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doi: 10.3934/dcds.2020375

Continuous and discrete Neumann systems on Stiefel varieties as matrix generalizations of the Jacobi–Mumford systems

1. 

Department of Mathematics, Polytechnic university of Catalonia, C. Pau Gargallo, 14, 08028 Barcelona, Spain

2. 

Mathematical Institute, Serbian Academy of Sciences and Arts, Kneza Mihaila 36, 11000 Belgrade, Serbia

* Corresponding author: Božidar Jovanović

Received  February 2020 Revised  September 2020 Published  November 2020

We study geometric and algebraic geometric properties of the continuous and discrete Neumann systems on cotangent bundles of Stiefel varieties $ V_{n,r} $. The systems are integrable in the non-commutative sense, and by applying a $ 2r\times 2r $–Lax representation, we show that generic complex invariant manifolds are open subsets of affine Prym varieties on which the complex flow is linear. The characteristics of the varieties and the direction of the flow are calculated explicitly. Next, we construct a family of multi-valued integrable discretizations of the Neumann systems and describe them as translations on the Prym varieties, which are written explicitly in terms of divisors of points on the spectral curve.

Citation: Yuri Fedorov, Božidar Jovanović. Continuous and discrete Neumann systems on Stiefel varieties as matrix generalizations of the Jacobi–Mumford systems. Discrete & Continuous Dynamical Systems, doi: 10.3934/dcds.2020375
References:
[1]

M. R. AdamsJ. Harnad and E. Previato, Isospectral Hamiltonian flows in finite and infinite dimensions. Ⅰ. Generalized Moser systems and moment maps into loop algebras, Comm. Math. Phys., 117 (1988), 451-500.  doi: 10.1007/BF01223376.  Google Scholar

[2]

M. R. AdamsJ. Harnad and J. Hurtubise, Isospectral Hamiltonian flows in finite and infinite dimensions Ⅱ. Integration of flows, Comm. Math. Phys., 134 (1990), 555-585.  doi: 10.1007/BF02098447.  Google Scholar

[3]

M. R. AdamsJ. Harnad and J. Hurtubise, Dual moment maps into loop algebras, Lett. Math. Phys., 20 (1990), 299-308.  doi: 10.1007/BF00626526.  Google Scholar

[4]

M. R. AdamsJ. Harnad and J. Hurtubise, Darboux coordinates on coadjoint orbits of Lie algebras, Lett. Math. Phys., 40 (1997), 41-57.  doi: 10.1023/A:1007355508426.  Google Scholar

[5]

M. Adler and P. van Moerbeke, Birkhoff strata, Bäckund transformations and regularization of isospectral operators, Advances in Math., 108 (1994), 140-204.  doi: 10.1006/aima.1994.1070.  Google Scholar

[6]

M. Adler, P. van Moerbeke and P. Vanheacke, Algebraic Integrability, Painleve Geometry and Lie Algebras, Springer, 2004. doi: 10.1007/978-3-662-05650-9.  Google Scholar

[7]

C. Athorne and A. Fordy, Generalized KdV and mKdV equations associated with symmetric spaces, J. Phys.A., 20 (1987), 1377-1386.  doi: 10.1088/0305-4470/20/6/021.  Google Scholar

[8]

E. D. Belokolos, A. I. Bobenko, V. Z. Enol'sii, A. R. Its and V. B. Matveev, Algebro-Geometric Approach to Nonlinear Integrable Equations, Springer Series in Nonlinear Dynamics. Springer–Verlag, 1994. Google Scholar

[9]

A. Beauville, Prym varieties and the Schottky problem, Invent. Math., 41 (1977), 149-196.  doi: 10.1007/BF01418373.  Google Scholar

[10]

A. Beauville, Jacobiennes des courbes spectrales et syst'emes hamiltoniens completement integrables, Acta Math., 164 (1990), 211-235.  doi: 10.1007/BF02392754.  Google Scholar

[11]

O. I. Bogoyavlenski, New integrable problem of classical mechanics, Comm. Math. Phys., 94 (1984), 255-269.  doi: 10.1007/BF01209304.  Google Scholar

[12]

