Geodesic
Three-webs $W(1,n,1)$ and associated systems of ordinary differential equations
Izvestiâ vysših učebnyh zavedenij. Matematika, no. 2 (2012), pp. 43-56

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We consider a three-web formed by two $n$-parameter families of curves and an one-parameter family of hypersurfaces on a smooth manifold. For such webs we define a family of adapted frames, formulate a system of structural equations, and study differential-geometric objects that arise in differential neighborhoods up to the third order. We prove that each system of ordinary differential equations (SODE) uniquely defines some three-web. This allows us to describe properties of SODE in terms of the corresponding three-web. In particular, we characterize autonomous SODE.
Keywords: multidimensional three-web, system of ordinary differential equations, affine connection. 
@article{IVM_2012_2_a4,
     author = {A. A. Duyunova},
     title = {Three-webs $W(1,n,1)$ and associated systems of ordinary differential equations},
     journal = {Izvesti\^a vys\v{s}ih u\v{c}ebnyh zavedenij. Matematika},
     pages = {43--56},
     publisher = {mathdoc},
     number = {2},
     year = {2012},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/IVM_2012_2_a4/}
}
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A. A. Duyunova. Three-webs $W(1,n,1)$ and associated systems of ordinary differential equations. Izvestiâ vysših učebnyh zavedenij. Matematika, no. 2 (2012), pp. 43-56. http://geodesic.mathdoc.fr/item/IVM_2012_2_a4/