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

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Speaker: Dmitri Alekseevsky

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

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) = \frac12 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.