A Remark on Almost-Quaternlon Substructures on the Sphere

Abstract

In [4] T. Onder has solved the existence problem of almostquaternion ^-substructures on the ^-sphere S" for all n and k except for n = l (mod 4)^5 and k=(n -l)/4. The purpose of this note is to solve it for this exceptional case. Let n and k be positive integers with 4:k^n. An almost-quaternion -substructure on an orientable n-manifold M is defined to be a reduction of the structural group of the tangent bundleit follows that S n has an almost-quaternion ^-substructure if and only if the associated fib ration SO(n)/Sp(k)Communicated by N.