On weak and strict relatives K\"ahler manifolds

TL;DR

This paper investigates the relationships between K"ahler manifolds that share submanifolds, establishing conditions under which weak relatives are also relatives, and introduces the concept of strict relatives with illustrative examples.

Contribution

It proves that weak relatives of projective K"ahler manifolds are actually relatives and introduces the new concept of strict relatives with examples.

Findings

Weak relatives of projective K"ahler manifolds are relatives.

Introduction of strict relatives K"ahler manifolds.

Several nontrivial examples of strict relatives.

Abstract

We study K\"ahler manifolds that are (weak) relatives, that is, K\"ahler manifolds which share a (locally isometric) submanifold. In particular, we prove that if two K\"ahler manifolds are weak relatives and one of them is projective, then they are relatives. Moreover, we introduce the notion of strict relatives K\"ahler manifolds and provide several nontrivial examples.