A Splitting Result for Real Submanifolds of a Kahler Manifold

Let (Z, omega) be a connected Kahler manifold with an holomorphic action of the complex reductive Lie group 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-C and let M be a G-invariant connected submanifold of Z. Let x is an element of M. If G is a real form of U-C, we investigate conditions such that G . x compact implies 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) generalizing a result proved in Gori and Podesta (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).