Symmetry, Integrability and Geometry: Methods and Applications (SIGMA)

SIGMA 14 (2018), 003, 14 pages      arXiv:1709.09682      https://doi.org/10.3842/SIGMA.2018.003
Contribution to the Special Issue on Modular Forms and String Theory in honor of Noriko Yui

Manifold Ways to Darboux-Halphen System

John Alexander Cruz Morales a, Hossein Movasati b, Younes Nikdelan c, Raju Roychowdhury d and Marcus A.C. Torres b
a) Departamento de Matemáticas, Universidad Nacional de Colombia, Bogotá, Colombia
b) Instituto Nacional de Matemática Pura e Aplicada (IMPA), Rio de Janeiro, Brazil
c) Instituto de Matemática e Estatística (IME), Universidade do Estado do Rio de Janeiro (UERJ), Rio de Janeiro, Brazil
d) Instituto de Física, Universidade de São Paulo (IF-USP), São Paulo, Brazil

Received September 29, 2017, in final form January 03, 2018; Published online January 08, 2018

Abstract
Many distinct problems give birth to Darboux-Halphen system of differential equations and here we review some of them. The first is the classical problem presented by Darboux and later solved by Halphen concerning finding infinite number of double orthogonal surfaces in $\mathbb{R}^3$. The second is a problem in general relativity about gravitational instanton in Bianchi IX metric space. The third problem stems from the new take on the moduli of enhanced elliptic curves called Gauss-Manin connection in disguise developed by one of the authors and finally in the last problem Darboux-Halphen system emerges from the associative algebra on the tangent space of a Frobenius manifold.

Key words: Darboux-Halphen system; Ramanujan system; Gauss-Manin connection; relativity and gravitational theory; Bianchi IX metric; Frobenius manifold; Chazy equation.

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