# Quaternion Kaehlerian manifold, definition

1. Apr 12, 2012

### DraganaMaric

Hello,
I am reading the paper of S. Ishihara, Quaternion Kaehlerian manifolds, I need it for understanding of totally complex submanifolds in quaternion Kaehlerian manifolds.

I am afraid that I don't understand well the definition of quaternion Kaehler manifold, that is my question is the next:
if (M,g,V) is an almost quaternion metric manifold and if the Riemannian connection $\nabla$ of M satisfies the condition:
a) if $\phi$ is a cross-section of the bundle V, then $\nabla\phi$ is also a cross-section of V,
then we say that (M,g,V) is a quaternion Kaehlerian manifold.
What I don't understand is why condition a) is equivalent to the next condition:
b) \nablaF = rG - qH
\nablaG = -rF + pH
\nablaH = qF - pG,
where {F, G, H} is a canonical local base of V in U.