Almost Coquaternion Structure

TL;DR

This paper introduces the concept of almost coquaternion structures on (4n+3)-dimensional manifolds, generalizing quaternion structures, and explores their properties and examples like spheres, extending Teleman's nonholonomic manifold ideas.

Contribution

It defines almost coquaternion structures, relates them to nonholonomic manifolds, and provides examples such as spheres, expanding the understanding of geometric structures in differential geometry.

Findings

Almost coquaternion structures generalize quaternion structures to (4n+3)-dimensional manifolds.

The sphere $S^{4n+3}$ admits an almost coquaternion structure.

Nonholonomic manifolds can be constructed using these structures, inspired by Teleman's work.

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 $(\phi_a, \xi_a, \eta_a)$, $a = 1,2,3$, 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 $(\phi_a, \xi_a, \eta_a)$, $a = 1,2,3$. 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 $V^{4n}_{4n+3}$ 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 $\eta_1 = 0$, $\eta_2 = 0$, $\eta_3 = 0$.