Foliations on Non-metrisable Manifolds

Gauld, David

doi:10.1007/978-981-287-257-9_8

Cited by 6 publications

(12 citation statements)

References 5 publications

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“…Our proof is almost the same as Connelly's in [2] and Gauld's version in [4,Theorem 3.10], except for some details. Some of them are just cosmetical and due to personal taste: we use partitions of unity, reverse some signs and try to avoid formulas, replacing them by a geometrical description.…”

Section: Theorem 23 (R Connelly)mentioning

confidence: 75%

“…Its boundary contains only the point 0, 0 , removing it we obtain the open longray L + . Chapter 1.2 in [4] is dedicated to proving almost all the elementary properties of L + , recall in particular that it is a normal countably compact space. We sometimes consider ω 1 as a subspace of L ≥0 by identifying α ∈ ω 1 with α, 0 ∈ L ≥0 .…”

Section: Example 31 (Trivial)mentioning

confidence: 99%

“…Despite the availability of very good texts about the general theory of non-metrisable manifolds (the aformentioned Gauld's book [4] and the older but less elementary article by P. Nyikos in the Handbook of Set-theoretical Topology [7], for instance), we are not aware of a reference gathering all the results and counter-examples pertaining to the collaring problem of the boundary, hence this small note. It can be thought of as a convenient reference for (the few) researchers using nonmetrisable manifolds and/or, for researchers in more usual areas of mathematics, as a catalog of the horrors awaiting those who dare to not include metrisability in the definition of a manifold 1 .…”

Section: Introduction and Definitionsmentioning

confidence: 99%

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Collared and non-collared manifold boundaries

Baillif

2020

Preprint

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Section: Theorem 23 (R Connelly)mentioning

confidence: 75%

Section: Example 31 (Trivial)mentioning

confidence: 99%

Section: Introduction and Definitionsmentioning

confidence: 99%

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Collared and non-collared manifold boundaries

Baillif

2020

Preprint

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“…Proof. Define M :" lim Ý Ñ N as the inductive limit [ML98] in the category of possibly non-metrizable topological manifolds [Gau14] where N goes through the BTZ-extensions of M. This inductive limit is well defined, since the Now, consider the spacelike circle C " Σ 0 X BS Ă M 0 and let C 1 (resp. C 2 ) be a spacelike circle in the shaft of S, in the future (resp.…”

Section: Proof Of Theoremmentioning

confidence: 99%

“…There are other ways to prove the BTZ lines are second countable in the proof above. One can also use the fact that every possibly non metrizable 1-manifolds are type I [Gau14] to show each BTZ-lines admits a neighborhood which is a type I submanifold of M ; then use separability of M to conclude using the fact that every type I separable manifolds are metrizable [Gau14].…”

Section: Maximal Common Sub-btz-extensionmentioning

confidence: 99%

Cauchy-compact flat spacetimes with extreme BTZ

Brunswic

2021

Preprint

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Cauchy-compact flat spacetimes with extreme BTZ are Lorentzian analogue of complete hyperbolic surfaces of finite volume. Indeed, the latter are 2-manifolds locally modeled on the hyperbolic plane, with group of isometries PSL 2 pRq, admitting finitely many cuspidal ends while the regular part of the former are 3-manifolds locally models on 3 dimensionnal Minkowski space, with group of isometries PSL 2 pRq Ṙ3 , admitting finitely many ends whose neighborhoods are foliated by cusps. We prove a Theorem akin to the classical parametrization result for finite volume complete hyperbolic surfaces: the tangent bundle of the Teichmüller space of a punctured surface parametrizes globally hyperbolic Cauchy-maximal and Cauchy-compact locally Minkowski manifolds with extreme BTZ. Previous results of Mess, Bonsante and Barbot provide already a satisfactory parametrization of regular parts of such manifolds, the particularity of the present work reside in the consideration of manifolds with a singular geometrical structure with a singularities modeled on extreme BTZ. We present a BTZ-extension procedure akin to the procedure compactifying finite volume complete hyperbolic surface by adding cusp points at infinity ;

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Global properties of physically interesting Lorentzian spacetimes

Nawarajan

Visser

2016

Int. J. Mod. Phys. D

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Abstract:Under normal circumstances most members of the general relativity community focus almost exclusively on the local properties of spacetime, such as the locally Euclidean structure of the manifold and the Lorentzian signature of the metric tensor. When combined with the classical Einstein field equations this gives an extremely successful empirical model of classical gravity and classical matter -at least as long as one does not ask too many awkward questions about global issues, (such as global topology and global causal structure). We feel however that this is a tactical error -even without invoking full-fledged "quantum gravity" we know that the standard model of particle physics is also an extremely good representation of some parts of empirical reality; and we had better be able to carry over all the good features of the standard model of particle physics -at least into the realm of semi-classical quantum gravity. Doing so gives us some interesting global features that spacetime should possess: On physical grounds spacetime should be space-orientable, time-orientable, and spacetime-orientable, and it should possess a globally defined tetrad (vierbein, or in general a globally defined vielbein/n-bein). So on physical grounds spacetime should be parallelizable. This strongly suggests that the metric is not the fundamental physical quantity; a very good case can be made for the tetrad being more fundamental than the metric. Furthermore, a globally-defined "almost complex structure" is almost unavoidable. Ideas along these lines have previously been mooted, but much is buried in the pre-arXiv literature and is either forgotten or inaccessible. We shall revisit these ideas taking a perspective very much based on empirical physical observation.

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Foliations on Non-metrisable Manifolds

Cited by 6 publications

References 5 publications

Collared and non-collared manifold boundaries

Collared and non-collared manifold boundaries

Cauchy-compact flat spacetimes with extreme BTZ

Global properties of physically interesting Lorentzian spacetimes

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