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Asymptotic symmetries of projectively compact order one Einstein manifolds

Jack Borthwick, Yannick Herfray

Abstract

We show that the boundary of a projectively compact Einstein manifold of dimension $n$ can be extended by a line bundle naturally constructed from the projective compactification. This extended boundary is such that its automorphisms can be identified with asymptotic symmetries of the compactification. The construction is motivated by the investigation of a new curved orbit decomposition for a $n+1$ dimensional manifold which we prove results in a line bundle over a projectively compact order one Einstein manifolds.

Asymptotic symmetries of projectively compact order one Einstein manifolds

Abstract

We show that the boundary of a projectively compact Einstein manifold of dimension can be extended by a line bundle naturally constructed from the projective compactification. This extended boundary is such that its automorphisms can be identified with asymptotic symmetries of the compactification. The construction is motivated by the investigation of a new curved orbit decomposition for a dimensional manifold which we prove results in a line bundle over a projectively compact order one Einstein manifolds.
Paper Structure (23 sections, 53 theorems, 149 equations, 7 figures)

This paper contains 23 sections, 53 theorems, 149 equations, 7 figures.

Table of Contents

  1. Introduction
  2. Projective tractors and the homogeneous model
  3. Projective tractor calculus
  4. The Model
  5. Curved orbit decomposition
  6. Resulting curved orbits
  7. Derivation of Theorem \ref{['Thrm: Curved orbits decomposition']}
  8. Derivation of Theorem \ref{['Thrm: Curved orbits decomposition: quotient']}
  9. Geometry of the open orbit $\mathcal{O}$
  10. Geometry of $\mathcal{Q}$
  11. Densities and tractors of $\mathcal{Q}$
  12. Densities on $\mathcal{Q}$
  13. Tractors of $\mathcal{Q}$
  14. The projective structure and the tractor connection of $\mathcal{Q}$
  15. The induced projective structure on $\mathcal{Q}$
  16. ...and 8 more sections

Key Result

Lemma 2.1

If $H^{AB}$ is invertible of inverse $\Phi_{AB}$ then $X^A X^B \Phi_{AB} =\det(H) \mathop{\mathrm{\textnormal{det}}}\nolimits(\zeta)$ and if $\det(H)=0$ then the kernel is generated by $D_{A}\tau$ with $\tau^2 = \frac{\mathop{\mathrm{\textnormal{det}}}\nolimits(\zeta)}{n+1}$.

Figures (7)

  • Figure 1: Fibration of $\mathbb{R}\textnormal{P}^{n+1}\setminus[I]$ over $\mathbb{R}\textnormal{P}^{n}$ and their corresponding ambient spaces.
  • Figure 2: Dimensionally reduced depiction of $\mathbb{R}\textnormal{P}^{n+1}\setminus[I]$. The oriented lines indicate the fibres of the projection $\mathbb{R}\textnormal{P}^{n+1}\setminus[I]\to\mathbb{R}\textnormal{P}^{n}$; apart from those along the extended boundary at infinity -- depicted here by a golden line -- the fibres cross successive surfaces of constant growing $\frac{\Phi(X,X)}{(\Phi(I,X)^2}$ (ranging from $-\infty$ to $+\infty$).
  • Figure 3: Defining diagram of $D_n$
  • Figure 4: Compatible splittings in adapted scales
  • Figure 5: Decomposition of $\mathscr{T}$
  • ...and 2 more figures

Theorems & Definitions (104)

  • Lemma 2.1
  • Theorem 3.1
  • Theorem 3.2
  • Lemma 3.1
  • proof
  • Proposition 3.1
  • proof
  • Proposition 3.2
  • proof
  • Proposition 3.3
  • ...and 94 more