Geodesic
On contact equivalence of Abel cubic differential equations of the second order
Izvestiâ vysših učebnyh zavedenij. Matematika, no. 3 (2016), pp. 93-96 Cet article a éte moissonné depuis la source Math-Net.Ru

Voir la notice de l'article

We study the geometry of cubic Abel differential equations on one-dimensional real curve. We prove that such an equation is the kernel of some non-linear differential operator. This operator is defined by a cubic on Cartan distribution in $1$-jet space. With the help of this observation we construct contact-invariant $\{e\}$-structure associated with non-degenerated Abel equation and obtain contact classification of such equations.
Keywords: cubic Abel equation, real curve
Mots-clés : jet space, contact pseudogroup, $\{e\}$-structure. 
@article{IVM_2016_3_a9,
     author = {P. V. Bibikov},
     title = {On contact equivalence of {Abel} cubic differential equations of the second order},
     journal = {Izvesti\^a vys\v{s}ih u\v{c}ebnyh zavedenij. Matematika},
     pages = {93--96},
     year = {2016},
     number = {3},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/IVM_2016_3_a9/}
}
TY  - JOUR
AU  - P. V. Bibikov
TI  - On contact equivalence of Abel cubic differential equations of the second order
JO  - Izvestiâ vysših učebnyh zavedenij. Matematika
PY  - 2016
SP  - 93
EP  - 96
IS  - 3
UR  - http://geodesic.mathdoc.fr/item/IVM_2016_3_a9/
LA  - ru
ID  - IVM_2016_3_a9
ER  - 
%0 Journal Article
%A P. V. Bibikov
%T On contact equivalence of Abel cubic differential equations of the second order
%J Izvestiâ vysših učebnyh zavedenij. Matematika
%D 2016
%P 93-96
%N 3
%U http://geodesic.mathdoc.fr/item/IVM_2016_3_a9/
%G ru
%F IVM_2016_3_a9
P. V. Bibikov. On contact equivalence of Abel cubic differential equations of the second order. Izvestiâ vysših učebnyh zavedenij. Matematika, no. 3 (2016), pp. 93-96. http://geodesic.mathdoc.fr/item/IVM_2016_3_a9/

[1] Bibikov P. V., “O geometrii kvadratichnykh obyknovennykh differentsialnykh uravnenii Abelya vtorogo poryadka”, Fundament. i prikl. matem. (to appear)

[2] Shurygin V. V. (jr.), The action of contact transformations pseudogroup on the second order ODEs which are cubic in second derivative, arXiv: 1211.6339v1

[3] Sternberg S., Lectures on differential geometry, Prentice Hall, Inc., Englewood Cliffs, N.J., 1964 | MR | Zbl

[4] Shurygin V. V. (jr.), “On the contact equivalence problem of second order ODEs which are quadratic with respect to the second order derivative”, Lobachevskii J. Math., 34:3 (2013), 264–271 | DOI | MR | Zbl

[5] Alekseevskii D. V., Vinogradov A. M., Lychagin V. V., Osnovnye idei i ponyatiya differentsialnoi geometrii, Itogi nauki i tekhn. Sovrem. probl. mat. Fundament. napravleniya, 28, VINITI, M., 1988 | MR | Zbl

[6] Akivis M. A., Goldberg V. V., Lychagin V. V., “Linearizability of $d$-webs, $d \geqslant4$, on two-dimensional manifolds”, Selecta Math., 10:4 (2004), 431–451 | DOI | MR | Zbl