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

Conditions for the integrability of an almost (para-)Hermitian structure on M are given. Assuming that the scalar curvature of M˜ is non-zero, we show that any almost (para-)Kähler submanifold is (para-)Kähler respectively and moreover that M is (para-)Kähler 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=J2∘h∈TM⊗S2T⁎M where J2∈Q is a compatible para-complex structure anticommuting with J1. This tensor, at any point x∈M, belongs to the first prolongation SJ(1) of the space SJ⊂EndTxM of symmetric endomorphisms anticommuting with J. When M˜4n is a symmetric manifold the condition for a (para-)Kähler submanifold M2n to be locally symmetric is given. In the case when M˜ is a para-quaternionic space form, it is shown, by using Gauss and Ricci equations, that a (para-)Kähler submanifold M2n is curvature invariant. Moreover it is a locally symmetric Hermitian submanifold iff the u(n)-valued 2-form [C,C] is parallel. Finally a characterization of parallel Kähler and para-Kähler submanifolds of maximal dimension is given.