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Let    $$(Z,\omega )$$  ( Z , ω ) be a connected Kähler manifold with an holomorphic action of the complex reductive Lie group    $$U^\mathbb {C}$$  U C , where  U  is a compact connected Lie group acting in a hamiltonian fashion. Let  G  be a closed compatible Lie group of    $$U^\mathbb {C}$$  U C and let  M  be a  G -invariant connected submanifold of  Z . Let    $$x\in M$$  x ∈ M . If  G  is a real form of    $$U^\mathbb {C}$$  U C , we investigate conditions such that    $$G\cdot x$$  G · x compact implies    $$U^\mathbb {C}\cdot x$$  U C · 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    $$(Z,\omega )$$  ( Z , ω ) 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).

A Splitting Result for Real Submanifolds of a Kähler Manifold