Manifold Ways to Darboux-Halphen System

Symmetry Integrability and Geometry-Methods and Applications

article

John Alexander Cruz Morales¹  Hossein Movasati²  Younes Nikdelan³  Raju Roychowdhury⁴  Marcus A.C. Torres² 

[1] Universidad Nacional de Colombia;Instituto Nacional de Matemática Pura e Aplicada (IMPA);Instituto de Matemática e Estatística (IME), Universidade do Estado do Rio de Janeiro (UERJ);Instituto de Física, Universidade de São Paulo (IF-USP)

关键词: Darboux–Halphen system;    Ramanujan system;    Gauss–Manin connection;    relativity and gravitational theory;    Bianchi IX metric;    Frobenius manifold;    Chazy equation;   

DOI  :  10.3842/SIGMA.2018.003

来源: National Academy of Science of Ukraine

【 摘 要 】

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.

【 授权许可】

Unknown