Generalized planar curves and quaternionic geometry. (English) Zbl 1097.53008

After giving some equivalent definitions of (almost) quaternionic manifolds \(M\), and Weyl connections on them, the authors study generalized planar curves and mappings. In particular, if \(M\) is equipped with an \(A\)-structure (i.e. an \(\ell\)-dimensional vector subbundle in \(T^*M\otimes TM\), such that for each \(x\in M\) the identity affinor \(\text{id}_{TM}\) restricted to \(T_xM\) belongs to \(A_xM\subset T_x^*M\otimes T_xM\)), the definition of an \(A\)-planar curve is given, as well as the notion of a diffeomorphism between quaternionic manifolds equipped with an \(A\)-structure.
The main result is that the natural class of \(\mathbb H\)-planar curves coincides with the class of all geodesics of the Weyl connections. Preserving this class turns out to be the necessary and sufficient condition on diffeomorphisms to become morphisms of almost quaternionic geometries.


53A40 Other special differential geometries
53C22 Geodesics in global differential geometry
58E10 Variational problems in applications to the theory of geodesics (problems in one independent variable)
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