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

Partially supported by SCyT (UNR)
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  • 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.

    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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