# Automorphisms of spatial curves

Archivum Mathematicum (1997)

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

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## Abstract

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Automorphisms of curves $y=y\left(x\right)$, $z=z\left(x\right)$ in ${𝐑}^{3}$ are investigated; i.e. invertible transformations, where the coordinates of the transformed curve $\overline{y}=\overline{y}\left(\overline{x}\right)$, $\overline{z}=\overline{z}\left(\overline{x}\right)$ 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 $\overline{x}=x$ and order $m\le 2$ are obtained. Finally, the problem of infinitesimal transformations is briefly mentioned.

## How to cite

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

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

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9. 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)
10. Pressley A., Segal G., Loop Groups, Clarendon Press, Oxford 1986, Russian translation Moscow, Mir, 1990. (1986) Zbl0618.22011MR1071737
11. Cartan E., Les systèmes différentiels extérieurs et leurs applications géometriques, Gauthier-Villars, Paris 1945, Russian translation Moscow University 1962. (1945) Zbl0063.00734MR0016174
12. Olver P., Applications of Lie Groups to Differential Equations, 1986, Springer-Verlag, Russian translation Moscow, Mir, 1989. (1989) Zbl0743.58003MR0836734
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