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## Main Question or Discussion Point

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!

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!