Global lightlike manifolds and harmonicity

Abstract

A new class of semi-Riemannian and lightlike manifolds (including globally null) is constructed by using a hypersurface of an orientable Riemannian manifold, endowed with the second fundamental form instead of a metric induced from the ambient space. We show the existence (or non-existence) of harmonic tensor fields and harmonic maps and extend to the semi-Riemannian and lightlike case a result of Chen-Nagano [4]. Then we deal with general lightlike submanifolds immersed in a semi-Riemannian manifold and propose a definition of minimal lightlike submanifolds, which generalize the one given in [7] in the Minkowski space R 4 1 . Several examples are given throughout.

Screen Isotropic Leaves on Lightlike Hypersurfaces of a Lorentzian Manifold

“…where trace| S(T M) (h) denotes the trace of h restricted to S(T M ) with respect to the degenerate metric g [5]. A lightlike hypersurface (M, g, S(T M )) is called screen locally conformal if the shape operators A N and A * ξ are related by (2.16) A N = ϕA * ξ , where ϕ is a non-vanishing smooth function on a neighborhood U in M [3].…”

GCR-Lightlike Submanifolds of a Semi-Riemannian Product Manifold

“…According to (2) considering the projection morphisms L and S of tr(T M ) on ltr(T M ) and S(T M ⊥ ) respectively, then (5) become…”

Nonexistence of Totally Contact Umbilical Slant Lightlike Submanifolds of Indefinite Cosymplectic Manifolds

“…Then in [15], a general notion of minimal lightlike submanifold of a semi-Riemannian manifold is introduced as follows. (ii) trace ℎ = 0, where trace is written with respect to restricted to ( ).…”