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7 - Pencils of Quadrics, Billiard Double-Reflection and Confocal Incircular Nets
- By Vladimir Dragović, Milena Radnović, Roger Fidèle Ranomenjanahary
Edited by Ron Donagi, University of Pennsylvania, Tony Shaska, Oakland University, Michigan
Book:

Integrable Systems and Algebraic Geometry

Published online:

19 March 2020

Print publication:

02 April 2020, pp 133-166
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Summary

We present recent results about double reflection and incircular nets. The building blocks are pencils of quadrics, related billiards and quad graphs.

5 - First Integrals and Integrable Systems
Andrei Agrachev, Scuola Internazionale Superiore di Studi Avanzati, Trieste, Davide Barilari, Université de Paris VII (Denis Diderot), Ugo Boscain, Centre National de la Recherche Scientifique (CNRS), Paris
Book:

A Comprehensive Introduction to Sub-Riemannian Geometry

Published online:

28 October 2019

Print publication:

31 October 2019, pp 149-170
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Summary

In this chapter we present some applications of theHamiltonian formalism developed in Chapter 4. Wegive a proof of the well-known Arnold–Liouvilletheorem and, as an application, we study thecomplete integrability of the geodesic flow on aspecial class of Riemannian manifolds.

Corrigendum: on a conjecture of Kontsevich and variants of Castelnuovo's lemma
J. M. Landsberg
Journal:

Compositio Mathematica / Volume 140 / Issue 4 / July 2004

Published online by Cambridge University Press:

04 December 2007, p. 1112

Print publication:

July 2004
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In the Mathematical Review of ‘On a conjecture of Kontsevich and variants of Castelnuovo's lemma’ [Compositio Mathematica 115 (1999), 205–230], Gizatullin pointed out that the proof as written is incomplete for the complex case because the possibility of a certain equation vanishing identically was ignored. This corrigendum addresses the missing case. Note that it is simply another iteration of the argument already present in the article.

On a Conjecture of Kontsevich and Variants of Castelnuovo‘s Lemma
J.M. LANDSBERG
Journal:

Compositio Mathematica / Volume 115 / Issue 2 / February 1999

Published online by Cambridge University Press:

04 December 2007, pp. 205-230

Print publication:

February 1999
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Let A=(aij) be an orthogonal matrix (over R or C) with no entries zero. Let B= (bij) be the matrix defined by bij= 1/ai j. M. Kontsevich conjectured that the rank of B is never equal to three. We interpret this conjecture geometrically and prove it. The geometric statement can be understood as variants of the Castelnuovo lemma and Brianchon‘s theorem.

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