# Differential Equations and Para-CR Structures

• Volume: 3, Issue: 1, page 25-91
• ISSN: 0392-4041

top

## Abstract

top
We study the local geometry of n dimensional manifolds which are equipped with two integrable distributions, one of dimension $r$ and one of dimension $s$, where $r$ and $s$ are allowed to be unequal. We call them para-CR structures of type $(k,r,s)$, with $k=n-r-s\geq 0$ being the para-CR codimension. When $r=s$ they are the real analogues of CR structures. In the general case these structures are the natural geometric setting in which to discuss the geometry of systems of ODE's, as well as the geometry of systems of PDE's of finite type. For particular small values of $k,r,s$ we determine the basic local invariants of such structures.

## How to cite

top

Hill, C. Denson, and Nurowski, Paweł. "Differential Equations and Para-CR Structures." Bollettino dell'Unione Matematica Italiana 3.1 (2010): 25-91. <http://eudml.org/doc/290641>.

@article{Hill2010,
abstract = {We study the local geometry of n dimensional manifolds which are equipped with two integrable distributions, one of dimension $r$ and one of dimension $s$, where $r$ and $s$ are allowed to be unequal. We call them para-CR structures of type $(k,r,s)$, with $k = n - r - s \ge 0$ being the para-CR codimension. When $r = s$ they are the real analogues of CR structures. In the general case these structures are the natural geometric setting in which to discuss the geometry of systems of ODE's, as well as the geometry of systems of PDE's of finite type. For particular small values of $k,r,s$ we determine the basic local invariants of such structures.},
author = {Hill, C. Denson, Nurowski, Paweł},
journal = {Bollettino dell'Unione Matematica Italiana},
language = {eng},
month = {2},
number = {1},
pages = {25-91},
publisher = {Unione Matematica Italiana},
title = {Differential Equations and Para-CR Structures},
url = {http://eudml.org/doc/290641},
volume = {3},
year = {2010},
}

TY - JOUR
AU - Hill, C. Denson
AU - Nurowski, Paweł
TI - Differential Equations and Para-CR Structures
JO - Bollettino dell'Unione Matematica Italiana
DA - 2010/2//
PB - Unione Matematica Italiana
VL - 3
IS - 1
SP - 25
EP - 91
AB - We study the local geometry of n dimensional manifolds which are equipped with two integrable distributions, one of dimension $r$ and one of dimension $s$, where $r$ and $s$ are allowed to be unequal. We call them para-CR structures of type $(k,r,s)$, with $k = n - r - s \ge 0$ being the para-CR codimension. When $r = s$ they are the real analogues of CR structures. In the general case these structures are the natural geometric setting in which to discuss the geometry of systems of ODE's, as well as the geometry of systems of PDE's of finite type. For particular small values of $k,r,s$ we determine the basic local invariants of such structures.
LA - eng
UR - http://eudml.org/doc/290641
ER -

