Canadian mathematical bulletin | |
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 | |
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
【 预 览 】
Files | Size | Format | View |
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RO201912050576459ZK.pdf | 36KB | download |