A splitting result for real submanifolds of a Kahler manifold

2021/01/24 by Biliotti, Leonardo
#22E45 #53D20 #Differential Geometry (math.DG) #FOS: Mathematics

paper · doi:10.48550/arxiv.2101.09733

Abstract

Let (Z,ω) be a connected Kahler 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 x∈ M. If G is a real form of U^{\mathbb C}, we investigate conditions such that G⋅ x compact implies U^{\mathbb 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,ω) generalizing a result proved in \citepg, see also \citebg,bs.

Related

• Compact orbits of Parabolic subgroups
• Remarks on Semistable Points and Nonabelian Convexity of Gradient Maps
• Properties of Gradient maps associated with Action of Real reductive Group
• Stability, analytic stability for real reductive Lie groups
• A remark on the gradient map