  In this paper, we first define the complexification of a real analytic map
between real analytic Koszul manifolds and show that the complexified map is
the holomorphic extension of the original map. Next we define an anti-Kaehler
metric compatible with the adapted complex structure on the complexification of
a real analytic pseudo-Riemannian manifold. In particular, for a
pseudo-Riemannian homogeneous space, we define another complexification and a
(complete) anti-Kaehler metric on the complexification. One of main purposes of
this paper is to find the interesting relation between these two
complexifications (equipped with the anti-Kaehler metrics) of a
pseudo-Riemannian homogeneous space. Another of main purposes of this paper is
to show that almost all principal orbits of some isometric action on the first
complexification (equipped with the anti-Kaehler metric) of a semi-simple
pseudo-Riemannian symmetric space are curvature-adapted isoparametric
submanifolds with flat section in the sense of this paper.
