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

{{MeetingTalk
| 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 <math>(M,g)</math>:

* 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 <math>M</math> of integral curves of the Hamiltoian flow on <math>(T^*M, \omega_{can})</math> with the quadratic Hamiltonian <math>H(p) = \frac12 g^{-1}(p,p),\, p \in T^*M</math>.

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

These definitions may be generalized to the sub-Riemannian manifold <math>(M,D,g)</math> where <math>g</math> is a sub-Riemannian metric, defined on a bracket generated distribution <math>D</math>. 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 <math>(D,g)</math> 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.
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| event = [[Local and Nonlocal Geometry of PDEs and Integrability]], 8-12 October 2018, SISSA, Trieste, Italy.<br>''The conference in honor of [[Joseph Krasil'shchik]]'s 70th birthday.''
| 79YY-MM-DD = 7981-89-91
}}