Para-Sasakian \(\phi -\)symmetric spaces

We study the Boothby–Wang fibration of para-Sasakian manifolds and introduce the class of para-Sasakian \(\phi \)-symmetric spaces, canonically fibering over para-Hermitian symmetric spaces. We remark that in contrast to the Hermitian setting the center of the isotropy group of a simple para-Hermitian symmetric space G/H can be either one- or two-dimensional, and prove that the associated metric is not necessarily the G-invariant extension of the Killing form of G. Using the Boothby–Wang fibration and the classification of semisimple para-Hermitian symmetric spaces, we explicitly construct semisimple para-Sasakian \(\phi \)-symmetric spaces fibering over semisimple para-Hermitian symmetric spaces. We provide moreover an example of non-semisimple para-Sasakian \(\phi \)-symmetric space.