On purely real surfaces in Kaehler surfaces and Lorentz surfaces in Lorentzian Kaehler surfaces

TL;DR

This paper surveys recent results on purely real surfaces in Kaehler and Lorentzian Kaehler surfaces, focusing on their geometric properties and classifications within indefinite Kaehler manifolds.

Contribution

It provides a comprehensive overview of recent advances in the study of purely real surfaces and Lorentz surfaces in indefinite Kaehler manifolds.

Findings

Classification results for purely real surfaces in Kaehler surfaces

Characterization of Lorentz surfaces in Lorentzian Kaehler surfaces

Summary of recent geometric properties and theorems

Abstract

An immersion $\phi \colon M \to \tilde M$ of a manifold $M$ into an indefinite Kaehler manifold $\tilde M$ is called purely real if the almost complex structure $J$ on $\tilde M$ carries the tangent bundle of $M$ into a transversal bundle. In this article we survey some recent results on purely real surfaces in Kaehler surfaces as well as on Lorentz surfaces in Lorentzian Kaehler surfaces.