A Splitting Result for Real Submanifolds of a Kähler Manifold

Abstract

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 xin M. If G is a real form of U^mathbb {C}, we investigate conditions such that Gcdot 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).