IJPAM: Volume 70, No. 4 (2011)

A SURVEY ON 3-STEP NILPOTENT
SUB-RIEMANNIAN GEOMETRY

F. Monroy-Pérez$^1$, A. Anzaldo-Meneses$^2$
$^{1,2}$Basic Sciences Departament
Metropolitan Autonomous University - Azcapotzalco
180, Av. San Pablo, Mexico D.F., 02200, MEXICO


Abstract. In this paper we survey on results about sub-Riemannian structures defined on 3-step nilpotent Lie groups which are solvable with solvability index equal to 2. We present the general Lie structure for both the Lie algebra and the Lie group and then we formulate the geodesic sub-Riemannian problem, as an optimal control problem consisting on the minimization of a quadratic functional among the solutions of a drift less control system which is affine in the control parameters. Necessary conditions for sub-Riemannian geodesics are given by the Pontryagin Maximum Principle. A general discussion for extremal curves on the cotangent bundle is carried out and then specialized to some low dimensional cases.

Received: February 3, 2011

AMS Subject Classification: 49K30, 49Q99, 93B29

Key Words and Phrases: nilpotent Lie algebras, extremal curves, sub-Riemannian geodesics, exponential mapping

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Source: International Journal of Pure and Applied Mathematics
ISSN printed version: 1311-8080
ISSN on-line version: 1314-3395
Year: 2011
Volume: 70
Issue: 4