Almost Coquaternion Structure

Our aim is to define and study a structure for some $(4n+3)$-dimensional manifolds which is named almost coquaternion structure. This structure is composed of three almost cocomplex structures $(ϕ_a, ξ_a, η_a)$, $a = 1,2,3$, which satisfy some relations and may be considered as analogous to the almost quaternion structure for $(4n+4)$-dimensional manifolds. The sphere $S^{4n+3}$ is a typical example of differentiable manifold which admits an almost coquaternion structure $(ϕ_a, ξ_a, η_a)$, $a = 1,2,3$. Using the 1-forms $η_a$ of the almost coquaternion structure of the sphere $S^{4n+3}$, C. Teleman defined and studied on $S^{4n+3}$ a nonholonomic manifold $V^{4n}_{4n+3}$ whose Riemannian metric is the one of a symmetric space of E. Cartan. Keeping in mind Teleman's idea, we observed that on an almost coquaternion manifold a nonholonomic (holonomic) manifold of codimension three can be defined and studied by nonintegrable (completely integrable) Pfaff's system $η_1 = 0$, $η_2 = 0$, $η_3 = 0$.

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