class turns out to be the necessary and sufficient condition on diffeomorphisms to become morphisms of almost quaternionic geometries. Various concepts generalizing geodesics of affine connections have been studied for almost quaternionic and similar geometries. Let us point out the generalized geodesics defined via generalizations of normal coordinates, cf. [2] and [3], or more recent [4, 11]. Another class of curves was studied in [10] for the hypercomplex structures with additional linear connections. The latter authors called a curve c quaternionic planar if the parallel transport of each of its tangent vectors ċ(t0) along c was quaternionic colinear with the tangent field ċ to the curve. Yet another natural class of curves is given by the set of all unparameterized geodesics of the so called Weyl connections, i.e. the connections compatible with the almost quaternionic structure with normalized minimal torsion. The latter connections have remarkably similar properties for all parabolic geometries, cf. [3], and so their name has been borrowed from the conformal case. In the setting of almost quaternionic structures, they were studied first in [7] and so they are also called Oproiu connections, see [1]. The first author showed in [6] that actually the concept of quaternionic planar curves was well defined for the almost quaternionic geometries and their Weyl connections. Moreover, it did not depend on the choice of a particular Weyl connection and it turned out that the quaternionic planar curves were just all unparameterized geodesics of all Weyl connections. The aim of this paper is to find further analogies of Mikeš’ classical results in the realm of the almost quaternionic geometry. On the way we