A. V. Bolsinov and B. Jovanović, Noncommutative integrability, moment map and geodesic flows, Annals of Global Analysis and Geometry, 23 (2003), 305–322, arXiv: math-ph/0109031. doi: 10.1023/A:1023023300665.  Google Scholar

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E. Casas-Alvero, Singularities of Plane Curves, London Math. Soc. Lecture Notes Series. 276 Cambridge University Press, 2000. doi: 10.1017/CBO9780511569326.  Google Scholar

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[15]

J. EilbeckV. Enol'skiV. Kuznetzov and A. Tsiganov, Linear $R$-matrix algebra for classical separable systems, J. Phys. A: Math. Gen., 27 (1994), 567-578.  doi: 10.1088/0305-4470/27/2/038.  Google Scholar

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P. A. Dirac, On generalized Hamiltonian dynamics, Can. J. Math., 2 (1950), 129-148.  doi: 10.4153/CJM-1950-012-1.  Google Scholar

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V. Dragović and B. Gajić, The Lagrange bitop on $so(4) \times so(4)$ and geometry of the Prym varieties, American J. of Math., 126 (2004), 981–1004, arXiv: math-ph/0201036. doi: 10.1353/ajm.2004.0035.  Google Scholar

[18]

B. A. Dubrovin, Completely integrable Hamiltonian systems associated with matrix operators and Abelian varieties, Funct.Anal.Appl., 11 (1977), 265-277.   Google Scholar

[19]

B. A Dubrovin, S. P. Novikov and I. M. Krichever, Integrable Systems. I,, in Itogi Nauki i Tekhniki. Sovr.Probl.Mat. Fund.Naprav., 4 (1985), 179–284. English transl.: Encyclopaedia of Math.Sciences, Vol. 4, Springer-Verlag, Berlin, 1989. doi: 10.1007/978-3-662-06791-8_3.  Google Scholar

[20]

J. Fay, Theta-functions on Riemann Surfaces, , Springer Lecture Notes, 352, Springer-Verlag, 1973.  Google Scholar

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Yu. Fedorov, Classical integrable systems related to generalized Jacobians, Acta Appl. Math., 55 (1999), 251-301.  doi: 10.1023/A:1006178224117.  Google Scholar

[22]

Yu. Fedorov, Bäcklund transformations on coadjoint orbits of the loop algebra $\widetilde {{\rm{gl}}}({r})$, Recent advances in integrable systems (Kowloon, Hong Kong, 2000). J. Nonlinear Math. Phys, 9 (2002), suppl. 1, 29–46. doi: 10.2991/jnmp.2002.9.s1.3.  Google Scholar

[23]

Yu. Fedorov, Integrable flows and Bäcklund transformations on extended Stiefel varieies with application to the Euler top on the Lie group $SO(3)$, J. Non. Math. Phys., 12 (2005), Suppl. 2, 77–94, arXiv: nlin/0505045. doi: 10.2991/jnmp.2005.12.s2.7.  Google Scholar

[24]

Yu. Fedorov and B. Jovanović, Geodesic flows and neumann systems on stiefel varieties: Geometry and integrability, Math. Z., 270 (2012), 659–698, arXiv: 1011.1835. doi: 10.1007/s00209-010-0818-y.  Google Scholar

[25]

Yu Fedorov and B. Jovanović, Three natural mechanical systems on Stiefel varieties, J. Phys. A., 45 (2012), 165204, (13pp), arXiv: 1202.1660. doi: 10.1088/1751-8113/45/16/165204.  Google Scholar

[26]

R. L. Fernandes and P. Vanhaecke, Hyperelliptic Prym varieties and integrable systems, Comm. Math. Phys., 221 (2001), 169–196, arXiv: math-ph/0011051. doi: 10.1007/s002200100476.  Google Scholar

[27]

L. Gavrilov, Generalized Jacobians of spectral curves and completely integrable systems, Math. Z., 230 (1999), 487-508.  doi: 10.1007/PL00004701.  Google Scholar

[28]

L. Gavrilov, Jacobians of singularized spectral curves and completely integrable systems, in The Kowalevski Property (Leeds, 2000), 59–68, CRM Proc. Lecture Notes, 32, Amer. Math. Soc., Providence, RI, 2002, arXiv: math/0111235.  Google Scholar

[29]

