Search Results for « Pseudo-Riemannian Manifolds »

Riemannian and pseudo-Riemannian symmetric spaces with semisimple transvection group are known and classified for a long time. Contrary to that the description of pseudo-Riemannian symmetric spaces with non-semisimple transvection group is an open problem. In the last years some progress on this problem was achieved. In this survey article we want to explain these results and some of their applications. Among other things, the material developed in our previous papers math.DG/0312243, math.DG/0408249, and math.DG/0503220 is presented in a unified way.

The subject of this paper is six-dimensional nearly (para-)Kähler geometry with pseudo-Riemannian metrics. Firstly, we derive the analogue of the well-known exterior differential system characterising a nearly Kähler manifold and prove applications to the automorphism group of a nearly (para-)Kähler structure. Secondly, we prove existence and uniqueness results for left-invariant nearly (para-)Kähler structures on Lie groups $G \times G$ where $G$ is three-dimensional and simple.

The geodesic orbit property has been studied intensively for Riemannian manifolds. Geodesic orbit spaces are homogeneous and allow simplifications of many structural questions using the Lie algebra of the isometry group. Weakly symmetric Riemannian manifolds are geodesic orbit spaces. Here we define "naturally reductive" for pseudo-Riemannian manifolds and note that they are geodesic orbit spaces. A few years ago two of the authors proved that weakly symmetric pseudo-Riemannian manifolds are geodesic orbit spaces. In particular these results apply to pseudo-Riemannian Lorentz manifolds. There our main results are Theorems 4.2 and 5.1. In the Riemannian case the nilpotent isometry group for a geodesic orbit nilmanifold is abelian or $2$-step nilpotent. Examples show that this fails dramatically in the pseudo-Riemannian case. Here we concentrate on the geodesic orbit property for Lorentz nilmanifolds $G/H$ with $G = N \rtimes H$ and $N$ nilpotent. When the metric is nondegenerate on $[\mathfrak{n},\mathfrak{n}]$, Theorem 4.2 shows that $N$ either is at most $2$-step nilpotent as in the Riemannian situation, or is $4$-step nilpotent, but cannot be $3$-step nilpotent. Examples show that these bounds are the best possible. Surprisingly, Theorem 5.1 shows that $N$ is at most $2$-step nilpotent when the metric is degenerate on $[\mathfrak{n},\mathfrak{n}]$. Both theorems give additional structural information and specialize to naturally reductive and to weakly symmetric Lorentz nilmanifolds. Key Words: Geodesic Orbit Space; Lorentz nilmanifold; Weakly Symmetric Space; Naturally Reductive Space; Pseudo-Riemannian Manifold.

We identify a class of left-invariant pseudo-Riemannian metrics on Lie groups for which the Laplace-Beltrami equation reduces to a first-order PDE and admits exact solutions. The defining condition is the existence of a commutative ideal $\mathfrak{h}$ in the Lie algebra $\mathfrak{g}$ whose orthogonal complement satisfies $\mathfrak{h}^\perp\subseteq\mathfrak{h}$. Using the noncommutative integration method based on the orbit method and generalized Fourier transforms, we reduce the Laplace--Beltrami equation to a first-order linear PDE, which can then be integrated explicitly. The symmetry of the reduced equation gives rise, via the inverse transform, to nonlocal symmetry operators for the original equation. These operators are generically integro-differential, contrasting with the polynomial symmetries appearing in previously studied classes. The method is illustrated by two examples: the Heisenberg group $\mathrm{H}_3(\mathbb{R})$ with a Lorentzian metric and a four-dimensional non-unimodular group with a metric of signature $(2,2)$. In the latter, classical separation of variables is not directly applicable, yet the noncommutative approach yields explicit solutions and reveals the predicted nonlocal symmetry.

A submanifold of a Riemannian manifold is called a parallel submanifold if its second fundamental form is parallel with respect to the van der Waerden–Bortolotti connection. From submanifold point of view, parallel submanifolds are the simplest Riemannian submanifolds next to totally geodesic ones. Parallel submanifolds form an important class of Riemannian submanifolds since extrinsic invariants of a parallel submanifold do not vary from point to point. In this paper, we provide a comprehensive survey on this important class of submanifolds.

Riemannian Einstein solvmanifolds can be described in terms of nilsolitons, namely nilpotent Lie groups endowed with a left-invariant Ricci soliton metric. This characterization does not extend to indefinite metrics; nonetheless, nilsolitons can be defined and used to construct Einstein solvmanifolds of a higher dimension in any signature. An Einstein solvmanifold obtained by this construction turns out to satisfy the pseudo-Iwasawa condition, meaning that its Lie algebra splits as the orthogonal sum of a nilpotent ideal and an abelian subalgebra, the latter acting by symmetric derivations. In this paper we construct a family of pseudo-Iwasawa solvmanifolds admitting a Killing spinor in any dimension and signature and prove that all pseudo-Iwasawa solvmanifolds admitting a Killing spinor, invariant or not, belong to this family. If in addition the metric is Einstein, we show that the only possibility is the hyperbolic half-space. As a byproduct, we prove that the only homogeneous Riemannian manifold admitting a Killing spinor with imaginary Killing constant is hyperbolic space.