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Let $(Z,\omega)$ be a connected Kahler manifold with an holomorphic action ofthe complex reductive Lie group $U^{\mathbb C}$, where $U$ is a compactconnected Lie group acting in a hamiltonian fashion. Let $G$ be a closedcompatible Lie group of $U^{\mathbb C}$ and let $M$ be a $G$-invariantconnected 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 alsoinvestigated. We also characterize $G$-invariant real submanifolds such thatthe norm square of the gradient map is constant. As an application, we prove asplitting result for real connected submanifolds of $(Z,\omega)$ generalizing aresult proved in \cite{pg}, see also \cite{bg,bs}.

A splitting result for real submanifolds of a Kahler manifold