Cones in the Euclidean space with vanishing scalar curvature

Abstract

Given a hypersurface M on a unit sphere of the Euclidean space, we define the cone based on M as the set of half-lines issuing from the origin and passing through M. By assuming that the scalar curvature of the cone vanishes, we obtain conditions under which bounded domains of such cone are stable or unstable.

Topology and signature changes in braneworlds

“…Another possible sources of interesting examples might come from Lorentzian version of the generalised helicoids considered by [40]. These are p + 1 dimensional submanifolds Σ p+1 invariant under a p-dimensional translation group R p .…”

“…The submanifold Σ p+1 may be thought of as a one parameter family of p-planes. A particularly intriguing case may be obtained from the work of section (1.6) of [40]. Let x(τ ) be a timelike curve γ in E 5,1 , where τ is propertime along the curve γ.…”