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Ricci-flat left-invariant Lorentzian metrics on 2-step nilpotent Lie groups

Mohammed Guediri, Mona Bin-Asfour (2014)

Archivum Mathematicum

The purpose of this paper is to investigate Ricci-flatness of left-invariant Lorentzian metrics on 2-step nilpotent Lie groups. We first show that if , is a Ricci-flat left-invariant Lorentzian metric on a 2-step nilpotent Lie group N , then the restriction of , to the center of the Lie algebra of N is degenerate. We then characterize the 2-step nilpotent Lie groups which can be endowed with a Ricci-flat left-invariant Lorentzian metric, and we deduce from this that a Heisenberg Lie group H 2 n + 1 can be...

Riesz potentials and amalgams

Michael Cowling, Stefano Meda, Roberta Pasquale (1999)

Annales de l'institut Fourier

Let ( M , d ) be a metric space, equipped with a Borel measure μ satisfying suitable compatibility conditions. An amalgam A p q ( M ) is a space which looks locally like L p ( M ) but globally like L q ( M ) . We consider the case where the measure μ ( B ( x , ρ ) of the ball B ( x , ρ ) with centre x and radius ρ behaves like a polynomial in ρ , and consider the mapping properties between amalgams of kernel operators where the kernel ker K ( x , y ) behaves like d ( x , y ) - a when d ( x , y ) ≤ 1 and like d ( x , y ) - b when d ( x , y ) ≥ 1 . As an application, we describe Hardy–Littlewood–Sobolev type regularity theorems...

Rigidity and L 2 cohomology of hyperbolic manifolds

Gilles Carron (2010)

Annales de l’institut Fourier

When X = Γ ∖ ℍ n is a real hyperbolic manifold, it is already known that if the critical exponent is small enough then some cohomology spaces and some spaces of L 2 harmonic forms vanish. In this paper, we show rigidity results in the borderline case of these vanishing results.

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