G. Jensen, Einstein metrics on principal fiber bundles, J. Diff. Geom., 8 (1973), 599-614.  doi: 10.4310/jdg/1214431962.  Google Scholar

[30]

B. Jovanović and Yu. Fedorov, Discrete geodesic flows on Stiefel manifolds, Tr. Mat. Inst. Steklova, 310 (2020) 176-–188, (Russian); English transl.: Proceedings of the Steklov Institute of Mathematics, 310 (2020), 163–-174. doi: 10.4213/tm4107.  Google Scholar

[31]

B. Jovanović and V. Jovanović, Virtual billiards in pseudo–Euclidean spaces: Discrete Hamiltonian and contact integrability, Discrete and Continuous Dynamical Systems–Series A, 37 (2017), 5163–5190, arXiv: 1510.04037. doi: 10.3934/dcds.2017224.  Google Scholar

[32]

B. Jovanović and V. Jovanović, Heisenberg model in pseudo-Euclidean spaces Ⅱ, Regular and Chaotic Dynamics, 23 (2018), 418–437, arXiv: 1808.10783. doi: 10.1134/S1560354718040044.  Google Scholar

[33]

A. N. Hone, V. B. Kuznetsov and O. Ragnisco, Bäcklund transformations for many-body systems related to KdV, J. Phys. A, 32 (1999), L299–L306, arXiv: solv-int/9904003. doi: 10.1088/0305-4470/32/27/102.  Google Scholar

[34]

R. InoueY. Konishi and T. Yamazaki, Jacobian variety and integrable system–-after Mumford, Beauville and Vanhaecke, J. Geom. Phys., 57 (2007), 815-831.  doi: 10.1016/j.geomphys.2006.06.004.  Google Scholar

[35]

S. Kapustin, The Neumann system on Stiefel varieties, Preprint, 1992 (Russian). Google Scholar

[36]

F. Kirwan, Complex Algebraic Curves, London Mathematical Society Student Texts, 23. Cambridge University Press, Cambridge, 1992. doi: 10.1017/CBO9780511623929.  Google Scholar

[37]

H. Knörrer, Geodesics on quadrics and a mechanical problem of C. Neumann, J. Reine Angew. Math., 334 (1982), 69-78.   Google Scholar

[38]

I. Krichiver, Methods of algebraic geometry in the theory of non-linear equations, Russ. Math. Surv., 32 (1977), 185-213.   Google Scholar

[39]

V. Kuznetsov and P. Vanhaecke, Bäcklund transformations for finite-dimensional integrable systems: A geometric approach, J. Geom. Phys., 44 (2002), 1–40, arXiv: nlin/0004003. doi: 10.1016/S0393-0440(02)00029-3.  Google Scholar

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A. S. Mishchenko and A. T. Fomenko, Generalized Liouville method of integration of Hamiltonian systems, Funct. Anal. Appl., 12 (1978), 113-121.   Google Scholar

[42]

P. van Moerbeke and D. Mumford, The spectrum of difference operators and algebraic curves, Acta Math., 143 (1979), 93-154.  doi: 10.1007/BF02392090.  Google Scholar

[43]

J. Moser, Geometry of quadric and spectral theory, in: Chern Symposium 1979, Berlin–Heidelberg–New York, 1980,147–188.  Google Scholar

[44]

J. Moser and A. Veselov, Discrete versions of some classical integrable systems and factorization of matrix polynomials, Comm. Math. Phys., 139 (1991), 217-243.  doi: 10.1007/BF02352494.  Google Scholar

[45]

D. Mumford, Tata Lectures on Theta II, Progress in Math., Birkhauser, 1984.  Google Scholar

[46]

N. N. Nekhoroshev, Action-angle variables and their generalization, Trans. Mosc. Math. Soc., 26 (1972), 180-198.   Google Scholar

[47]

C. Neumann, De probleme quodam mechanico, quod ad primam integralium ultra-ellipticoram classem revocatum, J. Reine Angew. Math., 56 (1859), 46-63.  doi: 10.1515/crll.1859.56.46.  Google Scholar

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M. Pedroni and P. Vanhaecke, A Lie algebraic generalization of the Mumford system, its symmetries and its multi-Hamiltonian structure, Regul. Chaotic Dyn., 3 (1998), 132-160.  doi: 10.1070/rd1998v003n03ABEH000086.  Google Scholar

