For Lorentzian two manifolds (Σ1, g1) and (Σ2, g2) we consider the two product para-Kähler structures (Formula Presented.) defined on the product four manifold Σ1 × Σ2, with ϵ = ±1. We show that the metric Gϵ is locally conformally flat (resp. Einstein) if and only if the Gauss curvatures κ1, κ2 of g1, g2, respectively, are both constants satisfying κ1 = -ϵκ2 (resp.κ1 = ϵκ2). We give the conditions on the Gauss curvatures for which every Lagrangian surface with parallel mean curvature vector is the product (Formula Presented.), where γ1 and γ2 are geodesics. We study Lagrangian surfaces in the product dS2 × dS2 with parallel mean curvature vector and finally, we explore the stability and Hamiltonian stability of certain minimal Lagrangian surfaces and H-minimal surfaces.
- Hamiltonian minimal surfaces
- Lorentzian surfaces
- Minimal Lagrangian surfaces
- Para-Kaehler structure
- Surfaces with parallel mean curvarture vector