Constant Scalar Curvature Metrics on Boundary Complexes of Cyclic Polytopes

Andrea Young; Andrew Marchese; Daniel Champion; Jacob Miller

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Constant Scalar Curvature Metrics on Boundary Complexes of Cyclic Polytopes

Daniel Champion, Andrew Marchese, Jacob Miller, Andrea Young

Differential Geometry · Submitted 15 Sep 2010

abstract compiled · 2026-09-23 · no language-model review

The authors' main claim We use combinatorial properties of cyclic polytopes to show that, for any number of vertices, these metrics have constant scalar curvature.

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arXiv ID1009.3061 (version 1)

Pith IDS2XI5YLK, Pith's permanent record of this paper and its review

CitationAndrea Young, Andrew Marchese, Daniel Champion, Jacob Miller. "Constant Scalar Curvature Metrics on Boundary Complexes of Cyclic Polytopes." arXiv:1009.3061 (2010). https://pith.science/paper/S2XI5YLK

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The paper's own sentences, arranged. Each label is drawn from a closed vocabulary; the number beside it is the probability the classifier gave that label. Numbers in the tables are the authors', checked by code. No sentence in this record was written by a model.

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Compiled by classifier-1.13-20260917 via OpenRouter, question bank abstract-p1-20260923, 2026-09-23: 1 request, 3,539 input tokens, $0.0001 reported.

What the abstract itself states, as probabilities: claims first 0.05 · states limitation 0.07 · human participants 0.01 · controlled comparison 0.01 · randomized assignment 0.01 · states quantitative result 0.09 · compares with prior results 0.27 · mentions code or data availability 0.02.

The abstract, sentence by sentence

arXiv listing abstract v1. Results and the main claim are underlined; show every role.

In [7], a notion of constant scalar curvature metrics on piecewise flat manifolds is defined. Such metrics are candidates for canonical metrics on discrete manifolds. In this paper, we define a class of vertex transitive metrics on certain triangulations of $\mathbb{S}^3$; namely, the boundary complexes of cyclic polytopes. We use combinatorial properties of cyclic polytopes to show that, for any number of vertices, these metrics have constant scalar curvature.

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What the classifier reports for each sentence. The number is the probability it gave the label.
#	sentence	role	the authors' own work	states the main conclusion	strength
s0	In [7], a notion of constant scalar curvature metrics on piecewise flat manifolds is defined.	background 1.00	0.03	0.02	asserted 1.00
s1	Such metrics are candidates for canonical metrics on discrete manifolds.	background 0.86	0.08	0.03	hedged possibility 0.77
s2	In this paper, we define a class of vertex transitive metrics on certain triangulations of $\mathbb{S}^3$; namely, the boundary complexes of cyclic polytopes.	contribution 1.00	0.94	0.09	asserted 0.99
s3	We use combinatorial properties of cyclic polytopes to show that, for any number of vertices, these metrics have constant scalar curvature.	result 0.97	0.96	0.90	asserted 1.00

What kind of paper the classifier reports this is: theoretical result 1.00 · evidence from proof or derivation 1.00.

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arXiv listing abstract v1

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full text
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references and citing sentences
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P abstract

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S sections
T body sentences
C citations
R cross-paper relations

Every quoted sentence on this record is the authors'. Every label is from Pith's closed vocabulary, with the probability shown beside it. Every number in the measurements table is the authors', checked by code. Record as JSON

The paper

Original abstract

In [7], a notion of constant scalar curvature metrics on piecewise flat manifolds is defined. Such metrics are candidates for canonical metrics on discrete manifolds. In this paper, we define a class of vertex transitive metrics on certain triangulations of $\mathbb{S}^3$; namely, the boundary complexes of cyclic polytopes. We use combinatorial properties of cyclic polytopes to show that, for any number of vertices, these metrics have constant scalar curvature.

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1 arXiv paper cite this in 2 sentences

Each citing sentence is taken from the citing paper's LaTeX source and typed by Pith's classifier. A label appears at probability 0.8 or more (0.95 for support), a contest at 0.7 or more; the quoted words are the citing authors'. How each label was checked · JSON

Papers citing it · 1 sentence without a confident label

Cites it as background 1 paper

In [ref] and many other papers, connections between discrete and continuous conformal geometry were investigated.

Conformally Covariant Operators and Conformal Invariants on Weighted Graphs · Discretization, and higher-dimensional complexes · 0.94

What it cites · 10 sentences without a confident label

Uses the method or tool of 1 paper

The condition for a metric can be described using Cayley-Menger determinants of the type described in [ref] .

Regge's Einstein-Hilbert Functional on the Double Tetrahedron · Piecewise flat manifolds · 0.84

Cites as background 2 papers

In this section, we will consider a certain conformal structure that has been studied in [ref] , [ref] , [ref] , [ref] .

Discrete conformal variations and scalar curvature on piecewise flat two and three dimensional manifolds · Discrete conformal structures and constant scalar curvature metrics · 0.95
In this section, we will consider a certain conformal structure that has been studied in [ref] , [ref] , [ref] , [ref] .

Regge's Einstein-Hilbert Functional on the Double Tetrahedron · Discrete conformal structures and constant scalar curvature metrics · 0.95

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