LOCALLY HOMOGENEOUS CURVES IN HOMOGENEOUS SPACES

BORIS DOUBROV

Homogeneous curves play an important role in study of arbitrary curves in a given homogeneous space as well as in Cartan geometries modeled by this homogeneous space. For example, any curve can be approximated up to a certain order by a homogeneous one, and this leads to the notion of osculating curves and differential invariants for arbitrary curves. For Cartan geometries, the concept of development allows to define distinguished curves of this geometry generalizing the notion of geodesics.

We introduce the notion of a symmetry algebra for any submanifold in a homogeneous space and show how to compute symmetry algebras of orbits of subgroups in terms of Lie algebras. In particular, we answer the question when a given homogeneous curve admits a projective reparametrization.

In case of a parabolic geometry G/P with a Lie algebra 𝔤 = ∑_{j=−k}^{k} 𝔤_{j} we prove that any trajectory of the one-parameter group exp(tX_{j}), X_{j} ∈ 𝔤_{j} (j < 0) through the origin o = eP admits a projective reparametrization.

Next, for a given locally homogeneous curve L we explicitly describe the stationary subalgebras of its i-jets [L]_{o}^{i} for any i ≥ 0. In particular, we show how compute the minimal jet order that determines L as an unparametrized curve.

Finally, we describe a classification algorithm of locally homogeneous curves and give several explicit examples for classical parabolic geometries.