Complete pseudo-Riemannian Manifold Research Articles

Helicoidal surfaces in some conformally flat pseudo-spaces of dimensional three

We introduce the complete pseudo-Riemannian manifold [Formula: see text] with a conformally flat pseudo-metric satisfying Einstein’s equation. First, we get a theorem that associates the curvatures of a non-degenerate surface belonging to conformally equivalent spaces, say [Formula: see text] and [Formula: see text]. Next, we evaluate the existence of non-degenerate helicoidal surfaces in some conformally flat pseudo-spaces with pseudo-metrics corresponding to particular conformal factors (for example, a certain rotational symmetry with translational symmetry). We determine these conformal factors according to the causal characters of the axis of rotation. Right after, we get a two-parameter family of non-degenerate helicoidal surfaces with prescribed extrinsic curvature or mean curvature given by smooth functions, corresponding to both spacelike and timelike axes of rotation. As for the lightlike axis, we discuss some particular cases.

On low-dimensional manifolds with isometric $\widetilde{\mathrm{U}} (p, q)$-actions

Denote by $\widetilde{\mathrm{U}}(p,q)$ the universal covering group of $\mathrm{U}(p,q)$, the linear group of isometries of the pseudo-Hermitian space $\mathbb{C}^{p,q}$ of signature $p,q$. Let $M$ be a connected analytic complete pseudo-Riemannian manifold that admits an isometric $\widetilde{\mathrm{U}}(p,q)$-action and that satisfies $\dim M \leq n(n+2)$ where $n = p+q$. We prove that if the action of $\widetilde{\mathrm{SU}}(p,q)$ (the connected derived group of $\widetilde{\mathrm{U}}(p,q)$) has a dense orbit and the center of $\widetilde{\mathrm{U}}(p,q)$ acts non-trivially, then $M$ is an isometric quotient of manifolds involving simple Lie groups with bi-invariant metrics. Furthermore, the $\widetilde{\mathrm{U}}(p,q)$-action is lifted to $\widetilde{M}$ to natural actions on the groups involved. As a particular case, we prove that when $\widetilde{M}$ is not a pseudo-Riemannian product, then its geometry and $\widetilde{\mathrm{U}}(p,q)$-action are obtained from one of the symmetric pairs $(\mathfrak{su}(p,q+1), \mathfrak{u}(p,q))$ or $(\mathfrak{su}(p+1,q), \mathfrak{u}(p,q))$.

Gallot–Tanno theorem for pseudo-Riemannian metrics and a proof that decomposable cones over closed complete pseudo-Riemannian manifolds do not exist

We generalize for complete pseudo-Riemannian metrics a classical result of Gallot (1979) [3] and Tanno (1978) [13]: we show that if a closed complete manifold admits a nonconstant function λ satisfying ∇ k ∇ j ∇ i λ + 2 ∇ k λ ⋅ g i j + ∇ i λ ⋅ g j k + ∇ j λ ⋅ g i k = 0 , then the metric is the Riemannian metric of constant curvature +1. We use this result to give a simple proof of a recent result of Alekseevsky, Cortes, Galaev and Leistner (2009) [1]. Certain generalizations for higher Gallot equations are given.

Cones over pseudo-Riemannian manifolds and their holonomy

By a classical theorem of Gallot (Ann. Sci. Éc. Norm. Sup. 12: 235–267, 1979), a Riemannian cone over a complete Riemannian manifold is either flat or has irreducible holonomy. We consider metric cones with reducible holonomy over pseudo-Riemannian manifolds. First we describe the local structure of the base of the cone when the holonomy of the cone is decomposable. For instance, we find that the holonomy algebra of the base is always the full pseudo-orthogonal Lie algebra. One of the global results is that a cone over a compact and complete pseudo-Riemannian manifold is either flat or has indecomposable holonomy. Then we analyse the case when the cone has indecomposable but reducible holonomy, which means that it admits a parallel isotropic distribution. This analysis is carried out, first in the case where the cone admits two complementary distributions and, second for Lorentzian cones. We show that the first case occurs precisely when the local geometry of the base manifold is para-Sasakian and that of the cone is para-Kählerian. For Lorentzian cones we get a complete description of the possible (local) holonomy algebras in terms of the metric of the base manifold.

The abstract boundary—a new approach to singularities of manifolds

A new scheme is proposed for dealing with the problem of singularities in General Relativity. The proposal is, however, much more general than this. It can be used to deal with manifolds of any dimension which are endowed with nothing more than an affine connection, and requires a family C of curves satisfying a bounded parameter property to be specified at the outset. All affinely parametrised geodesics are usually included in this family, but different choices of family C will in general lead to different singularity structures. Our key notion is the abstract boundary or a-boundary of a manifold, which is defined for any manifold M and is independent of both the affine connection and the chosen family C of curves. The a-boundary is made up of equivalence classes of boundary points of M in all possible open embeddings. It is shown that for a pseudo-Riemannian manifold ( M , g) with a specified family C of curves, the abstract boundary points can then be split up into four main categories—regular, points at infinity, unapproachable points and singularities. Precise definitions are also provided for the notions of a removable singularity and a directional singularity. The pseudo-Riemannian manifold will be said to be singularity-free if its abstract boundary contains no singularities. The scheme passes a number of tests required of any theory of singularities. For instance, it is shown that all compact manifolds are singularity-free, irrespective of the metric and chosen family C . All geodesically complete pseudo-Riemannian manifolds are also singularity-free if the family C simply consists of all affinely parametrised geodesics. Furthermore, if any closed region is excised from a singularity-free manifold then the resulting manifold is still singularity-free. Numerous examples are given throughout the text. Problematic cases posed by Geroch and Misner are discussed in the context of the a-boundary and are shown to be readily accommodated.

Proper homothetic maps and fixed points

Let f be a proper homothetic map of the pseudo-Riemannian manifold M and assume f has a fixed point p. If all of the eigenvalues of either f*p or f-1*p have absolute values less than unity, then M is topologically Rn and M has a flat metric. This yields three characterizations of Minkowski spacetime. In general, a homothetic map of a complete pseudo-Riemannian manifold need not have fixed points. Furthermore, an example shows the existence of a proper homothetic map with a fixed point does not imply M is flat. The scalar curvature vanishes at a fixed point, but some of the sectional curvatures may be nonzero.