The Geometry of $ d^{2}y^{1}/dt^{2} = f(y, dot{y},t) ; ext{and} ; d^{2}y^{2}/dt^{2} = g(y,dot{y},t)$, and Euclidean Spaces

Canadian mathematical bulletin

关键词: Mean curvature;    Ricci curvature;   

DOI  :  10.4153/CMB-2006-018-7

学科分类：数学（综合）

来源: University of Toronto Press * Journals Division

【 摘 要 】

This paper investigates the relationship between a system ofdifferential equations and the underlying geometry associated withit. The geometry of a surface determines shortest paths, orgeodesics connecting nearby points, which are defined as thesolutions to a pair of second-order differential equations: theEuler--Lagrange equations of the metric. We ask when the converseholds, that is, when solutions to a system of differentialequations reveals an underlying geometry. Specifically, when maythe solutions to a given pair of second order ordinarydifferential equations $d^{2}y^{1}/dt^{2} = f(y,dot{y},t)$ and$d^{2}y^{2}/dt^{2} = g(y,dot{y},t)$ be reparameterized by$tightarrow T(y,t)$ so as to give locally the geodesics of aEuclidean space? Our approach is based upon Cartan's method ofequivalence. In the second part of the paper, the equivalenceproblem is solved for a generic pair of second order ordinarydifferential equations of the above form revealing the existenceof 24 invariant functions.

【 授权许可】

Unknown