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show all references

References:
[1]

M. R. AdamsJ. Harnad and E. Previato, Isospectral Hamiltonian flows in finite and infinite dimensions. Ⅰ. Generalized Moser systems and moment maps into loop algebras, Comm. Math. Phys., 117 (1988), 451-500.  doi: 10.1007/BF01223376.  Google Scholar

[2]

M. R. AdamsJ. Harnad and J. Hurtubise, Isospectral Hamiltonian flows in finite and infinite dimensions Ⅱ. Integration of flows, Comm. Math. Phys., 134 (1990), 555-585.  doi: 10.1007/BF02098447.  Google Scholar

[3]

M. R. AdamsJ. Harnad and J. Hurtubise, Dual moment maps into loop algebras, Lett. Math. Phys., 20 (1990), 299-308.  doi: 10.1007/BF00626526.  Google Scholar

[4]

M. R. AdamsJ. Harnad and J. Hurtubise, Darboux coordinates on coadjoint orbits of Lie algebras, Lett. Math. Phys., 40 (1997), 41-57.  doi: 10.1023/A:1007355508426.  Google Scholar

[5]

M. Adler and P. van Moerbeke, Birkhoff strata, Bäckund transformations and regularization of isospectral operators, Advances in Math., 108 (1994), 140-204.  doi: 10.1006/aima.1994.1070.  Google Scholar

[6]

M. Adler, P. van Moerbeke and P. Vanheacke, Algebraic Integrability, Painleve Geometry and Lie Algebras, Springer, 2004. doi: 10.1007/978-3-662-05650-9.  Google Scholar

[7]

C. Athorne and A. Fordy, Generalized KdV and mKdV equations associated with symmetric spaces, J. Phys.A., 20 (1987), 1377-1386.  doi: 10.1088/0305-4470/20/6/021.  Google Scholar

[8]

E. D. Belokolos, A. I. Bobenko, V. Z. Enol'sii, A. R. Its and V. B. Matveev, Algebro-Geometric Approach to Nonlinear Integrable Equations, Springer Series in Nonlinear Dynamics. Springer–Verlag, 1994. Google Scholar

[9]

A. Beauville, Prym varieties and the Schottky problem, Invent. Math., 41 (1977), 149-196.  doi: 10.1007/BF01418373.  Google Scholar

[10]

A. Beauville, Jacobiennes des courbes spectrales et syst'emes hamiltoniens completement integrables, Acta Math., 164 (1990), 211-235.  doi: 10.1007/BF02392754.  Google Scholar

[11]

O. I. Bogoyavlenski, New integrable problem of classical mechanics, Comm. Math. Phys., 94 (1984), 255-269.  doi: 10.1007/BF01209304.  Google Scholar

[12]

A. V. Bolsinov and B. Jovanović, Noncommutative integrability, moment map and geodesic flows, Annals of Global Analysis and Geometry, 23 (2003), 305–322, arXiv: math-ph/0109031. doi: 10.1023/A:1023023300665.  Google Scholar

[13]

E. Casas-Alvero, Singularities of Plane Curves, London Math. Soc. Lecture Notes Series. 276 Cambridge University Press, 2000. doi: 10.1017/CBO9780511569326.  Google Scholar

[14]

A. Clebsch and P. Gordan, Theorie Der Abelschen Funktionen, Teubner, Leipzig, 1866. Google Scholar

[15]

J. EilbeckV. Enol'skiV. Kuznetzov and A. Tsiganov, Linear $R$-matrix algebra for classical separable systems, J. Phys. A: Math. Gen., 27 (1994), 567-578.  doi: 10.1088/0305-4470/27/2/038.  Google Scholar

[16]

P. A. Dirac, On generalized Hamiltonian dynamics, Can. J. Math., 2 (1950), 129-148.  doi: 10.4153/CJM-1950-012-1.  Google Scholar

[17]

V. Dragović and B. Gajić, The Lagrange bitop on $so(4) \times so(4)$ and geometry of the Prym varieties, American J. of Math., 126 (2004), 981–1004, arXiv: math-ph/0201036. doi: 10.1353/ajm.2004.0035.  Google Scholar

