Lorentzian manifolds admitting a Killing vector field have been studied in the literature from different points of view. Functional analysis, proper actions of Lie groups or Bochner’s technique in Differential Geometry, have been some of the very different tools involved to study them. The purpose of this paper is to review some of their properties, pointing out several techniques and supplying...

We give a short and simple proof that the Lorentzian 10j symbol, which forms a key part of the Barrett-Crane model of Lorentzian quantum gravity, is finite. The argument is very general, and applies to other integrals. For example, we show that the Lorentzian and Riemannian causal 10j symbols are finite, despite their singularities. Moreover, we show that integrals that arise in Cherrington’s w...

Superconformal algebras embedding space-time in any dimension and signature are considered. Different real forms of the R-symmetries arise both for usual space-time signature (one time) and for Euclidean or exotic signatures (more than one times). Application of these superalgebras are found in the context of supergravities with 32 supersymmetries, in any dimension D ≤ 11. These theories are re...

and Applied Analysis 3 2.2. Bitension Field. For smooth maps φ : (Mn, g) → (̃ Mm, ⟨, ⟩), the tension field τ(φ) is a section of the vector bundle φ∗T̃ M defined by τ (φ) = trace∇dφ = n