A Splitting Result for Real Submanifolds of a Kähler Manifold

Abstract

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).

Projective representations of real reductive Lie groups and the gradient map

“…By Borel-Weyl Theorem U C • v is the unique compact orbit of the U C -action on P(C n ) which coincides with the unique complex orbit of U [25]. Since P(R n ) is Lagrangian and G is a real form of U C , by [9,Proposition 9] it follows that G • v is compact as well and Lagrangian. By Proposition 37, G • v is a K-orbit.…”