[18]

B. A. Dubrovin, Completely integrable Hamiltonian systems associated with matrix operators and Abelian varieties, Funct.Anal.Appl., 11 (1977), 265-277.   Google Scholar

[19]

B. A Dubrovin, S. P. Novikov and I. M. Krichever, Integrable Systems. I,, in Itogi Nauki i Tekhniki. Sovr.Probl.Mat. Fund.Naprav., 4 (1985), 179–284. English transl.: Encyclopaedia of Math.Sciences, Vol. 4, Springer-Verlag, Berlin, 1989. doi: 10.1007/978-3-662-06791-8_3.  Google Scholar

[20]

J. Fay, Theta-functions on Riemann Surfaces, , Springer Lecture Notes, 352, Springer-Verlag, 1973.  Google Scholar

[21]

Yu. Fedorov, Classical integrable systems related to generalized Jacobians, Acta Appl. Math., 55 (1999), 251-301.  doi: 10.1023/A:1006178224117.  Google Scholar

[22]

Yu. Fedorov, Bäcklund transformations on coadjoint orbits of the loop algebra $\widetilde {{\rm{gl}}}({r})$, Recent advances in integrable systems (Kowloon, Hong Kong, 2000). J. Nonlinear Math. Phys, 9 (2002), suppl. 1, 29–46. doi: 10.2991/jnmp.2002.9.s1.3.  Google Scholar

[23]

Yu. Fedorov, Integrable flows and Bäcklund transformations on extended Stiefel varieies with application to the Euler top on the Lie group $SO(3)$, J. Non. Math. Phys., 12 (2005), Suppl. 2, 77–94, arXiv: nlin/0505045. doi: 10.2991/jnmp.2005.12.s2.7.  Google Scholar

[24]

Yu. Fedorov and B. Jovanović, Geodesic flows and neumann systems on stiefel varieties: Geometry and integrability, Math. Z., 270 (2012), 659–698, arXiv: 1011.1835. doi: 10.1007/s00209-010-0818-y.  Google Scholar

[25]

Yu Fedorov and B. Jovanović, Three natural mechanical systems on Stiefel varieties, J. Phys. A., 45 (2012), 165204, (13pp), arXiv: 1202.1660. doi: 10.1088/1751-8113/45/16/165204.  Google Scholar

[26]

R. L. Fernandes and P. Vanhaecke, Hyperelliptic Prym varieties and integrable systems, Comm. Math. Phys., 221 (2001), 169–196, arXiv: math-ph/0011051. doi: 10.1007/s002200100476.  Google Scholar

[27]

L. Gavrilov, Generalized Jacobians of spectral curves and completely integrable systems, Math. Z., 230 (1999), 487-508.  doi: 10.1007/PL00004701.  Google Scholar

[28]

L. Gavrilov, Jacobians of singularized spectral curves and completely integrable systems, in The Kowalevski Property (Leeds, 2000), 59–68, CRM Proc. Lecture Notes, 32, Amer. Math. Soc., Providence, RI, 2002, arXiv: math/0111235.  Google Scholar

[29]

G. Jensen, Einstein metrics on principal fiber bundles, J. Diff. Geom., 8 (1973), 599-614.  doi: 10.4310/jdg/1214431962.  Google Scholar

[30]

B. Jovanović and Yu. Fedorov, Discrete geodesic flows on Stiefel manifolds, Tr. Mat. Inst. Steklova, 310 (2020) 176-–188, (Russian); English transl.: Proceedings of the Steklov Institute of Mathematics, 310 (2020), 163–-174. doi: 10.4213/tm4107.  Google Scholar

[31]

B. Jovanović and V. Jovanović, Virtual billiards in pseudo–Euclidean spaces: Discrete Hamiltonian and contact integrability, Discrete and Continuous Dynamical Systems–Series A, 37 (2017), 5163–5190, arXiv: 1510.04037. doi: 10.3934/dcds.2017224.  Google Scholar

[32]

B. Jovanović and V. Jovanović, Heisenberg model in pseudo-Euclidean spaces Ⅱ, Regular and Chaotic Dynamics, 23 (2018), 418–437, arXiv: 1808.10783. doi: 10.1134/S1560354718040044.  Google Scholar

