The complexifications of pseudo-Riemannian manifolds and anti-Kaehler geometry

TL;DR

This paper explores the complexification of pseudo-Riemannian manifolds, introducing anti-Kaehler metrics and analyzing the geometric relations and properties of these complexified spaces, especially in the context of homogeneous spaces and isometric actions.

Contribution

It defines complexifications of real analytic maps and pseudo-Riemannian manifolds, introduces anti-Kaehler metrics, and investigates their relations and geometric properties, including curvature-adapted isoparametric submanifolds.

Findings

Two types of complexifications are related through anti-Kaehler metrics.

Most principal orbits form curvature-adapted isoparametric submanifolds.

The complexified spaces exhibit rich geometric structures with applications to symmetric spaces.

Abstract

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.