Quaternion Kaehlerian manifold, definition

[DraganaMaric]

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!

 

[jvicens]

Hello, thank you for your question. I will do my best to help clarify the definition of a quaternion Kaehler manifold for you.

First, let's start with the definition of an almost quaternion metric manifold. This is a Riemannian manifold (M,g) equipped with a rank-3 vector bundle V over M, called the quaternionic bundle, with a metric h on V that is compatible with the Riemannian metric g on M. This means that for any two vector fields X and Y on M and any two cross-sections F and G of V, we have h(F,G) = g(X,Y).

Now, let's move on to the definition of a quaternion Kaehlerian manifold. This is an almost quaternion metric manifold (M,g,V) where the Riemannian connection $\nabla$ satisfies the condition you mentioned, a). This condition basically means that the Riemannian connection preserves the quaternionic structure of the bundle V.

To understand why this condition is equivalent to the conditions in b), we need to look at the structure of the bundle V. In an almost quaternion metric manifold, the bundle V is always locally isomorphic to the tangent bundle of M. This means that there exists a local base {F, G, H} of V in a neighborhood U of M, such that each vector field in this base is a cross-section of V.

Now, let's look at the conditions in b). These conditions basically say that the Riemannian connection $\nabla$ applied to the vector fields in the base {F, G, H} results in a linear combination of those same vector fields. This is exactly what we want for a quaternionic structure - the Riemannian connection should preserve the quaternionic structure of the bundle V.

To summarize, the condition a) is equivalent to the conditions in b) because it ensures that the Riemannian connection preserves the quaternionic structure of the bundle V. I hope this helps clarify the definition for you. Let me know if you have any further questions.