AbstractLet $$(Z,\omega )$$
(
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
