Alekseevsky D. Shortest and straightest geodesics of an invariant sub-Riemanniasn metric on a flag manifold (abstract)

Speaker: Dmitri Alekseevsky

Title: Shortest and straightest geodesics of an invariant sub-Riemanniasn metric on a flag manifold

Abstract:
There are different equivalent definitions of geodesics of a Riemannian manifold $(M,g)$:

• According to Euler-Lagrange variational definition, geodesics are extremals of length functional or functional of kinetic energy that is shortest curves, joint two closed points.

• According to Hamilton approach, geodesics are projection to $M$ of integral curves of the Hamiltoian flow on $(T^{*}M,\omega _{can})$ with the quadratic Hamiltonian $H(p)={\frac {1}{2}}g^{-1}(p,p),\,p\in T^{*}M$.

• According to Levi-Civita, geodesics are straightest curves, i.e. curves $\gamma (s)$ whose tangent field ${\dot {\gamma }}(s)$ is parallel with respect to Levi-Civita connection.

These definitions may be generalized to the sub-Riemannian manifold $(M,D,g)$ where $g$ is a sub-Riemannian metric, defined on a bracket generated distribution $D$. However , as it was remarked by A.M.Vershik and L.D.Faddeev, variational definition of sub-Riemannian geodesics as "shortest" horisontal curves (used in control theory and many applications) is different from definition of geodesics as straightes curves , used in non-holonomic mechanics. Moreover, they proved that generically shortest geodesics are different from straightes geodesics and indicate examples when these to notions are equivalent.

We recall a Schouten-Wagner description of straightest sub-Riemannian geodesics as geodesics of a partial connection and definition of Wagner curvature tensor of such connection.

Then we classify invariant sub-Riemannian structures $(D,g)$ on flag manifolds (i.e. adjoint orbits of a compact simple Lie group) and study relations between shortest and strainghest sub-Riemannian geodesics of such sub-Riemannian homogeneous manifold.

Slides: AlekseevskyTrieste2018slides.pdf

Event: Local and Nonlocal Geometry of PDEs and Integrability, 8-12 October 2018, SISSA, Trieste, Italy.
The conference in honor of Joseph Krasil'shchik's 70th birthday.