A splitting result for real submanifolds of a Kahler manifold

TL;DR

This paper studies the conditions under which the orbits of certain real forms in a Kahler manifold are compact, and establishes a splitting theorem for real submanifolds, extending previous results in the field.

Contribution

It provides new criteria for orbit compactness and a splitting theorem for real submanifolds in Kahler manifolds, generalizing earlier work.

Findings

Conditions linking compactness of $G$-orbits to $U^{ ext{C}}$-orbits.

Characterization of $G$-invariant submanifolds with constant gradient map norm.

A generalized splitting theorem for real submanifolds in Kahler manifolds.

Abstract

Let $(Z,\omega)$ be a connected Kahler manifold with an holomorphic action of the complex reductive Lie group $U^{\mathbb C}$, where $U$ is a compact connected Lie group acting in a hamiltonian fashion. Let $G$ be a closed compatible Lie group of $U^{\mathbb C}$ and let $M$ be a $G$-invariant connected submanifold of $Z$. Let $x\in M$. If $G$ is a real form of $U^{\mathbb C}$, we investigate conditions such that $G\cdot x$ compact implies $U^{\mathbb C} \cdot x$ is compact as well. The vice-versa is also investigated. We also characterize $G$-invariant real submanifolds such that the norm square of the gradient map is constant. As an application, we prove a splitting result for real connected submanifolds of $(Z,\omega)$ generalizing a result proved in \cite{pg}, see also \cite{bg,bs}.