  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}.
