On the curvature of piecewise flat spaces

Cheeger, Jeff; Müller, Werner; Schrader, Robert

doi:10.1007/bf01210729

Cited by 339 publications

(411 citation statements)

References 23 publications

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“…Rigorous justification for this formula followed in [3], where it was shown that it converges to the continuum form of the action, in the sense of measures, provided that certain conditions on the fatness of the simplices are satisfied. Friedberg and Lee [4] approached the problem from the opposite direction, deriving the Regge action from a sequence of continuum spaces approaching a discrete one.…”

Section: A Basic Formalismmentioning

confidence: 99%

Discrete structures in gravity

Regge

Williams²

2000

Journal of Mathematical Physics

164

293

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Discrete approaches to gravity, both classical and quantum, are reviewed briefly, with emphasis on the method using piecewise-linear spaces. Models of 3-dimensional quantum gravity involving 6j-symbols are then described, and progress in generalising these models to four dimensions is discussed, as is the relationship of these models in both three and four dimensions to topological theories. Finally, the repercussions of the generalisations are explored for the original formulation of discrete gravity using edge-length variables. I Introduction to discrete gravity A Basic formalismThe original motivation for the development of a discrete formalism for gravity [1] arose from a number of problems with the continuum formulation of general relativity. These included the difficulty of solving Einstein's equations for general systems without a large degree of symmetry, the problems of representing complicated topologies and the need for considerable geometric insight and capacity for visualisation. It turned out, as we shall see, that the discretisation scheme to be described not only helped with these problems but also found a vital rôle in numerical relativity and in attempts at a formulation of quantum gravity.The related branches of mathematics which found their application to physics in this formulation of gravity are those of piecewise-linear spaces and topology and the geometric notion of intrinsic curvature on polyhedra. The immediate aim was to develop an approach to general relativity which avoided the 1

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Section: A Basic Formalismmentioning

confidence: 99%

Discrete structures in gravity

Regge

Williams²

2000

Journal of Mathematical Physics

164

293

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“…Alternatively, of course the summation can be done over the points p, instead of the simplices s, (as in the exact expression of ref. [20] This second choice for the definition of C 2 (sum over sites) appears in fact more natural when one considers the coupling of gravity to matter fields, which will be represented here for simplicity by a scalar field ~(x). In the continuum an invariant action, up to terms quadratic in ~, is [4] Imatter = f d4x v/g [½g ~ 0fl~ 0~b +½m2@ 2 + gtR¢~ 2 -' 1"-g2 R~ 0tt~b 01, ~ n t-... ].…”

Section: Rtor(j) -[ A-~sv U~"u~ I(o[ # Up°uoomentioning

confidence: 99%

Simplicial quantum gravity with higher derivative terms: Formalism and numerical results in four dimensions

Hamber

Williams

1986

Nuclear Physics B

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“…The version of the combinatorial Gauss-Bonnet theorem due to Cheeger, Müller and Schrader is essentially a quick calculation [38]. See also the paper by Charney and Davis [36] where they review this calculation and then use the resulting formula to find a combinatorial analogue of the Hopf conjecture in the context of nonpositively curved piecewise Euclidean n-manifolds.…”

Section: Combinatorial Gauss-bonnetmentioning

confidence: 99%

Constructing non-positively curved spaces and groups

McCammond¹

2009

Geometric and Cohomological Methods in Group Theory

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Abstract. The theory of non-positively curved spaces and groups is tremendously powerful and has enormous consequences for the groups and spaces involved. Nevertheless, our ability to construct examples to which the theory can be applied has been severely limited by an inability to test -in real timewhether a random finite piecewise Euclidean complex is non-positively curved. In this article I focus on the question of how to construct examples of nonpositively curved spaces and groups, highlighting in particular the boundary between what is currently do-able and what is not yet feasible. Since this is intended primarily as a survey, the key ideas are merely sketched with references pointing the interested reader to the original articles.Over the past decade or so, the consequences of non-positive curvature for geometric group theorists have been throughly investigated, most prominently in the book by Bridson and Haefliger [26]. See also the recent review article by Kleiner in the Bulletin of the AMS [59] and the related books by Ballmann [4], BallmannGromov-Schroeder [5] and the original long article by Gromov [48]. In this article I focus not on the consequences of the theory, but rather on the question of how to construct examples to which it applies. The structure of the article roughly follows the structure of the lectures I gave during the Durham symposium with the four parts corresponding to the four talks. Part 1 introduces the key problems and presents some basic decidability results, Part 2 focuses on practical algorithms and low-dimensional complexes, Part 3 presents case studies involving special classes of groups such as Artin groups, small cancellation groups and ample-twisted face pairing 3-manifolds. In Part 4 I explore the weaker notion of conformal non-positive curvature and introduce the notion of an angled n-complex. As will become clear, the topics covered have a definite bias towards research in which I have played some role. These are naturally the results with which I am most familiar and I hope the reader will pardon this lack of a more impartial perspective. Part 1. Negative curvatureAlthough the definitions of δ-hyperbolic and CAT(0) spaces / groups are wellknown, it is perhaps under-appreciated that there are at least seven potentially distinct classes of groups which all have some claim to the description "negativelycurved." Sections 1 and 2 briefly review the definitions needed in order to describe these classes, the known relationships between them, and the current status of the (rather optimisitic) conjecture that all seven of these classes are identical. Since the emphasis in this article is on the construction of examples, finite piecewise

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On the curvature of piecewise flat spaces

Cited by 339 publications

References 23 publications

Discrete structures in gravity

Discrete structures in gravity

Simplicial quantum gravity with higher derivative terms: Formalism and numerical results in four dimensions

Constructing non-positively curved spaces and groups

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