Schwarzschild Spacetimes: Topology
Round 1
Reviewer 1 Report
The authors work within the geometric theory of the Schwarzschild spacetime. The main resut of the paper is the proof of the existence and the characterization of all Schwarzschild metrics on two topologically non-equivalent manifolds, namely R × (R3 \ {(0, 0, 0)}) and S1 × (R3 \ {(0, 0, 0)}).
They classify SO(3)-invariant, time-translation invariant and time-reflection invariant metrics on R × (R3 \ {(0, 0, 0)}) . An interesting result is the determination of submanifolds where the metric is not defined, the so-called Schwarzschild spheres, and the proof of the existence of a global metric whose Schwarzschild sphere is empty. They also define a mapping of the real line R onto the circle S1 enabling to transfer the classification results to the S1 × (R3 \ {(0, 0, 0)}) case. The paper is detailed, self contained and well written; the results are interesting and they can be useful for deeper understanding of cosmological models. Therefore I do recommend its publication in Axioms.
Author Response
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Author Response File: 
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Reviewer 2 Report
The authors investigate solutions of vacuum Einstein's equations in dimension 4, on $S^1\times S^2\times (0,+\infty)$ or $R\times S^2\times (0,+\infty)$ admitting a complete Killing vector field and SO(3) symmetry.
In my humble opinion, the authors should try to motivate better the aim and the novelty of their work since, at a first glance, it might appear that the manuscript does not contain new results (for example the family of solutions that the authors have found, reflects diffeomorphism invariance of general relativity and the fact that they can be transferred on $S^1\times S^2\times (0,+\infty)$ it is well-known).
Author Response
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