January  2022, 42(1): 327-352. doi: 10.3934/dcds.2021119

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

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

Received  April 2020 Revised  May 2021 Published  January 2022 Early access  September 2021

Fund Project: Partially supported by SCyT (UNR)

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.

Citation: Gabriela P. Ovando. The geodesic flow on nilpotent Lie groups of steps two and three. Discrete & Continuous Dynamical Systems, 2022, 42 (1) : 327-352. doi: 10.3934/dcds.2021119
References:
[1]

R. Abraham and J. Marsden, Foundations of Mechanics, 2$^{nd}$ edition, Benjamin/Cummings Publishing Co., Inc., Advanced Book Program, Reading, Mass., 1978.  Google Scholar

[2]

V. del Barco and A. Moroianu, Symmetric Killing tensors on nilmanifolds, Bull. Soc. Math. France, 148 (2020), 411-438.  doi: 10.24033/bsmf.2811.  Google Scholar

[3]

W. Bauer and D. Tarama, On the complete integrability of the geodesic flow of pseudo-H-type Lie groups, Anal. Math. Phys., 8 (2018), 493-520.  doi: 10.1007/s13324-018-0250-8.  Google Scholar

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A. V. Bolsinov and I. A. Taimanov, Integrable geodesic flows with positive topological entropy, Invent. math., 140 (2000), 639-650.  doi: 10.1007/s002220000066.  Google Scholar

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L. Butler, Integrable geodesic flows with wild first integrals: The case of two-step nilmanifolds, Ergodic Theory Dynam. Systems, 23 (2003), 771-797.  doi: 10.1017/S0143385702001517.  Google Scholar

[6]

S. G. Dani and M. G. Mainkar, Anosov automorphisms on compact nilmanifolds associated with graphs, Trans. Amer. Math. Soc., 357 (2005), 2235-2251.  doi: 10.1090/S0002-9947-04-03518-4.  Google Scholar

[7]

R. DecosteL. Demeyer and M. Mainkar, Graphs and metric 2-step nilpotent Lie algebras, Adv. Geom., 18 (2018), 265-284.  doi: 10.1515/advgeom-2017-0052.  Google Scholar

[8]

P. Eberlein, Geometry of 2-step nilpotent Lie groups, Modern Dynamical Systems, Cambridge University Press, (2004), 67–101.  Google Scholar

[9]

P. Eberlein, Left invariant geometry of Lie groups, Cubo, 6 (2004), 427-510.   Google Scholar

[10]

R. Gornet and M. Mast, The length spectrum of riemannian two-step nilmanifolds, Ann. Scient. École Norm. Sup., 33 (2000), 181-209.  doi: 10.1016/S0012-9593(00)00111-7.  Google Scholar

[11]

W. A. de Graaf, Classification of 6-dimensional nilpotent Lie algebras over fields of characteristic not 2, J. Algebra, 309 (2007), 640-653.  doi: 10.1016/j.jalgebra.2006.08.006.  Google Scholar

[12]

K. HeilA. Moroianu and U. Semmelmann, Killing and conformal Killing tensors, J. Geom. Phys., 106 (2016), 383-400.  doi: 10.1016/j.geomphys.2016.04.014.  Google Scholar

[13]

S. Helgasson, Differential Geometry, Lie groups and Symmetric Spaces, American Mathematical Society, Providence, RI, 2001. doi: 10.1090/gsm/034.  Google Scholar

[14]

A. KocsardG. Ovando and S. Reggiani, On first integrals of the geodesic flow on Heisenberg nilmanifolds, Diff. Geom. Appl., 49 (2016), 496-509.  doi: 10.1016/j.difgeo.2016.08.004.  Google Scholar

[15]

B. Kostant, The solution to a generalized Toda lattice and representation theory, Adv. in Math., 34 (1979), 195-338.  doi: 10.1016/0001-8708(79)90057-4.  Google Scholar

[16]

V. V. Kozlov, Topological obstructions to the integrability of natural mechanical systems, Dokl. Akad. Nauk SSSR, 249 (1979), 1299-1302.   Google Scholar

[17]

J. Lauret and C. Will, Nilmanifolds of dimension $\leq$ 8 admitting Anosov diffeomorphisms, Trans. Am. Math. Soc., 361 (2009), 2377-2395.  doi: 10.1090/S0002-9947-08-04757-0.  Google Scholar

[18]

M. Mainkar and C. Will, Examples of Anosov Lie algebras, Discrete Contin. Dyn. Syst., 18 (2007), 39-52.  doi: 10.3934/dcds.2007.18.39.  Google Scholar

[19]

G. Ovando, The geodesic flow on nilmanifolds associated to graphs, Rev. Un. Mat. Argentina, 61 (2020), 315-338.  doi: 10.33044/revuma.v61n2a09.  Google Scholar

[20]

T. L. Payne, Anosov automorphisms of nilpotent Lie algebras, J. Mod. Dyn., 3 (2009), 121-158.  doi: 10.3934/jmd.2009.3.121.  Google Scholar

[21]