## References

top
1. ALEKSEEVSKY, D. V. - MEDORI, C. - TOMASSINI, A., Maximally homogeneous para-CR manifolds of semisimple type, to appear in Handbook of Pseudo-Riemannian geometry and Supersymmetry (2008), arXiv:0808.0431 MR2681601DOI10.4171/079-1/16
2. CARTAN, E., Les systemes de Pfaff a cinq variables et les equations aux derivees partielles du seconde ordre, Ann. Sc. Norm. Sup., 27 (1910), 109-192. Zbl41.0417.01MR1509120
3. CARTAN, E., Varietés à connexion projective, Bull. Soc. Math., LII (1924), 205-241. Zbl50.0500.02MR1504846
4. CHERN, S. S., The geometry of the differential equations $y^{\prime\prime\prime}=F(x,y,y^{\prime}y^{\prime\prime})$, Sci. Rep. Nat. Tsing Hua Univ., 4 (1940), 97-111. MR4538
5. FEFFERMAN, C. - GRAHAM, C. R., Conformal invariants, in Elie Cartan et mathematiques d'aujourd'hui, Asterisque, hors serie (Societe Mathematique de France, Paris) (1985), 95-116. MR837196
6. FRITELLI, S. - KOZAMEH, C. N. - NEWMAN, E. T., GR via characteristic surfaces, J. Math. Phys., 36 (1995), 4984-. Zbl0848.53045MR1347127DOI10.1063/1.531210
7. FRITELLI, S. - NEWMAN, E. T. - NUROWSKI, P., Conformal Lorentzian metrics on the spaces of curves and 2-surfaces, Class. Q. Grav., 20 (2003), 3649-3659. Zbl1050.83021MR2001687DOI10.1088/0264-9381/20/16/308
8. GODLINSKI, M., Geometry of Third-Order Ordinary Differential Equations and Its Applications in General Relativity, PhD Thesis, Warsaw University (2008), arXiv: 0810.2234.
9. GODLINSKI, M. - NUROWSKI, P., Geometry of third-order ODEs (2009), arXiv: 0902.4129. MR2501740DOI10.1016/j.geomphys.2008.11.015
10. GOVER, A. R. - NUROWSKI, P., Obstructions to conformally Einstein metrics in n dimensions, Journ. Geom. Phys., 56 (2006), 450-484. Zbl1098.53014MR2171895DOI10.1016/j.geomphys.2005.03.001
11. LIE, S., Klassifikation und Integration von gewohnlichen Differentialgleichungen zwischen x, y, die eine Gruppe von Transformationen gestatten III, in Gesammelte Abhandlungen, Vol. 5 (Teubner, Leipzig, 1924). Zbl15.0751.03
12. LEISTNER, TH. - NUROWSKI, P., Ambient metrics for n-dimensional pp-waves (2008), arXiv:0810.2903. Zbl1207.53028MR2628825DOI10.1007/s00220-010-0995-x
13. LEWANDOWSKI, J., Reduced holonomy group and Einstein equations with a cosmological constant, Class. Q. Grav., 9 (1992), L147-L151. Zbl0773.53051MR1184492
14. NUROWSKI, P., Differential equations and conformal structures, Journ. Geom. Phys., 55 (2005), 19-49. Zbl1082.53024MR2157414DOI10.1016/j.geomphys.2004.11.006
15. NUROWSKI, P. - ROBINSON, D. C., Intrinsic geometry of a null hypersurface, Class. Q. Grav., 17 (2000), 4065-4084. Zbl1085.53504MR1789400DOI10.1088/0264-9381/17/19/308
16. NUROWSKI, P. - SPARLING, G. A. J., Three dimensional Cauchy-Riemann structures and second order ordinary differential equations, Class. Q. Grav., 20 (2003), 4995-5016. Zbl1051.32019MR2024797DOI10.1088/0264-9381/20/23/004
17. OLVER, P. J., Equivalence Invariants and Symmetry, Cambridge University Press (Cambridge, 1996). Zbl1156.58002MR1337276DOI10.1017/CBO9780511609565
18. PERKINS, K., The Cartan-Weyl conformal geometry of a pair of second-order partial-differential equations, PhD Thesis, Department of Physics & Astronomy (University of Pittsburgh, 2006).
19. TANAKA, N., On affine symmetric spaces and the automorphism groups of product manifolds, Hokkaido Math. J., 14 (1985), 277-351. Zbl0585.53044MR808817DOI10.14492/hokmj/1381757644
20. TRESSE, M. A., Determinations des invariants ponctuels de l'equation differentielle ordinaire du second ordre $y^{\prime\prime}=\omega(x,y,y^{\prime})$, Hirzel (Leipzig, 1896). Zbl27.0254.01
21. WUÈ NSCHMANN, K., Über Beruhrungsbedingungen bei Differentialgleichungen, Dissertation (Greifswald, 1905).

## 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.