The geodesic flow on nilpotent Lie groups of steps two and three

Gabriela P. Ovando

doi:10.3934/dcds.2021119

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2022, Volume 42, Issue 1: 327-352. Doi: 10.3934/dcds.2021119

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The geodesic flow on nilpotent Lie groups of steps two and three

Gabriela P. Ovando

CONICET- Universidad Nacional de Rosario, Departamento de Matemática, ECEN - FCEIA, Pellegrini 250, 2000 Rosario, Santa Fe, Argentina

Received: March 31, 2020

Revised: April 30, 2021

Early access: September 2021

Published: January 2022

Partially supported by SCyT (UNR)

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Abstract

The goal of this paper is the study of the integrability of the geodesic flow on $ k $-step nilpotent Lie groups, k = 2, 3, when equipped with a left-invariant metric. Liouville integrability is proved in low dimensions. Moreover, it is shown that complete families of first integrals can be constructed with Killing vector fields and symmetric Killing 2-tensor fields. This holds for dimension $ m\leq 5 $. The situation in dimension six is similar in most cases. Several algebraic relations on the Lie algebra of first integrals are explicitly written. Also invariant first integrals are analyzed and several involution conditions are shown.

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Mathematics Subject Classification: Primary: 53D25, 70H06; Secondary: 22E25, 70G65, 70H05, 22E60.

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Table 1.

Lie algebra	Non-zero Lie brackets	First integrals
$\mathfrak{h}_3$	$[X_1,Y_1]=Z$	$f_Z, {\rm{E}}, f_{X_1^*}$
$\mathfrak{n}_2$	$[e_1,e_2]=e_3, [e_1,e_3]=e_4$	${\rm{E}}, f_{e_2^}, f_{e_3^}, f_{e_4^*}$
$\mathfrak{h}_5$	$[X_1,Y_1]=[X_2,Y_2]=Z$	$f_Z, f_{X_1^}, f_{X_2^}, g_{S_1}, g_{S_2}$
$\mathfrak{n}_1$	$[e_1,e_2]=e_3, [e_1,e_3]=[e_2,e_4]=e_5$	${\rm{E}}, f_{e_3^}, f_{e_4^}, f_{e_5}, f_{D^*}$
$\mathfrak{n}_3$	$[e_1,e_2]=e_4, [e_1,e_3]=e_5$	${\rm{E}}, f_{e_2^}, f_{e_3^}, f_{e_4}, f_{e_5}$
$\mathfrak{n}_{2,3}$	$[e_1,e_2]=e_3, [e_1,e_3]=e_4, [e_2,e_3]=e_5$	${\rm{E}}, f_{e_3^*}, f_{e_4}, f_{e_5}, g_S$

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