A Splitting Result for Real Submanifolds of a Kähler Manifold

Let $$(Z,\omega )$$ be a connected Kähler 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 Gori and Podestà (Ann Global Anal Geom 26: 315–318, 2004), see also Bedulli and Gori (Results Math 47: 194–198, 2005), Biliotti (Bull Belg Math Soc Simon Stevin 16: 107–116 2009).