The conformal group of a compact simply connected Lorentzian manifold

We show presence a special torse-forming vector field (a particular form of torse-forming of a vector field) on generalized Robertson–Walker (GRW) spacetime, which is an eigenvector of the de Rham–Laplace operator. This paves the way to showing that the presence of a time-like special torse-forming vector field ξ with potential function ρ on a Lorentzian manifold (M,g), dimM>5, which is an eigenvector of the de Rham Laplace operator, gives a characterization of a GRW-spacetime. We show that if, in addition, the function ξ(ρ) is nowhere zero, then the fibers of the GRW-spacetime are compact. Finally, we show that on a simply connected Lorentzian manifold (M,g) that admits a time-like special torse-forming vector field ξ, there is a function f called the associated function of ξ. It is shown that if a connected Lorentzian manifold (M,g), dimM>4, admits a time-like special torse-forming vector field ξ with associated function f nowhere zero and satisfies the Fischer–Marsden equation, then (M,g) is a quasi-Einstein manifold.

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Conformal (2 + 4)-braneworld

International Journal of Modern Physics D ◽

10.1142/s0218271817501255 ◽

2017 ◽

Vol 26 (11) ◽

pp. 1750125

Author(s):

Merab Gogberashvili

Keyword(s):

Geometrical Structure ◽

Conformal Group ◽

Effective Dimension ◽

Brane Model ◽

Embedding Theory ◽

Fifth Dimension ◽

The Stability ◽

Simply Connected ◽

Additional Reduction ◽

Matter Fields

The 6D brane model is considered, where matter is trapped on the surface of a (2+4)-hyperboloid, as is suggested by the geometrical structure behind the 4D conformal group. The effective dimension of the bulk spacetime for matter fields is five, with the extra space-like and time-like domains. Using the embedding theory, the presence of the familiar factorizable 5D brane metrics in both domains is shown. These metrics with exponential warp factors are able to provide with the additional reduction of the effective spacetime dimensions down to four. It is demonstrated that the extra (1+1)-space is not simply connected and there is a gap in the range of the extra coordinates. This can explain the stability of the model in the domain with the time-like effective fifth dimension and the appearance of the cosmological constant due to the tachyon condensation. It is found that the model exhibits orbifold symmetry and thus is free from the fermion chirality problem.

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Analysis of the Brylinski-Kostant Model for Spherical Minimal Representations

Canadian Journal of Mathematics ◽

10.4153/cjm-2012-011-9 ◽

2012 ◽

Vol 64 (4) ◽

pp. 721-754 ◽

Cited By ~ 3

Author(s):

Dehbia Achab ◽

Jacques Faraut

Keyword(s):

Holomorphic Functions ◽

Conformal Group ◽

View Point ◽

Dimensional Representation ◽

Real Form ◽

Complex Vector ◽

Finite Dimensional Representation ◽

Finite Dimensional ◽

Minimal Representations ◽

Simply Connected

Abstract We revisit with another view point the construction by R. Brylinski and B. Kostant of minimal representations of simple Lie groups. We start froma pair (V,Q), where V is a complex vector space and Q a homogeneous polynomial of degree 4 on V. The manifold is an orbit of a covering of Conf(V,Q), the conformal group of the pair (V,Q), in a finite dimensional representation space. By a generalized Kantor-Koecher-Tits construction we obtain a complex simple Lie algebra 𝔤, and furthermore a real form 𝔤ℝ. The connected and simply connected Lie group Gℝ with Lie(Gℝ) = 𝔤ℝ acts unitarily on a Hilbert space of holomorphic functions defined on the manifold .

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Some groups with T1 primitive ideal spaces

Journal of the Australian Mathematical Society. Series A. Pure Mathematics and Statistics ◽

10.1017/s144678870002259x ◽

1985 ◽

Vol 38 (1) ◽

pp. 55-64 ◽

Cited By ~ 2

Author(s):

A. L. Carey ◽

W. Moran

Keyword(s):

Normal Subgroup ◽

Lie Groups ◽

Lie Group ◽

Locally Compact Group ◽

Primitive Ideal ◽

Ideal Space ◽

Solvable Lie Groups ◽

Locally Compact ◽

Simply Connected ◽

Connected Lie Group

AbstractLet G be a second countable locally compact group possessing a normal subgroup N with G/N abelian. We prove that if G/N is discrete then G has T1 primitive ideal space if and only if the G-quasiorbits in Prim N are closed. This condition on G-quasiorbits arose in Pukanzky's work on connected and simply connected solvable Lie groups where it is equivalent to the condition of Auslander and Moore that G be type R on N (-nilradical). Using an abstract version of Pukanzky's arguments due to Green and Pedersen we establish that if G is a connected and simply connected Lie group then Prim G is T1 whenever G-quasiorbits in [G, G] are closed.

