SCREEN CONFORMAL SUBMERSIONS BETWEEN LIGHTLIKE MANIFOLDS AND SEMI-RIEMANNIAN MANIFOLDS AND THEIR HARMONICITY

Abstract

In this paper, we introduce a new submersion ϕ : M → N, namely screen conformal submersion, between a lightlike manifold M and a semi-Riemann manifold N. We give examples and show that the lightlike manifold M is shear free under a condition if ϕ : M → N is a screen conformal submersion. Also we define special screen submersions: radical and screen homothetic submersions. We obtain that the radical homothetic submersion ϕ : M → N implies that M is a Reinhart lightlike manifold. Moreover we define and study the lightlike versions of O'Neill's tensors for a horizontally conformal submersion and show that these tensors have different properties from the Riemannian case. Using these tensors, we investigate the relationships between the curvatures of base and total manifolds. Finally, since the trace of a smooth mapping is not meaningful on the radical part of a lightlike manifold, we introduce lightlike harmonic map between lightlike manifolds and semi-Riemannian manifolds, supported by an example. We also give a characterization for a screen conformal submersion to be lightlike harmonic.