A Splitting Result for Real Submanifolds of a Kähler Manifold

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).

A characterization for totally real submanifolds using self-adjoint differential operator

<abstract><p>In this article, we study totally real submanifolds in Kaehler product manifold with constant scalar curvature using self-adjoint differential operator $ \Box $. Under this setup, we obtain a characterization result. Moreover, we discuss $ \delta- $invariant properties of such submanifolds and get an obstruction result as an application of the inequality derived. The results in the article are supported by non-trivial examples.</p></abstract>

Submanifolds in Normal Complex Contact Manifolds

In the present article we initiate the study of submanifolds in normal complex contact metric manifolds. We define invariant and anti-invariant ( C C -totally real) submanifolds in such manifolds and start the study of their basic properties. Also, we establish the Chen first inequality and Chen inequality for the invariant δ ( 2 , 2 ) for C C -totally real submanifolds in a normal complex contact space form and characterize the equality cases. We also prove the minimality of C C -totally real submanifolds of maximum dimension satisfying the equalities.