We deal with complete submanifolds with weighted Poincaré inequality. By assuming the submanifold is -stable or has sufficiently small total curvature, we establish two vanishing theorems for harmonic -forms, which are extensions of the results of Dung-Seo and Cavalcante-Mirandola-Vitório.
We study lightlike hypersurfaces of an indefinite Kaehler manifold of quasi-constant curvature subject to the condition that the characteristic vector field of is tangent to . First, we provide a new result for such a lightlike hypersurface. Next, we investigate such a lightlike hypersurface of such that (1) the screen distribution is totally umbilical or (2) is screen conformal.
Lorentzian geometry in the large has certain similarities and certain fundamental differences from Riemannian geometry in the large. The Morse index theory for timelike geodesics is quite similar to the corresponding theory for Riemannian manifolds. However, results on completeness for Lorentzian manifolds are quite different from the corresponding results for positive definite manifolds. A generalization of global hyperbolicity known as pseudoconvexity is described. It has important implications...
An (m+2)-dimensional Lorentzian similarity manifold M is an affine flat manifold locally modeled on (G,ℝm+2), where G = ℝm+2 ⋊ (O(m+1, 1)×ℝ+). M is also a conformally flat Lorentzian manifold because G is isomorphic to the stabilizer of the Lorentzian group PO(m+2, 2) of the Lorentz model S m+1,1. We discuss the properties of compact Lorentzian similarity manifolds using developing maps and holonomy representations.