Totally complex submanfiolds in quaternionic Kaehlerian manifolds Hello,
I would like to know some examples of totally complex submanifolds in quaternionic Kaehlerian manifolds, is there any references in which I could find them? So far I could not find. 
Also, the interesting example would be a holomorphic submanifold that is not totally complex in quaternionic Kaehlerian manifolds.
Thank you in advance!
 A: These examples are easy to come by:  Let $Q\to M$ be the canonical twistor bundle over the Q-K manifold $M^{4n}$.  Then $Q$ is a holomorphic contact manifold of complex dimension $2n{+}1$.  Now let $L\subset Q$ be any Legendrian (or sub-Legendrian) holomorphic submanifold and project it into $M$.  This will be a manifold of the desired type.
For example, if $M$ is quaternionic projective space, then $Q$ is a complex projective space, and there are many algebraic Legendrian submanifolds.
A: I am not sure that projecting a legendrian submanifold  in $Q$  to $M$  you get again a submanifold, many singularities can occur. For example if you take one of the simplest inhomogeneous Legendrian submanifolds considered by Buczynski in "HYPERPLANE SECTIONS OF LEGENDRIAN SUBVARIETIES", (if I remember correctly) or in Landsberg Manivel "Legendrian varieties", it seems to me that the projection is not smooth.
Several examples of Totally complex submanifolds can be found in:
M. Takeuchi: Totally complex submanifolds of quaternionic symmetric spaces, Japan. J.
Math., (N.S.) 12 (1986) 161–189.
K. Tsukada: Parallel submanifolds in a Quaternion projective space, Osaka J. Math 22
(1985), 187–241.
K. Tsukada: Einstein-K ̈ahler submanifolds in a quaternion projective space, Bull. London
Math. Soc. 36 (2004), 527–536.
Bedulli-Gori-podestà "Maximal totally complex submanifolds of $\mathbb{H}\mathbb{P}^n$: homogeneity and normal holonomy" Bullettin of LMS 41 (2009) 10029--1040\
A: I think this paper is also useful: arXiv:math/0308283 Complex Forms of Quaternionic Symmetric Spaces by Joseph A. Wolf. There are also several papers by Alekseevsky and Marchiafava.
Liviu