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Exploring the landscape of CHL strings on Td

Journal of High Energy Physics ◽

10.1007/jhep08(2021)095 ◽

2021 ◽

Vol 2021 (8) ◽

Author(s):

Anamaría Font ◽

Bernardo Fraiman ◽

Mariana Graña ◽

Carmen A. Núñez ◽

Héctor Parra De Freitas

Keyword(s):

Gauge Group ◽

Moduli Space ◽

Dynkin Diagram ◽

Fundamental Groups ◽

Computer Algorithms ◽

Abelian Gauge ◽

Reduced Rank ◽

Systematic Exploration ◽

String Models ◽

Simply Connected

Abstract Compactifications of the heterotic string on special Td/ℤ2 orbifolds realize a landscape of string models with 16 supercharges and a gauge group on the left-moving sector of reduced rank d + 8. The momenta of untwisted and twisted states span a lattice known as the Mikhailov lattice II(d), which is not self-dual for d > 1. By using computer algorithms which exploit the properties of lattice embeddings, we perform a systematic exploration of the moduli space for d ≤ 2, and give a list of maximally enhanced points where the U(1)d+8 enhances to a rank d + 8 non-Abelian gauge group. For d = 1, these groups are simply-laced and simply-connected, and in fact can be obtained from the Dynkin diagram of E10. For d = 2 there are also symplectic and doubly-connected groups. For the latter we find the precise form of their fundamental groups from embeddings of lattices into the dual of II(2). Our results easily generalize to d > 2.

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R-GROUP AND WHITTAKER SPACE OF SOME GENUINE REPRESENTATIONS

Journal of the Institute of Mathematics of Jussieu ◽

10.1017/s1474748021000128 ◽

2021 ◽

pp. 1-61

Author(s):

Fan Gao

Keyword(s):

Series Representation ◽

Principal Series ◽

Symplectic Groups ◽

Principal Series Representation ◽

Connected Group ◽

Covering Group ◽

Simply Connected

Abstract For a unitary unramified genuine principal series representation of a covering group, we study the associated R-group. We prove a formula relating the R-group to the dimension of the Whittaker space for the irreducible constituents of such a principal series representation. Moreover, for certain saturated covers of a semisimple simply connected group, we also propose a simpler conjectural formula for such dimensions. This latter conjectural formula is verified in several cases, including covers of the symplectic groups.

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Horosphere slab separation theorems in manifolds without conjugate points

Journal of the Egyptian Mathematical Society ◽

10.1186/s42787-019-0038-5 ◽

2019 ◽

Vol 27 (1) ◽

Author(s):

Sameh Shenawy

Keyword(s):

Euclidean Space ◽

Hyperbolic Space ◽

Existence And Uniqueness ◽

Convex Set ◽

Sufficient Conditions ◽

Conjugate Points ◽

Separation Theorems ◽

Farthest Points ◽

Simply Connected ◽

Convex Subsets

Abstract Let $\mathcal {W}^{n}$ W n be the set of smooth complete simply connected n-dimensional manifolds without conjugate points. The Euclidean space and the hyperbolic space are examples of these manifolds. Let $W\in \mathcal {W}^{n}$ W ∈ W n and let A and B be two convex subsets of W. This note aims to investigate separation and slab horosphere separation of A and B. For example,sufficient conditions on A and B to be separated by a slab of horospheres are obtained. Existence and uniqueness of foot points and farthest points of a convex set A in $W\in \mathcal {W}$ W ∈ W are considered.

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TOPOLOGICAL 4-MANIFOLDS WITH 4-DIMENSIONAL FUNDAMENTAL GROUP

Glasgow Mathematical Journal ◽

10.1017/s0017089521000215 ◽

2021 ◽

pp. 1-8

Author(s):

DANIEL KASPROWSKI ◽

MARKUS LAND

Keyword(s):

Fundamental Group ◽

Homotopy Equivalent ◽

Poincaré Duality ◽

Poincare Duality ◽

Canonical Map ◽

Duality Space ◽

Simply Connected Manifolds ◽

Simply Connected ◽

Poincaré Duality Space

Abstract Let $\pi$ be a group satisfying the Farrell–Jones conjecture and assume that $B\pi$ is a 4-dimensional Poincaré duality space. We consider topological, closed, connected manifolds with fundamental group $\pi$ whose canonical map to $B\pi$ has degree 1, and show that two such manifolds are s-cobordant if and only if their equivariant intersection forms are isometric and they have the same Kirby–Siebenmann invariant. If $\pi$ is good in the sense of Freedman, it follows that two such manifolds are homeomorphic if and only if they are homotopy equivalent and have the same Kirby–Siebenmann invariant. This shows rigidity in many cases that lie between aspherical 4-manifolds, where rigidity is expected by Borel’s conjecture, and simply connected manifolds where rigidity is a consequence of Freedman’s classification results.

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