Flat pseudo-Riemannian Manifolds Research Articles

A classification of conformally flat generalized Ricci recurrent pseudo-Riemannian manifolds

Conformally flat pseudo-Riemannian manifolds with generalized Ricci recurrent, [Formula: see text] structure are completely classified in this short report. A conformally flat generalized Ricci recurrent pseudo-Riemannian manifold is shown to be an Einstein manifold. In particular, a conformally flat generalized Ricci recurrent spacetime must be either a de Sitter spacetime or an anti-de Sitter spacetime.

Anti-Kählerian Geometry on Lie Groups

Let $G$ be a Lie group of even dimension and let $(g,J)$ be a left invariant anti-K\"ahler structure on $G$. In this article we study anti-K\"{a}hler structures considering the distinguished cases where the complex structure $J$ is abelian or bi-invariant. We find that if $G$ admits a left invariant anti-K\"ahler structure $(g,J)$ where $J$ is abelian then the Lie algebra of $G$ is unimodular and $(G,g)$ is a flat pseudo-Riemannian manifold. For the second case, we see that for any left invariant metric $g$ for which $J$ is an anti-isometry we obtain that the triple $(G, g, J)$ is an anti-K\"ahler manifold. Besides, given a left invariant anti-Hermitian structure on $G$ we associate a covariant $3$-tensor $\theta$ on its Lie algebra and prove that such structure is anti-K\"ahler if and only if $\theta$ is a skew-symmetric and pure tensor. From this tensor we classify the real 4-dimensional Lie algebras for which the corresponding Lie group has a left invariant anti-K\"ahler structure and study the moduli spaces of such structures (up to group isomorphisms that preserve the anti-K\"ahler structures).

Holonomy groups of complete flat pseudo-Riemannian homogeneous spaces

We show that a complete flat pseudo-Riemannian homogeneous manifold with non-abelian linear holonomy is of dimension ≥14. Due to an example constructed in a previous article (Baues and Globke, 2012 [2]), this is a sharp bound. Also, we give a structure theory for the fundamental groups of complete flat pseudo-Riemannian manifolds in dimensions ≤6. Finally, we observe that every finitely generated torsion-free 2-step nilpotent group can be realized as the fundamental group of a complete flat pseudo-Riemannian manifold with abelian linear holonomy.

Flat pseudo-Riemannian homogeneous spaces with non-abelian holonomy group

We construct homogeneous flat pseudo-Riemannian manifolds with non-abelian fundamental group. In the compact case, all homogeneous flat pseudo-Riemannian manifolds are complete and have abelian linear holonomy group. To the contrary, we show that there do exist non-compact and non-complete examples, where the linear holonomy is non-abelian, starting in dimensions ≥ 8 \geq 8 , which is the lowest possible dimension. We also construct a complete flat pseudo-Riemannian homogeneous manifold of dimension 14 with non-abelian linear holonomy. Furthermore, we derive a criterion for the properness of the action of an affine transformation group with transitive centralizer.

On the theory of algebraic invariants of vector spaces of Killing tensors

We employ a hybrid version of the classical Cartan method of moving frames to develop a practical algorithm for the generation of isometry group invariants and covariants for arbitrary vector spaces of Killing tensors defined on any flat pseudo-Riemannian manifold. We then apply our algorithm to construct a set of fundamental covariants for the space of valence-two Killing tensors defined in three-dimensional Euclidean space and use them to invariantly characterize the associated eleven orthogonally separable webs.

Bi-conformal vector fields and the local geometric characterization of conformally separable pseudo-Riemannian manifolds II

In this paper we continue the study of bi-conformal vector fields started in [A. García-Parrado, J.M.M. Senovilla, Class. Quant. Grav. 21 (2004) 2153–2177]. These are vector fields defined on a pseudo-Riemannian manifold by the differential conditions £ ξ → P a b = ϕ P a b , £ ξ → Π a b = χ Π a b , where P a b , Π a b are orthogonal and complementary projectors with respect to the metric tensor g a b . In a previous paper we explained how the analysis of these differential conditions enabled us to derive local geometric characterizations of the most relevant cases of conformally separable (also called double twisted) pseudo-Riemannian manifolds. In this paper we carry on this analysis further and provide local invariant characterizations of conformally separable pseudo-Riemannian manifolds with conformally flat leaf metrics. These characterizations are rather similar to that existing for conformally flat pseudo-Riemannian manifolds but instead of the Weyl tensor, we must demand the vanishing of certain four rank tensors constructed from the curvature of an affine, non-metric, connection (bi-conformal connection). We also speculate with possible applications to finding results for the existence of foliations by conformally flat hypersurfaces in any pseudo-Riemannian manifold.

Curvature homogeneous spacelike Jordan Osserman pseudo-Riemannian manifolds

Let s ⩾ 2. We construct Ricci flat pseudo-Riemannian manifolds of signature (2s, s) which are not locally homogeneous but whose curvature tensors nevertheless exhibit a number of important symmetry properties. They are curvature homogeneous; their curvature tensor is modelled on that of a local symmetric space. They are spacelike Jordan Osserman with a Jacobi operator which is nilpotent of order 3; they are not timelike Jordan Osserman. They are k-spacelike higher-order Jordan Osserman for 2 ⩽ k ⩽ s; they are k-timelike higher-order Jordan Osserman for s + 2 ⩽ k ⩽ 2s, and they are not k-timelike higher-order Jordan Osserman for 2 ⩽ k ⩽ s + 1.

Minimal immersion of pseudo-Riemannian manifolds

j=3+l where (xu x2,･■■, xn)is a rectangular coordinate system of E%. (E%, g)isa flat Pseudo-Riemannian manifold of signature (q,n―q). Let c be a point in E%+1 (or Effi) and r>0. We put S%(c,r)= (iE￡,a+1: g(x-c, x-c)=r2} H≫(ctr)={x^E≫#'.g{x-c, x-c)=-r2). Itis known that S%(c,r) and H%(c, r)arecomplete Pseudo-Riemannian manifolds ofsignature(q,n―q)and respective constant sectionalcurvatures r~2and ―r~2. Sq(c,r) and H%(c, r) are calledthe Pseudo-Riemannian sphere and the Pseudohyperbolic space,respectively. The point c is calledthe center of S%(c,r) and H*{c, r). In the following, SJ(O,r) and Hj(0, r) are simply denoted by S%(r) and Hq(r), respectively. AT^ denotes the Pseudo-Riemannian manifold with metric tensor of signature (p, n―p). The Pseudo-Riemannian manifold, the Pseudo-Euclidean space,thePseudo-Riemannian sphere and the Pseudo-hyperbolic space are simply denoted by the P―R manifold, the P―E space, the P―R sphere and theP―h space. The P―R manifold A/?is calledthe Lorentz manifoldand the P―E space E＼is calledthe Minkowski space. Let /: M ^>Ni be an isometric immersion of a P― R manifold M in another P―R manifold A^. That is f*g=g, where g and g are theindefinite metric tensors of M$ and A7￡,respectively. T(M%) and TL{Mf) denote the tangent bundle and the normal bundle of M|＼ 7, 7 and 1L denote theRiemannian connections and thenormal connection on Mf, N% and TX(M^), respectively. Then for any vectorfieldsX, Y<bT(M$), v(=Tl{M ),we have theGauss formula VYY=VrY + B(X.Y).

Flat Pseudo-Reimannian manifolds with a nilpotent transitive group of isometries

Flat pseudo-Riemannian manifolds with a nilpotent transitive group of isometries are shown to be complete. Also flat pseudo-Riemannian homogeneous manifolds with non-trivial holonomy are shown to contain a complete geodesic.