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Register a new userOn the restricted conformal group of the (1+n)-Einstein static universe
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Explicit models for the restricted conformal group of the Einstein static universe of dimension greater than two and for its universal covering group are constructed. Based on these models, as an application we determine all oriented and time-oriented conformal Lorentz manifolds whose restricted conformal group has maximal dimension. They amount to the Einstein static universe itself and two countably infinite series of its compact quotients.
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We study when the Jacobi operator associated to the Weyl conformal curvature tensor has constant eigenvalues on the bundle of unit spacelike or timelike tangent vectors. This leads to questions in the conformal geometry of pseudo-Riemannian manifolds which generalize the Osserman conjecture to this setting. We also study similar questions related to the skew-symmetric curvature operator defined by the Weyl conformal curvature tensor.
We show using covariant techniques that the Einstein static universe containing a perfect fluid is always neutrally stable against small inhomogeneous vector and tensor perturbations and neutrally stable against adiabatic scalar density inhomogeneities so long as c_{s}^2>1/5, and unstable otherwise. We also show that the stability is not significantly changed by the presence of a self-interacting scalar field source, but we find that spatially homogeneous Bianchi type IX modes destabilise an Einstein static universe. The implications of these results for the initial state of the universe and its pre-inflationary evolution are also discussed.
We endow the group of automorphisms of an exact Courant algebroid over a compact manifold with an infinite dimensional Lie group structure modelled on the inverse limit of Hilbert spaces (ILH). We prove a slice theorem for the action of this Lie group on the space of generalized metrics. As an application, we show that the moduli space of generalized metrics is stratified by ILH submanifolds and relate it to the moduli space of usual metrics. Finally, we extend these results to odd exact Courant algebroids.
We consider static cosmological solutions along with their stability properties in the framework of a recently proposed theory of massive gravity. We show that the modifcation introduced in the cosmological equations leads to several new solutions, only sourced by a perfect fluid, generalizing the Einstein Static Universe found in General Relativity. Using dynamical system techniques and numerical analysis, we show that the found solutions can be either neutrally stable or unstable against spatially homogeneous and isotropic perturbations.
We introduce three non-trivial 2-cocycles $c_k$, k=0,1,2, on the Lie algebra $S^3H=Map(S^3,H)$ with the aid of the corresponding basis vector fields on $S^3$, and extend them to 2-cocycles on the Lie algebra $S^3gl(n,H)=S^3H otimes gl(n,C)$. Then we have the corresponding central extension $S^3gl(n,H)oplus oplus_k (Ca_k)$. As a subalgebra of $S^3H$ we have the algebra $C[phi]$ of the Laurent polynomial spinors on $S^3$. Then we have a Lie subalgebra $hat{gl}(n, H)=C[phi] otimes gl(n, C)$ of $S^3gl(n,H)$, as well as its central extension by the 2-cocycles ${c_k}$ and the Euler vector field $d$: $hat{gl}=hat{gl}(n, H) oplus oplus_k(Ca_k)oplus Cd$ . The Lie algebra $hat{sl}(n,H)$ is defined as a Lie subalgebra of $hat{gl}(n,H)$ generated by $C[phi]otimes sl(n,C))$. We have the corresponding central extension of $hat{sl}(n,H)$ by the 2-cocycles ${c_k}$ and the derivation $d$, which becomes a Lie subalgebra $hat{sl}$ of $hat{gl}$. Let $h_0$ be a Cartan subalgebra of $sl(n,C)$ and $hat{h}=h_0 oplus oplus_k(Ca_k)oplus Cd$. The root space decomposition of the $ad(hat{h})$-representation of $hat{sl}$ is obtained. The set of roots is $Delta ={ m/2 delta + alpha ; alpha in Delta_0, m in Z} bigcup {m/2 delta ; m in Z }$ . And the root spaces are $hat{g}_{m/2 delta+ alpha}= C[phi ;m] otimes g_{alpha}$, for $alpha eq 0$ , $hat{g}_{m/2 delta}= C[phi ;m] otimes h_0$, for $m eq 0$, and $hat{g}_{0 delta}= hat{h}$, where $C[phi ;m]$ is the subspace with the homogeneous degree m. The Chevalley generators of $hat{sl}$ are given.
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