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Correction to: Geometry and topology of the Kerr photon region in the phase space

  • Carla Cederbaum
  • Sophia JahnsEmail author
Correction
  • 144 Downloads

1 Correction to: General Relativity and Gravitation (2019) 51:79  https://doi.org/10.1007/s10714-019-2561-y

The proof of Theorem 10 can be considerably simplified, as was pointed out to us by Gregory J. Galloway: Indeed, it is unnecessary to rule out L(2; 0); since 0 and 2 are not coprime, this case does not need to be considered, which makes it superfluous to argue that \(P_0\) admits a Seifert fibration without exceptional fibers (Proposition 15). Thus, one is left with the case \(P_0\approx L(2;1)\approx SO(3)\).

From a slightly different point of view, one may also argue as follows: Since \(P_0\) is a closed 3-dimensional manifold with \(\pi _1(P_0)=\mathbb Z_2\), it is doubly covered by an \(\mathbb S^3\) (by the Poincaré conjecture). By the elliptization conjecture, this \(\mathbb S^3\) can be taken to be the standard 3-sphere and the group \(\mathbb Z_2\) as a subgroup of SO(3) acting on it. (For the statements of the Poincaré and the elliptization conjecture, see  [4, 5]; for the proofs covering these conjectures see [1, 2, 3].) Hence, \(P_0\) is the quotient \(\mathbb S^3 /\mathbb Z_2 \approx \mathbb R P^3\approx SO(3)\).

Recalling how \(P_0\) was obtained as a slice \(P\cap \{t=0, p_0=-1\}\) of the photon region in the phase space, this proves Theorem 10.

Notes

References

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    Perelman, G.: The entropy formula for the Ricci flow and its geometric applications (2002)Google Scholar
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    Perelman, G.: Finite extinction time for the solutions to the Ricci flow on certain three-manifolds (2003)Google Scholar
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    Perelman, G.: Ricci flow with surgery on three-manifolds (2003)Google Scholar
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    Thurston, W.: The Geometry and Topology of Three-Manifolds. Princeton lecture notes on geometric structures on 3-manifolds (1980)Google Scholar
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    Scott, P.: The geometries of 3-manifolds. Bull. Lond. Math. Soc. 15(5), 401–487 (1983)MathSciNetCrossRefGoogle Scholar

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© Springer Science+Business Media, LLC, part of Springer Nature 2019

Authors and Affiliations

  1. 1.University of TübingenTübingenGermany

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