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.

Thank you in advance!

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) $\nabla F = rG - qH$;

$\nabla G = -rF + pH$;

$\nabla H = qF - pG$,

where $\{F, G, H\}$ is a canonical local base of $V$ in $U$.

Thank you in advance!