General-affine invariants of plane curves and space curves

Shimpei Kobayashi; Takeshi Sasaki

Czechoslovak Mathematical Journal (2020)

  • Volume: 70, Issue: 1, page 67-104
  • ISSN: 0011-4642

Abstract

top
We present a fundamental theory of curves in the affine plane and the affine space, equipped with the general-affine groups GA ( 2 ) = GL ( 2 , ℝ ) ⋉ ℝ 2 and GA ( 3 ) = GL ( 3 , ℝ ) ⋉ ℝ 3 , respectively. We define general-affine length parameter and curvatures and show how such invariants determine the curve up to general-affine motions. We then study the extremal problem of the general-affine length functional and derive a variational formula. We give several examples of curves and also discuss some relations with equiaffine treatment and projective treatment of curves.

How to cite

top

Kobayashi, Shimpei, and Sasaki, Takeshi. "General-affine invariants of plane curves and space curves." Czechoslovak Mathematical Journal 70.1 (2020): 67-104. <http://eudml.org/doc/297149>.

@article{Kobayashi2020,
abstract = {We present a fundamental theory of curves in the affine plane and the affine space, equipped with the general-affine groups $\{\rm GA\}(2)=\{\rm GL\}(2,\{\mathbb \{R\}\})\ltimes \{\mathbb \{R\}\}^2$ and $\{\rm GA\}(3)=\{\rm GL\}(3,\{\mathbb \{R\}\})\ltimes \{\mathbb \{R\}\}^3$, respectively. We define general-affine length parameter and curvatures and show how such invariants determine the curve up to general-affine motions. We then study the extremal problem of the general-affine length functional and derive a variational formula. We give several examples of curves and also discuss some relations with equiaffine treatment and projective treatment of curves.},
author = {Kobayashi, Shimpei, Sasaki, Takeshi},
journal = {Czechoslovak Mathematical Journal},
keywords = {plane curve; space curve; general-affine group; general-affine curvature; variational problem},
language = {eng},
number = {1},
pages = {67-104},
publisher = {Institute of Mathematics, Academy of Sciences of the Czech Republic},
title = {General-affine invariants of plane curves and space curves},
url = {http://eudml.org/doc/297149},
volume = {70},
year = {2020},
}

TY - JOUR
AU - Kobayashi, Shimpei
AU - Sasaki, Takeshi
TI - General-affine invariants of plane curves and space curves
JO - Czechoslovak Mathematical Journal
PY - 2020
PB - Institute of Mathematics, Academy of Sciences of the Czech Republic
VL - 70
IS - 1
SP - 67
EP - 104
AB - We present a fundamental theory of curves in the affine plane and the affine space, equipped with the general-affine groups ${\rm GA}(2)={\rm GL}(2,{\mathbb {R}})\ltimes {\mathbb {R}}^2$ and ${\rm GA}(3)={\rm GL}(3,{\mathbb {R}})\ltimes {\mathbb {R}}^3$, respectively. We define general-affine length parameter and curvatures and show how such invariants determine the curve up to general-affine motions. We then study the extremal problem of the general-affine length functional and derive a variational formula. We give several examples of curves and also discuss some relations with equiaffine treatment and projective treatment of curves.
LA - eng
KW - plane curve; space curve; general-affine group; general-affine curvature; variational problem
UR - http://eudml.org/doc/297149
ER -

