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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!

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  • $\begingroup$ I edited the LaTeX to make it pretty. $\endgroup$
    – David Roberts ♦
    Commented Apr 12, 2012 at 7:43
  • $\begingroup$ Have you tried looking at other papers or books that explain what a quaternionic Kahler manifold is? There are different ways to describe it, and you might find one of the other explanations easier to understand. Once you do, it should be easy to match this explanation with the one you prefer. $\endgroup$
    – Deane Yang
    Commented Apr 12, 2012 at 8:04
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    $\begingroup$ In your notation, $V$ is the rank $3$ fundamental bundle which is a subbundle of $\Lambda^2T\M$. A `canonical base' here is a local frame for $V$ consisting of anticommuting almost complex structures. Your three equations in b) say that if you differentiate a section of $V$ you get another section of $V$, so I guess you're asking why the coefficients form a $3 \times 3$ skew-symmetric matrix of $1$-forms. You can prove this easily by taking inner products, e.g. of $\nabla F$ with $G$. This corresponds to the $\mathfrak{sp}_1 = \mathfrak{so}_3$ part of the curvature. A good reference is Besse. $\endgroup$
    – Paul Reynolds
    Commented Apr 12, 2012 at 10:12
  • $\begingroup$ Is there some reference where I can find the examples of totally complex submanifolds in, for example, complex space $C^{n}$? And also holomorphic submanifolds. I am trying to find some example of submanifold that is holomorphic but not totally complex in complex space. Since that is not my field, I am not quite sure how will I define almost complex structures $K$ and $L$. They should be orthogonal to naturally defined structure $J$ which is parallel, and map tangent bundle of submanifold to normal bundle of submanifold. $\endgroup$
    – Mirjana
    Commented Apr 17, 2012 at 9:16

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