Toric para-Kaehler-Einstein manifolds immersed in para-Kaehler space forms

TL;DR

This paper classifies mutually non-isometric toric para-Kaehler-Einstein manifolds immersed in para-Kaehler space forms, extending classical results from Kaehler to para-Kaehler geometry.

Contribution

It provides a complete list of such manifolds in the para-Kaehler setting, addressing a longstanding problem in differential geometry.

Findings

Classified all mutually non-isometric toric para-Kaehler-Einstein manifolds

Extended classical Kaehler-Einstein classification to para-Kaehler case

Identified explicit conditions for immersions into para-Kaehler space forms

Abstract

A classical and long-staying problem addressed, among others, by Calabi and Chern, is that to find a complete list of mutually non-isometric Kaehler-Einstein manifolds immersed in a finite-dimensional Kaehler space form. We address the same problem in the para-Kaehler context and, then, we find a list of mutually non-isometric toric para-Kaehler manifolds analytically immersed in a finite-dimensional para-Kaehler space form