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 and 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.

[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.

[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.

[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.

[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.

[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.

[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.

[8]

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

[9]

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

[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.

[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.

[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.

[13]

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

[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.

[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.

[16]

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

[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.

[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.

[19]

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

[20]

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

[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.

[22]

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

[23]

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

[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.

[25]

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

[26]

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

[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.

[28]

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

[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.

[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.

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.

[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.

[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.

[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.

[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.

[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.

[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.

[8]

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

[9]

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

[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.

[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.

[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.

[13]

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

[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.

[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.

[16]

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

[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.

[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.

[19]

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

[20]

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

[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.

[22]

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

[23]

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

[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.

[25]

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

[26]

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

[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.

[28]

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

[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.

[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.

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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