Boris Doubrov, Igor Zelenko
On Geometry of Curves of Flags of Constant Type
Preprint series: ESI preprints

MSC:

53B25 Local submanifolds, See also {53C40}

53B15 Other connections

53A55 Differential invariants (local theory), geometric objects

17B70 Graded Lie algebras

Abstract: We develop an algebraic version of Cartan method of equivalence
or an analog of Tanaka prolongation for the (extrinsic) geometry
of curves of flags of a vector space $W$ with respect to the
action of a subgroup $G$ of the $GL(W)$. Under some natural
assumptions on the subgroup $G$ and on the flags, one can pass
from the filtered objects to the corresponding graded objects
and describe the construction of canonical bundles of moving
frames for these curves in the language of pure Linear Algebra.
The scope of applicability of the theory includes geometry of
natural classes of curves of flags with respect to
reductive linear groups or their parabolic subgroups. As simplest
examples, this includes the projective and affine geometry of curves.
The case of classical groups is considered in more detail.


Keywords: curves and submanifolds in flag varieties, equivalence problem, bundles of moving frames, graded Lie algebras, Tanaka prolongation, sl_2-representations