Almost Coquaternion Structure

Abstract

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 $({\varphi }_a,{\xi }_a,{\eta }_a)$, $a=1,2,3$, 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 $({\varphi }_a,{\xi }_a,{\eta }_a)$, $a=1,2,3$. Using the 1-forms ${\eta }_a$ of the almost coquaternion structure of the sphere $S^{4n+3}$, C. Teleman defined and studied on $S^{4n+3}$ a nonholonomic manifold $V_{4n+3}^{4n}$ 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 ${\eta }_1=0$, ${\eta }_2=0$, ${\eta }_3=0$.