Is the Universe finite or infinite, and what shape does it have? These fundamental questions, of which relatively little is known, are typically studied within the context of the standard model of cosmology where the Universe is assumed to be homogeneous and isotropic. Here we address the above questions in highly general cosmological models, with the only assumption being that the average flow of matter is irrotational. Using techniques from differential geometry, specifically extensions of the Bonnet–Myers theorem, we derive a condition which implies a finite Universe and yields a bound for its diameter. Furthermore, under a weaker condition involving the interplay between curvature and diameter, together with the assumption that the Universe is finite (i.e. has closed spatial slices), we provide a concise list of possible topologies. Namely, the spatial sections then would be either the ring topologies
Chiral and helical Majorana fermions are two archetypal edge excitations in twodimensional topological superconductors. They emerge from systems of different Altland–Zirnbauer symmetries and characterized by
 Award ID(s):
 1707484
 Publication Date:
 NSFPAR ID:
 10303260
 Journal Name:
 New Journal of Physics
 Volume:
 21
 Issue:
 12
 Page Range or eLocationID:
 Article No. 123014
 ISSN:
 13672630
 Publisher:
 IOP Publishing
 Sponsoring Org:
 National Science Foundation
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