Generalized Planar Curves

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’s classical results in the realm of the almost quaternionic geometry. On the way we 1 2 JAROSLAV HRDINA AND JAN SLOVÁK simplify, recover, and extend results on generalized planar mappings, explain results from [6], and finally we show that morphisms of almost quaternionic geometries are just those diffeomorphisms which leave invariant the class of all unparameterized geodesics of Weyl connections. Acknowledgments. The authors have been supported by GACR, grants Nr. 201/05/H005 and 201/05/2117, respectively. The second author also gratefully acknowledges support from the Royal Society of New Zealand via Marsden Grant no. 02-UOA-108 during writing essential part of this paper. The authors are also grateful to Josef Mikeš for numerous discussions and to Dmitri Alekseevsky for helpful comments. 1. Motivation and background on quaternionic geometry There are many equivalent definitions of almost quaternionic geometry to be found in the literature. Let us start with the following one: Definition 1.1. Let M be a smooth manifold of dimension 4n. An almost hypercomplex structure on M is a triple (I, J,K) of smooth affinors in Γ(T M ⊗ TM) satisfying I = J = −E, K = I ◦ J = −J ◦ I where E = idTM . An almost quaternionic structure is a rank four subbundle Q ⊂ T M ⊗ TM locally generated by the identity E and a hypercomplex structure. An almost complex geometry on a 2m–dimensional manifold M is given by the choice of the affinor J satisfying J = −E. Let us observe, that such a J is uniquely determined within the rank two subbundle 〈E, J〉 ⊂ TM , up to its sign. Indeed, if Ĵ = aE+bJ , then the condition Ĵ = −E implies a = 0 and b = ±1. Thus we may view the almost quaternionic geometry as a straightforward generalization of this case. Here, a similar simple computation reveals that the rank three subbundle 〈I, J,K〉 is invariant of the choice of the generators and this is the definition we may find in [1]. More explicitly, different choices will always satisfy Î = aI + bJ + cK with a + b + d = 1, and similarly for J and K. Let us also remark that the 4–dimensional almost quaternionic geometry coincides with 4-dimensional conformal Riemannian geometries. 1.2. The frame bundles. Equivalently, we can define an almost quaternionic structure Q onM as a reduction of the linear frame bundle P M to an appropriate structure group, i.e. as a G–structure with the structure group of all automorphisms preserving the subbundle Q. We may GENERALIZED PLANAR CURVES AND QUATERNIONIC GEOMETRY 3 view such frames as linear mappings TxM → H n which carry over the multiplications by i, j, k ∈ H onto some of the possible choices for I, J,K. Thus, a further reduction to a fixed hypercomplex structure leads to the structure group GL(n,H) of all quaternionic linear mappings on H. Additionally, we have to allow morphisms which do not leave the affinors I, J , K invariant but change them within the subbundle Q. As well known, the resulting group is (1) G0 = GL(n,H)×Z2 Sp(1) where Sp(1) are the unit quaternions in GL(1,H), see e.g. [8]. We shall write G0 ⊂ P M for this principal G0–bundle defining our structure. The simplest example of such a structure is well understood as the homogeneous space PnH = G/P where G0 ⊂ P are the subgroups in G = SL(n+ 1,H) G0 = {(