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Publications [#318260] of Robert Bryant
search www.ams.org.Papers Published
 BRYANT, RL, Conformal geometry and 3plane fields on 6manifolds,
in Developments of Cartan Geometry and Related Mathematical Problems, RIMS Symposium Proceedings, vol. 1502 (Developments of Cartan Geometry an
(July, 2006),
pp. 115, Kyoto University [math.DG/0511110]
(last updated on 2017/12/15)
Author's Comments: For the proceedings of the RIMS
Symposium "Developments of Cartan geometry and related
mathematical problems" (2427 October 2005)
Abstract: The purpose of this note is to provide yet another
example of the link between certain conformal geometries
and ordinary differential equations, along the lines of the
examples discussed by Nurowski in math.DG/0406400.
In this particular case, I consider the equivalence problem
for 3plane fields D on 6manifolds M that satisfy the
nondegeneracy condition that D+[D,D]=TM
I give a solution of the equivalence problem for such D (as
Tanaka has previously), showing that it defines a so(4,3)
valued Cartan connection on a principal right Hbundle
over M where H is the subgroup of SO(4,3) that stabilizes
a null 3plane in R^{4,3}. Along the way, I observe that
there is associated to each such D a canonical conformal
structure of split type on M, one that depends on two
derivatives of the plane field D.
I show how the primary curvature tensor of the Cartan
connection associated to the equivalence problem for D
can be interpreted as the Weyl curvature of the associated
conformal structure and, moreover, show that the split
conformal structures in dimension 6 that arise in this
fashion are exactly the ones whose so(4,4)valued Cartan
connection admits a reduction to a spin(4,3)connection. I
also discuss how this case has features that are analogous
to those of Nurowski's examples.
Keywords: differential invariants


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