On Some Geometric Properties of the Sphere Sn

It is known that the sphere 2n ? or 6n ? . In this paper, we show that the sphere S admits an almost complex structure only when n Abstract S is a space of constant sectional curvature and using the results of T. Sato in [4], we determine the scalar curvature and the *-scalar curvature of S . We shall also prove that S is a non-K?hler nearly K?hler manifold using the Levi-Civita connection on 6 S defined by H. Hashimoto and K. Sekigawa [3]. In [2], A. Gray and L. Hervella defined sixteen classes of almost Hermitian manifolds. We shall 6 n 6 define quasi-Hermitian, a class of almost Hermitian manifolds and partially characterize almost Hermitian manifolds that belong to this class. Finally, under certain conditions, we shall show the sphere S is quasi-Hermitian.