Da rigidez de variedades kählerianas munidas de campo vetorial conforme fechado.

Abstract

The goal of this work is to demonstrate two results about the rigidity of kählerian manifolds under certain conditions. In the first result, we show that if a connected compact kählerian surface M with nonpositive gaussian curvature is endowed with a closed conformal vector field ξ whose singular points are isolated, then M has necessarily zero gaussian curvature, ξ is parallel and M is isometric to a flat torus. In the second result, we consider a connected complete kählerian manifold M, of complex dimension n > 1 and equipped with a nontrivial closed conformal vector field ξ. In this case, if the distribution D in M \ ξ −1 (0), generated by ξ and Jξ, has one compact leaf Σ with nonpositive holomorphic sectional curvature and Hol ⊥ (Σ) is a torsion group, then ξ −1 (0) = ∅, ξ and Jξ are parallel in M, the leafs of D are isometric to the flat torus and the leafs of D ⊥ are isometric to a kählerian manifold of complex dimension n − 1. In particular, the universal covering of M is a cartesian product of R2 with a connected, simply connected, complete kählerian manifold.