(Para-)Hermitian and (para-)Kähler Submanifolds of a para-quaternionic Kähler manifold

Abstract: On a para-quaternionic K\"ahler manifold $(\widetilde M^{4n},Q,\widetilde g)$, which is first of all a pseudo-Riemannian manifold, a natural definition of (almost) K\"ahler and (almost) para-K\"ahler submanifold $(M^{2m},\mathcal{J},g)$ can be given where $\mathcal{J}=J_1|_M$ is a (para-)complex structure on $M$ which is the restriction of a section $J_1$ of the para-quaternionic bundle $Q$. In this paper, we extend to such a submanifold $M$ most of the results proved by Alekseevsky and Marchiafava, 2001, where Hermitian and K\"ahler submanifolds of a quaternionic K\"ahler manifold have been studied. Conditions for the integrability of an almost (para-)Hermitian structure on $M$ are given. Assuming that the scalar curvature of $\widetilde M$ is non zero, we show that any almost (para-)K\"ahler submanifold is (para-)K\"ahler and moreover that $M$ is (para-)K\"ahler iff it is totally (para-)complex. Considering totally (para-)complex submanifolds of maximal dimension $2n$, we identify the second fundamental form $h$ of $M$ with a tensor $C= J_2 \circ h \in TM\otimes S^2T^*M$ where $J_2 \in Q$ is a compatible para-complex structure anticommuting with $J_1$. When $\widetilde M^{4n}$ is a symmetric manifold the condition for a (para-)K\"ahler submanifold $M^{2n}$ to be locally symmetric is given. In the case when $\widetilde M$ is a para-quaternionic space form, it is shown, by using Gauss and Ricci equations, that a (para-)K\"ahler submanifold $M^{2n}$ is curvature invariant. Moreover it is a locally symmetric Hermitian submanifold iff the $\mathfrak u(n)$-valued 2-form $[C,C]$ is parallel. %$$[C,C]: X\wedge Y \mapsto [C_X,C_Y], \qquad \quad X,Y \in TM$$ %(which satisfies the first and the second Bianchi identity) is parallel. Finally a characterization of \textit{parallel} K\"ahler and para-K\"ahler submanifold of maximal dimension is given.