D. Schueth, Integrability of geodesic flows and isospectrality of Riemannian manifolds, Math. Z., 260 (2008), 595-613.  doi: 10.1007/s00209-007-0290-5.  Google Scholar

[22]

U. Semmelmann, Conformal Killing forms on Riemannian manifolds, Math. Z., 245 (2003), 503-527.  doi: 10.1007/s00209-003-0549-4.  Google Scholar

[23]

W. Symes, Systems of Toda type, inverse spectral problems and representation theory, Invent. Math., 59 (1980), 13-51.  doi: 10.1007/BF01390312.  Google Scholar

[24]

I. A. Taimanov, Topological obstructions to the integrability of geodesic flows on non-simply-connected manifolds, Math. USSR-Izv., 30 (1988), 403-409.  doi: 10.1070/IM1988v030n02ABEH001021.  Google Scholar

[25]

I. A. Taimanov, Topology of Riemannian manifolds with integrable geodesic flows, Tr. Mat. Inst. Steklova, 205 (1994), 150-163.   Google Scholar

[26]

A. Thimm, Integrable geodesic flows on homogeneous spaces, Ergodic Theory Dynam. Systems, 1 (1981), 495-517.  doi: 10.1017/S0143385700001401.  Google Scholar

[27]

V. Varadarajan, Lie Groups, Lie Algebras and Their Representations, Reprint of the 1974 Edition. Graduate Texts in Mathematics, Springer-Verlag, New York, 1984. doi: 10.1007/978-1-4612-1126-6.  Google Scholar

[28]

E. Wilson, Isometry groups on homogeneous nilmanifolds, Geom. Dedicata, 12 (1982), 337-346.  doi: 10.1007/BF00147318.  Google Scholar

[29]

J. Wolf, On locally symmetric spaces of non-negative curvature and certain other locally homogeneous spaces, Comment. Math. Helv., 37 (1962/1963), 266-295.  doi: 10.1007/BF02566977.  Google Scholar

[30]

N. M. J. Woodhouse, Killing tensors and the separation of the Hamilton-Jacobi equation, Commun. Math. Phys., 44 (1975), 9-38.  doi: 10.1007/BF01609055.  Google Scholar

show all references

References:
[1]

R. Abraham and J. Marsden, Foundations of Mechanics, 2$^{nd}$ edition, Benjamin/Cummings Publishing Co., Inc., Advanced Book Program, Reading, Mass., 1978.  Google Scholar

[2]

V. del Barco and A. Moroianu, Symmetric Killing tensors on nilmanifolds, Bull. Soc. Math. France, 148 (2020), 411-438.  doi: 10.24033/bsmf.2811.  Google Scholar

[3]

W. Bauer and D. Tarama, On the complete integrability of the geodesic flow of pseudo-H-type Lie groups, Anal. Math. Phys., 8 (2018), 493-520.  doi: 10.1007/s13324-018-0250-8.  Google Scholar

[4]

A. V. Bolsinov and I. A. Taimanov, Integrable geodesic flows with positive topological entropy, Invent. math., 140 (2000), 639-650.  doi: 10.1007/s002220000066.  Google Scholar

[5]

L. Butler, Integrable geodesic flows with wild first integrals: The case of two-step nilmanifolds, Ergodic Theory Dynam. Systems, 23 (2003), 771-797.  doi: 10.1017/S0143385702001517.  Google Scholar

[6]

S. G. Dani and M. G. Mainkar, Anosov automorphisms on compact nilmanifolds associated with graphs, Trans. Amer. Math. Soc., 357 (2005), 2235-2251.  doi: 10.1090/S0002-9947-04-03518-4.  Google Scholar

[7]

R. DecosteL. Demeyer and M. Mainkar, Graphs and metric 2-step nilpotent Lie algebras, Adv. Geom., 18 (2018), 265-284.  doi: 10.1515/advgeom-2017-0052.  Google Scholar

[8]

P. Eberlein, Geometry of 2-step nilpotent Lie groups, Modern Dynamical Systems, Cambridge University Press, (2004), 67–101.  Google Scholar

[9]

P. Eberlein, Left invariant geometry of Lie groups, Cubo, 6 (2004), 427-510.   Google Scholar

[10]

R. Gornet and M. Mast, The length spectrum of riemannian two-step nilmanifolds, Ann. Scient. École Norm. Sup., 33 (2000), 181-209.  doi: 10.1016/S0012-9593(00)00111-7.  Google Scholar

[11]

W. A. de Graaf, Classification of 6-dimensional nilpotent Lie algebras over fields of characteristic not 2, J. Algebra, 309 (2007), 640-653.  doi: 10.1016/j.jalgebra.2006.08.006.  Google Scholar

[12]

K. HeilA. Moroianu and U. Semmelmann, Killing and conformal Killing tensors, J. Geom. Phys., 106 (2016), 383-400.  doi: 10.1016/j.geomphys.2016.04.014.  Google Scholar

[13]

S. Helgasson, Differential Geometry, Lie groups and Symmetric Spaces, American Mathematical Society, Providence, RI, 2001. doi: 10.1090/gsm/034.  Google Scholar

[14]

