# Automorphisms of spatial curves

Archivum Mathematicum (1997)

- Volume: 033, Issue: 3, page 213-243
- ISSN: 0044-8753

## Access Full Article

top## Abstract

top## How to cite

topBradáč, Ivan. "Automorphisms of spatial curves." Archivum Mathematicum 033.3 (1997): 213-243. <http://eudml.org/doc/18498>.

@article{Bradáč1997,

abstract = {Automorphisms of curves $y= y(x)$, $z=z(x)$ in $\{\mathbf \{R\}\}^3$ are investigated; i.e. invertible transformations, where the coordinates of the transformed curve $\bar\{y\}=\bar\{y\}(\bar\{x\})$, $\bar\{z\}= \bar\{z\}(\bar\{x\})$ depend on the derivatives of the original one up to some finite order $m$. While in the two-dimensional space the problem is completely resolved (the only possible transformations are the well-known contact transformations), the three-dimensional case proves to be much more complicated. Therefore, results (in the form of some systems of partial differential equations for the functions, determining the automorphisms) only for the special case $\bar\{x\} =x$ and order $m\le 2$ are obtained. Finally, the problem of infinitesimal transformations is briefly mentioned.},

author = {Bradáč, Ivan},

journal = {Archivum Mathematicum},

keywords = {automorphisms of curves; infinite-dimensional space; contact forms; automorphisms of curves; contact forms},

language = {eng},

number = {3},

pages = {213-243},

publisher = {Department of Mathematics, Faculty of Science of Masaryk University, Brno},

title = {Automorphisms of spatial curves},

url = {http://eudml.org/doc/18498},

volume = {033},

year = {1997},

}

TY - JOUR

AU - Bradáč, Ivan

TI - Automorphisms of spatial curves

JO - Archivum Mathematicum

PY - 1997

PB - Department of Mathematics, Faculty of Science of Masaryk University, Brno

VL - 033

IS - 3

SP - 213

EP - 243

AB - Automorphisms of curves $y= y(x)$, $z=z(x)$ in ${\mathbf {R}}^3$ are investigated; i.e. invertible transformations, where the coordinates of the transformed curve $\bar{y}=\bar{y}(\bar{x})$, $\bar{z}= \bar{z}(\bar{x})$ depend on the derivatives of the original one up to some finite order $m$. While in the two-dimensional space the problem is completely resolved (the only possible transformations are the well-known contact transformations), the three-dimensional case proves to be much more complicated. Therefore, results (in the form of some systems of partial differential equations for the functions, determining the automorphisms) only for the special case $\bar{x} =x$ and order $m\le 2$ are obtained. Finally, the problem of infinitesimal transformations is briefly mentioned.

LA - eng

KW - automorphisms of curves; infinite-dimensional space; contact forms; automorphisms of curves; contact forms

UR - http://eudml.org/doc/18498

ER -

## References

top- Lie S., Geometrie der Berührungstransformationen, erster Band, Leipzig 1896. Zbl0406.01015
- Anderson R., Ibragimov N., Lie-Bäcklund transformations in applications, Philadelphia 1979. (1979) Zbl0447.58001MR0520395
- Ibragimov N., Transformation groups in mathematical physics, Moscow, Nauka, 1983 (Russian) (1983) Zbl0529.53014MR0734307
- Carathèodory C., Variationsrechnung und partielle Differentialgleichungen erster Ordnung, Band I, Theorie der partielen Differentialgleichungen erster Ordnung, Zweite Auflage, Leipzig 1956. (1956) Zbl0070.31601MR0089338
- Shlomo Sternberg, Lectures on Differential Geometry, Prentice-Hall, Inc. Englewood Cliffs, New Jersey, 1965. (1965) MR0193578
- Chrastina J., From Elementary Algebra to Bäcklund Transformations, Czechoslovak Mathematical Journal, 40 (115) 1990, Praha. (1990) Zbl0726.58041MR1046292
- Chrastina J., Formal theory of differential equations, (to appear). Zbl0906.35002MR1656843
- Chrastina J., On the Equivalence of Variational Problems, I, Journal of Differential Equations, Vol. 98, No. 1, July 1992. (1992) Zbl0764.49008MR1168972
- Stormark O., Formal and local solvability of partial differential equations, Trita-Mat-1989-11, Mathematics, ch. 1–12, Royal Institute of Technology, Stockholm 1989. (1989)
- Pressley A., Segal G., Loop Groups, Clarendon Press, Oxford 1986, Russian translation Moscow, Mir, 1990. (1986) Zbl0618.22011MR1071737
- Cartan E., Les systèmes différentiels extérieurs et leurs applications géometriques, Gauthier-Villars, Paris 1945, Russian translation Moscow University 1962. (1945) Zbl0063.00734MR0016174
- Olver P., Applications of Lie Groups to Differential Equations, 1986, Springer-Verlag, Russian translation Moscow, Mir, 1989. (1989) Zbl0743.58003MR0836734
- Vinogradov A. M., Krasilščik I. S., Lygačin V. V., Introduction into the geometry of nonlinear differential equations, Moscow 1986 (Russian). (1986)

## NotesEmbed ?

topTo embed these notes on your page include the following JavaScript code on your page where you want the notes to appear.