On combinatorial Gauss-Bonnet Theorem for general Euclidean simplicial complexes

Klaus, Stephan

doi:10.1007/s11464-016-0575-2

Cited by 3 publications

(5 citation statements)

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“…where χ(M) = ∑ 3 i=0 (−1) i N i is the Euler characteristic of the simplicial complex M [79]. Thus, a non-vanishing cosmological constant implies a non-trivial topology of spacetime.…”

Section: The Cosmological Constant Problemmentioning

confidence: 99%

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Spacetime as a Complex Network and the Cosmological Constant Problem

Nesterov

2023

Universe

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We propose a promising model of discrete spacetime based on nonassociative geometry and complex networks. Our approach treats space as a simplicial 3-complex (or complex network), built from “atoms” of spacetime and entangled states forming n-dimensional simplices (n=1,2,3). At large scales, a highly connected network is a coarse, discrete representation of a smooth spacetime. We show that, for high temperatures, the network describes disconnected discrete space. At the Planck temperature, the system experiences phase transition, and for low temperatures, the space becomes a triangulated discrete space. We show that the cosmological constant depends on the Universe’s topology. The “foamy” structure, analogous to Wheeler’s “spacetime foam”, significantly contributes to the effective cosmological constant, which is determined by the Euler characteristic of the Universe.

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“…where χ(M) = ∑ 3 i=0 (−1) i N i is the Euler characteristic of the simplicial complex M [79]. Thus, a non-vanishing cosmological constant implies a non-trivial topology of spacetime.…”

Section: The Cosmological Constant Problemmentioning

confidence: 99%

“…Remark 1. If M is a simplicial decomposition of the three-dimensional manifold, then χ(M) = 0 [79]. Thus, a non-trivial topology of the Universe is mandatory for a non-vanishing cosmological constant.…”

Section: The Cosmological Constant Problemmentioning

confidence: 99%

Spacetime as a Complex Network and the Cosmological Constant Problem

Nesterov

2023

Universe

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“…What we are not considering here, are combinatorial analogs of the Gauss-Bonnet theorem (though [14] was an incentive for the present consideration). For example, let Z be a finite cell complex in R n (the cells are convex polytopes), and let ∆ k denote the set of k-dimensional cells.…”

Section: Definitionmentioning

confidence: 99%

“…where |Z| = n k=0 Z∈∆ k Z denotes the underlying space of Z. In a recent paper by Klaus [14], Theorem 2.1 is formula (3) above for the special case of simplicial complexes. Theorem 5.1 of that paper uses, for Euclidean simplicial complexes, a differently defined vertex curvature, involving internal angles.…”

Section: Definitionmentioning

confidence: 99%

“…The Euler characteristic of P can conveniently be calculated via the simplicial complex, but depends only on P and not on the chosen triangulation. However, if we want to determine the Euler characteristic of P by summing vertex curvatures according to one of the methods in [18,14], then the vertex curvatures do depend on the special triangulation. In contrast, a polyhedral vertex curvature of P at x according to Definition 1 depends only on the set P , in arbitrarily small neighborhoods of x.…”

Section: Introductionmentioning

confidence: 99%

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Polyhedral Gauss–Bonnet theorems and valuations

Schneider

2017

Beitr Algebra Geom

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The Gauss-Bonnet theorem for a polyhedron (a union of finitely many compact convex polytopes) in n-dimensional Euclidean space expresses the Euler characteristic of the polyhedron as a sum of certain curvatures, which are different from zero only at the vertices of the polyhedron. This note suggests a generalization of these polyhedral vertex curvatures, based on valuations, and thus obtains a variety of polyhedral Gauss-Bonnet theorems.

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Spacetime as a Complex Network and the Cosmological Constant Problem

Nesterov¹

2023

Preprint

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We propose a promising model of discrete spacetime based on nonassociative geometry and complex networks. Our approach treats space as a simplicial 3-complex (or complex network), built from "atoms" of spacetime and entangled states forming n-dimensional simplices (n=1,2,3). At large scales, a highly connected network is a coarse, discrete representation of a smooth spacetime. We show that for high temperatures, the network describes disconnected discrete space. At the Planck temperature, the system experiences the phase transition, and for low temperatures, the space becomes a triangulated discrete space. We show that the cosmological constant depends on the universe’s topology. The "foamy" structure, analogous to Wheeler’s "spacetime foam," significantly contributes to the effective cosmological constant, which is determined by the Euler characteristic of the universe.

show abstract

On combinatorial Gauss-Bonnet Theorem for general Euclidean simplicial complexes

Cited by 3 publications

References 13 publications

Spacetime as a Complex Network and the Cosmological Constant Problem

Spacetime as a Complex Network and the Cosmological Constant Problem

Polyhedral Gauss–Bonnet theorems and valuations

Spacetime as a Complex Network and the Cosmological Constant Problem

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