[33]

A. N. Hone, V. B. Kuznetsov and O. Ragnisco, Bäcklund transformations for many-body systems related to KdV, J. Phys. A, 32 (1999), L299–L306, arXiv: solv-int/9904003. doi: 10.1088/0305-4470/32/27/102.  Google Scholar

[34]

R. InoueY. Konishi and T. Yamazaki, Jacobian variety and integrable system–-after Mumford, Beauville and Vanhaecke, J. Geom. Phys., 57 (2007), 815-831.  doi: 10.1016/j.geomphys.2006.06.004.  Google Scholar

[35]

S. Kapustin, The Neumann system on Stiefel varieties, Preprint, 1992 (Russian). Google Scholar

[36]

F. Kirwan, Complex Algebraic Curves, London Mathematical Society Student Texts, 23. Cambridge University Press, Cambridge, 1992. doi: 10.1017/CBO9780511623929.  Google Scholar

[37]

H. Knörrer, Geodesics on quadrics and a mechanical problem of C. Neumann, J. Reine Angew. Math., 334 (1982), 69-78.   Google Scholar

[38]

I. Krichiver, Methods of algebraic geometry in the theory of non-linear equations, Russ. Math. Surv., 32 (1977), 185-213.   Google Scholar

[39]

V. Kuznetsov and P. Vanhaecke, Bäcklund transformations for finite-dimensional integrable systems: A geometric approach, J. Geom. Phys., 44 (2002), 1–40, arXiv: nlin/0004003. doi: 10.1016/S0393-0440(02)00029-3.  Google Scholar

[40]

H. P. McKean, Variation on a theme of Jacobi, Comm. Pure Appl. Math., 38 (1985), 669-678.  doi: 10.1002/cpa.3160380514.  Google Scholar

[41]

A. S. Mishchenko and A. T. Fomenko, Generalized Liouville method of integration of Hamiltonian systems, Funct. Anal. Appl., 12 (1978), 113-121.   Google Scholar

[42]

P. van Moerbeke and D. Mumford, The spectrum of difference operators and algebraic curves, Acta Math., 143 (1979), 93-154.  doi: 10.1007/BF02392090.  Google Scholar

[43]

J. Moser, Geometry of quadric and spectral theory, in: Chern Symposium 1979, Berlin–Heidelberg–New York, 1980,147–188.  Google Scholar

[44]

J. Moser and A. Veselov, Discrete versions of some classical integrable systems and factorization of matrix polynomials, Comm. Math. Phys., 139 (1991), 217-243.  doi: 10.1007/BF02352494.  Google Scholar

[45]

D. Mumford, Tata Lectures on Theta II, Progress in Math., Birkhauser, 1984.  Google Scholar

[46]

N. N. Nekhoroshev, Action-angle variables and their generalization, Trans. Mosc. Math. Soc., 26 (1972), 180-198.   Google Scholar

[47]

C. Neumann, De probleme quodam mechanico, quod ad primam integralium ultra-ellipticoram classem revocatum, J. Reine Angew. Math., 56 (1859), 46-63.  doi: 10.1515/crll.1859.56.46.  Google Scholar

[48]

A. M. Perelomov, Some remarks on the integrability of the equations of motion of a rigid body in an ideal fluid, Funct. Anal. Appl., 15 (1981), 144-146.   Google Scholar

[49]

M. Pedroni and P. Vanhaecke, A Lie algebraic generalization of the Mumford system, its symmetries and its multi-Hamiltonian structure, Regul. Chaotic Dyn., 3 (1998), 132-160.  doi: 10.1070/rd1998v003n03ABEH000086.  Google Scholar

[50]

J. Potter, Matrix quadratic solutions, J. SIAM Appl. Math., 14 (1966), 496-501.  doi: 10.1137/0114044.  Google Scholar

[51]

E. Previato, Flows on $r$-gonal Jacobians,, in: The legacy of Sonya Kovalevskaya (Cambridge, Mass., and Amherst, Mass., 1985), Contemp. Math., 64, Amer. Math. Soc., Providence, RI, 1987,153–180. doi: 10.1090/conm/064/881461.  Google Scholar

[52]

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