References

top
  1. Bagderina, Y. Y., 10.1111/j.1467-9590.2008.00400.x, Stud. Appl. Math. 120 (2008), 293-332. (2008) Zbl1196.34047MR2406822DOI10.1111/j.1467-9590.2008.00400.x
  2. Berzolari, L., 10.1007/BF02346203, Annali di Math., Ser 2 Italian 26 (1897), 1-58 9999JFM99999 28.0584.04. (1897) DOI10.1007/BF02346203
  3. Blaschke, W., Vorlesungen über Differentialgeometrie und geometrische Grundlagen von Einsteins Relativitätstheorie. II. Affine Differentialgeometrie, bearbeitet von K. Reidemeister, Springer, Berlin German (1923),9999JFM99999 49.0499.01. (1923) MR0015247
  4. Bol, G., Projektive Differentialgeometrie. I. Teil, Vandenhoeck & Ruprecht, Göttingen German (1950). (1950) Zbl0035.23401MR0034066
  5. Calugareanu, G., Gheorghiu, G. T., Sur l'interprétation géométrique des invariants différentiels fondamentaux en géométrie affine et projective des courbes planes, Bull. Math. Soc. Roum. Sci. 43 French (1941), 69-83. (1941) Zbl0060.36603MR0012877
  6. Cartan, E., Sur un problème du calcul des variations en géométrie projective plane, Moscou, Rec. Math. 34 French (1927), 349-364 9999JFM99999 53.0486.01. (1927) 
  7. Cartan, E., Lecons sur la théorie des espaces à connexion projective, Cahiers scient. 17, Gauthier-Villars. VI, Paris French (1937). (1937) Zbl0016.07603MR0041456
  8. Chazy, J., 10.1007/BF02393131, Acta Math. 34 French (1911), 317-385 9999JFM99999 42.0340.03. (1911) MR1555070DOI10.1007/BF02393131
  9. Chou, K.-S., Qu, C., 10.1016/S0167-2789(01)00364-5, Physica D 162 (2002), 9-33. (2002) Zbl0987.35139MR1882237DOI10.1016/S0167-2789(01)00364-5
  10. Fubini, G., Čech, E., Introduction à la géométrie projective différentielle des surfaces, Gauthier-Villars and Cie VI, Paris French (1931). (1931) Zbl0005.31102MR1562218
  11. Griffiths, P. A., 10.1007/978-1-4615-8166-6, Progress in Mathematics 25, Birkhäuser/Springer, Basel (1983). (1983) Zbl0512.49003MR0684663DOI10.1007/978-1-4615-8166-6
  12. Halphen, G. H., Sur les invariants différentielles, Oeuvre II Gauthier-Villars, Paris French (1918), 197-257 9999JFM99999 46.1418.01. (1918) 
  13. Izumiya, S., Sano, T., 10.1017/S0013091500019672, Proc. Edinb. Math. Soc., II. Ser. 41 (1998), 315-324. (1998) Zbl0965.53013MR1626425DOI10.1017/S0013091500019672
  14. Kimpara, M., 10.11429/ppmsj1919.19.0_977, Proc. Phys.-Math. Soc. Japan, III. Ser. 19 French (1937), 977-983. (1937) Zbl0018.08802DOI10.11429/ppmsj1919.19.0_977
  15. Lane, E. P., A Treatise on Projective Differential Geometry, University of Chicago Press, Chicago (1942). (1942) Zbl0063.03443MR0007286
  16. Beffa, G. Marí, 10.1016/j.physd.2008.08.009, Physica D 238 (2009), 100-115. (2009) Zbl1163.37023MR2571970DOI10.1016/j.physd.2008.08.009
  17. Mihăilescu, T., Géométrie différentielle affine des courbes planes, Czech. Math. J. 9 French (1959), 265-288. (1959) Zbl0089.17002MR0105704
  18. Mihăilescu, T., Sobre la variacion del arco afin de las curvas planas, Math. Notae 17 Spanish (1961), 59-81. (1961) Zbl0108.34202MR0146691
  19. Mihăilescu, T., Geometria diferencial afin general de las curvas alabeadas, Math. Notae 18 Spanish (1963), 23-70. (1963) Zbl0114.37102MR0156269
  20. Monge, G., Sur les équations différentielles des courbes du second degré, Corresp. sur l'École imp. Polytechnique Klostermann, Paris M. Hachette French (1810), 51-54. (1810) 
  21. Musso, E., 10.3842/SIGMA.2012.030, SIGMA, Symmetry Integrability Geom. Methods Appl. 8 (2012), paper 030, 20 pages. (2012) Zbl1246.53012MR2942809DOI10.3842/SIGMA.2012.030
  22. Musso, E., Grant, J. D. E., 10.1016/j.geomphys.2003.10.005, J. Geom. Phys. 50 (2004), 303-338. (2004) Zbl1076.58011MR2078230DOI10.1016/j.geomphys.2003.10.005
  23. Musso, E., Nicolodi, L., 10.1515/form.2005.17.4.569, Forum Math. 17 (2005), 569-590. (2005) Zbl1084.53012MR2154420DOI10.1515/form.2005.17.4.569
  24. Nomizu, K., Sasaki, T., Affine Differential Geometry, Cambridge Tracts in Mathematics 111, Cambridge University Press, Cambridge (1994). (1994) Zbl0834.53002MR1311248
  25. Olver, P. J., Sapiro, G., Tannenbaum, A., Classification and uniqueness of invariant geometric flows, C. R. Acad. Sci., Paris, Sér. I 319 (1994), 339-344. (1994) Zbl0863.53008MR1289308
  26. Ovsienko, V., Tabachnikov, S., 10.1017/CBO9780511543142, Cambridge Tracts in Mathematics 165, Cambridge University Press, Cambridge (2005). (2005) Zbl1073.53001MR2177471DOI10.1017/CBO9780511543142
  27. Polyanin, A. D., Zaitsev, V. F., 10.1201/9781420035339, CRC Press, Boca Raton (1995). (1995) Zbl0855.34001MR1396087DOI10.1201/9781420035339
  28. Sasaki, S., 10.4099/jjm1924.13.0_473, Jap. J. Math. 13 (1937), 473-481. (1937) Zbl0018.17003DOI10.4099/jjm1924.13.0_473
  29. Sasaki, T., Projective Differential Geometry and Linear Homogeneous Differential Equations, Rokko Lectures in Mathematics 5, Kobe University (1999). (1999) 
  30. Schirokow, P. A., Schirokow, A. P., Affine Differentialgeometrie, B. G. Teubner, Leipzig German (1962). (1962) Zbl0106.14703MR0150666
  31. Thorbergsson, G., Umehara, M., 10.1017/S0027763000025435, Nagoya Math. J. 167 (2002), 55-94. (2002) Zbl1088.53049MR1924719DOI10.1017/S0027763000025435
  32. Verpoort, S., 10.1007/s10587-011-0064-4, Czech. Math. J. 61 (2011), 419-435. (2011) Zbl1249.49028MR2905414DOI10.1007/s10587-011-0064-4
  33. Wilczynski, E. J., 10.1007/BF01736764, B. G. Teubner, Leipzig German (1906),9999JFM99999 37.0620.02. (1906) MR0131232DOI10.1007/BF01736764

NotesEmbed ?

top

You must be logged in to post comments.

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

Only the controls for the widget will be shown in your chosen language. Notes will be shown in their authored language.

Tells the widget how many notes to show per page. You can cycle through additional notes using the next and previous controls.

    
                

                
Note: Best practice suggests putting the JavaScript code just before the closing </body> tag.