A. KocsardG. Ovando and S. Reggiani, On first integrals of the geodesic flow on Heisenberg nilmanifolds, Diff. Geom. Appl., 49 (2016), 496-509.  doi: 10.1016/j.difgeo.2016.08.004.  Google Scholar

[15]

B. Kostant, The solution to a generalized Toda lattice and representation theory, Adv. in Math., 34 (1979), 195-338.  doi: 10.1016/0001-8708(79)90057-4.  Google Scholar

[16]

V. V. Kozlov, Topological obstructions to the integrability of natural mechanical systems, Dokl. Akad. Nauk SSSR, 249 (1979), 1299-1302.   Google Scholar

[17]

J. Lauret and C. Will, Nilmanifolds of dimension $\leq$ 8 admitting Anosov diffeomorphisms, Trans. Am. Math. Soc., 361 (2009), 2377-2395.  doi: 10.1090/S0002-9947-08-04757-0.  Google Scholar

[18]

M. Mainkar and C. Will, Examples of Anosov Lie algebras, Discrete Contin. Dyn. Syst., 18 (2007), 39-52.  doi: 10.3934/dcds.2007.18.39.  Google Scholar

[19]

G. Ovando, The geodesic flow on nilmanifolds associated to graphs, Rev. Un. Mat. Argentina, 61 (2020), 315-338.  doi: 10.33044/revuma.v61n2a09.  Google Scholar

[20]

T. L. Payne, Anosov automorphisms of nilpotent Lie algebras, J. Mod. Dyn., 3 (2009), 121-158.  doi: 10.3934/jmd.2009.3.121.  Google Scholar

[21]

D. Schueth, Integrability of geodesic flows and isospectrality of Riemannian manifolds, Math. Z., 260 (2008), 595-613.  doi: 10.1007/s00209-007-0290-5.  Google Scholar

[22]

U. Semmelmann, Conformal Killing forms on Riemannian manifolds, Math. Z., 245 (2003), 503-527.  doi: 10.1007/s00209-003-0549-4.  Google Scholar

[23]

W. Symes, Systems of Toda type, inverse spectral problems and representation theory, Invent. Math., 59 (1980), 13-51.  doi: 10.1007/BF01390312.  Google Scholar

[24]

I. A. Taimanov, Topological obstructions to the integrability of geodesic flows on non-simply-connected manifolds, Math. USSR-Izv., 30 (1988), 403-409.  doi: 10.1070/IM1988v030n02ABEH001021.  Google Scholar

[25]

I. A. Taimanov, Topology of Riemannian manifolds with integrable geodesic flows, Tr. Mat. Inst. Steklova, 205 (1994), 150-163.   Google Scholar

[26]

A. Thimm, Integrable geodesic flows on homogeneous spaces, Ergodic Theory Dynam. Systems, 1 (1981), 495-517.  doi: 10.1017/S0143385700001401.  Google Scholar

[27]

V. Varadarajan, Lie Groups, Lie Algebras and Their Representations, Reprint of the 1974 Edition. Graduate Texts in Mathematics, Springer-Verlag, New York, 1984. doi: 10.1007/978-1-4612-1126-6.  Google Scholar

[28]

E. Wilson, Isometry groups on homogeneous nilmanifolds, Geom. Dedicata, 12 (1982), 337-346.  doi: 10.1007/BF00147318.  Google Scholar

[29]

J. Wolf, On locally symmetric spaces of non-negative curvature and certain other locally homogeneous spaces, Comment. Math. Helv., 37 (1962/1963), 266-295.  doi: 10.1007/BF02566977.  Google Scholar

[30]

N. M. J. Woodhouse, Killing tensors and the separation of the Hamilton-Jacobi equation, Commun. Math. Phys., 44 (1975), 9-38.  doi: 10.1007/BF01609055.  Google Scholar

Table 1.   
Lie algebra Non-zero Lie brackets First integrals
$\mathfrak{h}_3$ $[X_1,Y_1]=Z$ $f_Z, {\rm{E}}, f_{X_1^*}$
$\mathfrak{n}_2$ $[e_1,e_2]=e_3, [e_1,e_3]=e_4$ ${\rm{E}}, f_{e_2^*}, f_{e_3^*}, f_{e_4^*}$
$\mathfrak{h}_5$ $[X_1,Y_1]=[X_2,Y_2]=Z$ $f_Z, f_{X_1^*}, f_{X_2^*}, g_{S_1}, g_{S_2}$
$\mathfrak{n}_1$ $[e_1,e_2]=e_3, [e_1,e_3]=[e_2,e_4]=e_5$ ${\rm{E}}, f_{e_3^*}, f_{e_4^*}, f_{e_5}, f_{D^*}$
$\mathfrak{n}_3$ $[e_1,e_2]=e_4, [e_1,e_3]=e_5$ ${\rm{E}}, f_{e_2^*}, f_{e_3^*}, f_{e_4}, f_{e_5}$
$\mathfrak{n}_{2,3}$ $[e_1,e_2]=e_3, [e_1,e_3]=e_4, [e_2,e_3]=e_5$ ${\rm{E}}, f_{e_3^*}, f_{e_4}, f_{e_5}, g_S$
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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