论文标题
几乎每个路径结构都不是变化的
Almost every path structure is not variational
论文作者
论文摘要
鉴于一个平稳的无参数曲线家族，使得在每个方向的每个点都完全通过一条曲线，是否存在一个Lagrangian，而极端人正是这个家庭？众所周知，在维度2中，答案是积极的。在维度3中，从道格拉斯（Douglas）的工作来看，答案通常是负面的。我们将此结果推广到所有更高的维度，并表明对几乎每个这样的曲线系列（也称为路径结构或路径几何形状）的答案实际上是负面的。另一方面，我们考虑具有无限对称性的路径几何形状，并表明具有次最大对称尺寸的路径和投射结构是变化的。请注意，所谓的Egorov结构具有次最大对称代数的投影结构，不可能伪METRIZIAN；我们证明它在Kropina伪量表的类别中是可以Metriz的，并明确构建了相应的Kropina Lagrangian。
Given a smooth family of unparameterized curves such that through every point in every direction there passes exactly one curve, does there exist a Lagrangian with extremals being precisely this family? It is known that in dimension 2 the answer is positive. In dimension 3, it follows from the work of Douglas that the answer is, in general, negative. We generalise this result to all higher dimensions and show that the answer is actually negative for almost every such a family of curves, also known as path structure or path geometry. On the other hand, we consider path geometries possessing infinitesimal symmetries and show that path and projective structures with submaximal symmetry dimensions are variational. Note that the projective structure with the submaximal symmetry algebra, the so-called Egorov structure, is not pseudo-Riemannian metrizable; we show that it is metrizable in the class of Kropina pseudo-metrics and explicitly construct the corresponding Kropina Lagrangian.
