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Quaternion Kaehlerian manifold, definition

  1. Apr 12, 2012 #1
    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!
     
  2. jcsd
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