Differential equations and conformal structures TeX Export

@misc{citeulike:803337, abstract = {We provide five examples of conformal geometries which are naturally associated with ordinary differential equations (ODEs). The first example describes a one-to-one correspondence between the Wuenschmann class of 3rd order ODEs considered modulo contact transformations of variables and (local) 3-dimensional conformal Lorentzian geometries. The second example shows that every point equivalent class of 3rd order ODEs satisfying the Wuenschmann and the Cartan conditions define a 3-dimensional Lorentzian Einstein-Weyl geometry. The third example associates to each point equivalence class of 3rd order ODEs a 6-dimensional conformal geometry of neutral signature. The fourth example exhibits the one-to-one correspondence between point equivalent classes of 2nd order ODEs and 4-dimensional conformal Fefferman-like metrics of neutral signature. The fifth example shows the correspondence between undetermined ODEs of the Monge type and conformal geometries of signature \$(3,2)\$. The Cartan normal conformal connection for these geometries is reducible to the Cartan connection with values in the Lie algebra of the noncompact form of the exceptional group \$G\_2\$. All the examples are deeply rooted in Elie Cartan's works on exterior differential systems.}, archivePrefix = {arXiv}, author = {Nurowski, Pawel}, citeulike-article-id = {803337}, citeulike-linkout-0 = {http://arxiv.org/abs/math.DG/0406400}, citeulike-linkout-1 = {http://arxiv.org/pdf/math.DG/0406400}, day = {16}, eprint = {math.DG/0406400}, keywords = {classical-gravity, geometry, gravity, mathematics, physics, printed}, month = {Aug}, posted-at = {2006-08-17 21:37:39}, priority = {4}, title = {Differential equations and conformal structures}, url = {http://arxiv.org/abs/math.DG/0406400}, year = {2006} }

TY - ELEC ID - citeulike:803337 L3 - citeulike-article-id:803337 N2 - We provide five examples of conformal geometries which are naturally associated with ordinary differential equations (ODEs). The first example describes a one-to-one correspondence between the Wuenschmann class of 3rd order ODEs considered modulo contact transformations of variables and (local) 3-dimensional conformal Lorentzian geometries. The second example shows that every point equivalent class of 3rd order ODEs satisfying the Wuenschmann and the Cartan conditions define a 3-dimensional Lorentzian Einstein-Weyl geometry. The third example associates to each point equivalence class of 3rd order ODEs a 6-dimensional conformal geometry of neutral signature. The fourth example exhibits the one-to-one correspondence between point equivalent classes of 2nd order ODEs and 4-dimensional conformal Fefferman-like metrics of neutral signature. The fifth example shows the correspondence between undetermined ODEs of the Monge type and conformal geometries of signature $(3,2)$. The Cartan normal conformal connection for these geometries is reducible to the Cartan connection with values in the Lie algebra of the noncompact form of the exceptional group $G_2$. All the examples are deeply rooted in Elie Cartan's works on exterior differential systems. TI - Differential equations and conformal structures KW - classical-gravity KW - geometry KW - gravity KW - mathematics KW - physics KW - printed AU - Nurowski, Pawel PY - 2006/Aug/16/ UR - http://arxiv.org/abs/math.DG/0406400 ER -