# Quaternion Kaehlerian manifold, definition

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 $U$.

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. – Paul Reynolds Apr 12 '12 at 10:12
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. – Mirjana Apr 17 '12 at 9:16