Abstract: Let M be an almost split quaternionic manifold on
which its almost split quaternionic structure is defined by a three
dimensional subbundle V of ( T M) T (M)
*
Ôèù and
{F,G,H} be a local basis for V . Suppose that the (global)
(1, 2) tensor field defined[V ,V ]is defined by
[V,V ] = [F,F]+[G,G] + [H,H], where [,] denotes
the Nijenhuis bracket. In ref. [7], for the almost split-hypercomplex
structureH = J α,α =1,2,3, and the Obata
connection ÔêçH
vanishes if and only if H is split-hypercomplex.
In this study, we give a prof, in particular, prove that if either
M is a split quaternionic Kaehler manifold, or if M is a splitcomplex
manifold with almost split-complex structure F , then the
vanishing [V ,V ] is equivalent to that of all the Nijenhuis brackets
of {F,G,H}. It follows that the bundle V is trivial if and only if
[V ,V ] = 0 .
Abstract: In this work some characterizations of semi Riemannian curvature tensor on almost split quaternion Kaehler manifolds and some characterizations of Ricci tensor on almost split quaternion Kaehler manifolds are given.