Surfaces of genus $g\geq 1$ in 3D contact sub-Riemannian manifolds

Eugenio Bellini; Ugo Boscain

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Surfaces of genus $g\geq 1$ in 3D contact sub-Riemannian manifolds

Eugenio Bellini, Ugo Boscain

Differential Geometry · Submitted 5 May 2023

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

The authors' main claim In this paper we study closed surfaces of genus larger than 1, proving that such surfaces can be embedded in such a way that the induced distance is finite or infinite.

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Compiled by classifier-1.13.0 via TypeSafe API, question bank abstract-p1-20260923, 2026-09-23: 1 request, 3,633 input tokens.

What the abstract itself states, as probabilities: claims first 0.06 · states limitation 0.25 · human participants 0.01 · controlled comparison 0.03 · randomized assignment 0.01 · states quantitative result 0.10 · compares with prior results 0.88 · mentions code or data availability 0.01.

The abstract, sentence by sentence

arXiv listing abstract v2. Results and the main claim are underlined; show every role.

We consider surfaces embedded in a 3D contact sub-Riemannian manifold and the problem of the finiteness of the induced distance (i.e., the infimum of the length of horizontal curves that belong to the surface). Recently it has been proved that for a surface having the topology of a sphere embedded in a tight co-orientable structure, the distance is always finite. In this paper we study closed surfaces of genus larger than 1, proving that such surfaces can be embedded in such a way that the induced distance is finite or infinite. We then study the structural stability of the finiteness/not-finiteness of the distance.

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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	We consider surfaces embedded in a 3D contact sub-Riemannian manifold and the problem of the finiteness of the induced distance (i.e., the infimum of the length of horizontal curves that belong to the surface).	background 0.95	0.32	0.03	unclear 0.58
s1	Recently it has been proved that for a surface having the topology of a sphere embedded in a tight co-orientable structure, the distance is always finite.	background 0.99	0.04	0.04	asserted 1.00
s2	In this paper we study closed surfaces of genus larger than 1, proving that such surfaces can be embedded in such a way that the induced distance is finite or infinite.	result 0.78	0.93	0.87	asserted 0.97
s3	We then study the structural stability of the finiteness/not-finiteness of the distance.	contribution 1.00	0.86	0.07	unclear 0.50

What kind of paper the classifier reports this is: theoretical result 1.00 · evidence from proof or derivation 0.99.

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arXiv listing abstract v2

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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

We consider surfaces embedded in a 3D contact sub-Riemannian manifold and the problem of the finiteness of the induced distance (i.e., the infimum of the length of horizontal curves that belong to the surface). Recently it has been proved that for a surface having the topology of a sphere embedded in a tight co-orientable structure, the distance is always finite. In this paper we study closed surfaces of genus larger than 1, proving that such surfaces can be embedded in such a way that the induced distance is finite or infinite. We then study the structural stability of the finiteness/not-finiteness of the distance.

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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

What it cites

Cites as background 1 paper

See for instance [ref] for Carnot groups, [ref] for generic structures, [ref] for Gauss-Bonnet theorems and [ref] for stochastic evolution equations.

Limit of Gaussian and normal curvatures of surfaces in Riemannian approximation scheme for sub-Riemannian three dimensional manifolds and Gauss-Bonnet theorem · Introduction · 0.96

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