Geometry of curves in generalized flag varieties

Abstract

Dedicated to Mike Eastwood on the occasion of his 60th birthday.Abstract. The current paper is devoted to the study of integral curves of constant type in generalized flag varieties. We construct a canonical moving frame bundle for such curves and give a criterion when it turns out to be a Cartan connection. Generalizations to parametrized curves, to higher-dimensional submanifolds and to parabolic geometries are discussed.2010 Mathematics Subject Classification. 53B25, 53B15, 53A55, 17B70.

Geometry of rank 2 distributions with nonzero Wilczynski invariants*

“…Later, we developed a different method, leading to the construction of canonical bundles of moving frames and invariants for quite general curves in Grassmannians and flag varieties [14,15]. The geometry of Jacobi curves J γ in the case of rank 2 distributions can be reduced to the geometry of the so-called self-dual curves in the projective space PW γ .…”

“…to the lower order derivatives of this sections. For the more algebraic point of view, based on Tanaka-like theory of curves of flags and sl 2 -representations see [9,14].…”

“…It was shown by Wilczynski that a curve in a projective space is self-dual if an only if all Wilczynski invariant of odd order vanish (for the modern Lie-algebraic interpretation of this fact see [14]). The remaining n − 4 Wilczynski invariants of even order, W 2i , 1 ≤ i ≤ n − 4, constitute the fundamental set of symplectic invariants of the unparametrized curve J Taking the Wilczynski invariants W 2i for the Jacobi curves of all abnormal extremals living in the set R D we obtain the invariants of the distribution D, called the generalized Wilczynski invariants of D of order 2(i + 1) and denoted also by W 2i .…”

Conformal theory of curves with tractors

“…, 2r+1, where ρ j are in general nonzero while all ε j vanish except for ε 2r+1 = (−1) r−1 . Secondly, the choice of metric may be adjusted so that the bottom slot of U ′ vanishes and the expressions of previous tractors are unchanged: according to (6), the corresponding Υ a has to satisfy Υ c U ′c = ρ 1 and Υ c U c = 0. Hence U ′′ = 0 U ′′a ρ2 , i.e.…”

“…. Finally, we may consider a rescaling so that the middle slot of the last tractor vanishes: according to (6), the corresponding Υ a equals to (−1) r U (2r+1)a along the curve. Hence the statement follows.…”