Spectra and geodesic flows on nilmanifolds Reductions of Hamiltonian systems and differential operators

Ruishi KUWABARA

doi:10.2969/jmsj/04610119

Spectra and geodesic flows on nilmanifolds

Reductions of Hamiltonian systems and differential operators

Ruishi KUWABARA

著者情報

ジャーナルフリー

1994 年 46 巻 1 号 p. 119-137

DOI https://doi.org/10.2969/jmsj/04610119

詳細

Published: 1994 Received: 1991/11/18 Available on J-STAGE: 2006/10/20 Accepted: - Advance online publication: - Revised: -
訂正情報
訂正日: 2006/10/20 訂正理由: - 訂正箇所: 論文タイトル訂正内容: Wrong : Spectra and geodesic flows on nilmanifolds: Reductions of Hamiltonian systems and differential operators
Right : Spectra and geodesic flows on nilmanifolds

訂正日: 2006/10/20 訂正理由: - 訂正箇所: 論文サブタイトル訂正内容: Right : Reductions of Hamiltonian systems and differential operators

訂正日: 2006/10/20 訂正理由: - 訂正箇所: 引用文献情報訂正内容: Wrong : 1) R. Abraham and J. E. Marsden, Foundations of Mechanics, 2nd ed., Benjamin/Cummings, Reading MA, 1978.
2) P. Bérard, Variétés riemanniennes isospectrales non isométriques, Séminaire BOURBAKI, 1988-89, 705, 1989.
3) L. Corwin and F. P. Greenleaf, Representations of nilpotent Lie groups and their applications. Part I: Basic theory and examples, Cambridge Univ. Press, 1989.
4) D. M. DeTurck and C. S. Gordon, Isospectral deformation I: Riemannian structures on two-step nilspaces, Comm. Pure Appl. Math., 40 (1987), 367-387.
5) D. M. DeTurck and C. S. Gordon, lsospectral deformation II: trace formulas, metrics, and potentials, Comm. Pure Appl. Math., 42 (1989), 1067-1095.
6) D. DeTurck, H. Gluck, C. Gordon and D. Webb, How can a drum change shape, while sounding the same? Part II, to appear in Mechanics, Analysis and Geometry 200 years after Lagrange.
7) C. S. Gordon, The Laplace spectra versus the length spectra of Riemannian manifolds, Contemp. Math., 51 (1986), 63-80.
8) C. S. Gordon, You can't hear the geodesic flow, preprint.
9) C. S. Gordon and E. N. Wilson, lsospectral deformations of compact solvmanifolds, J. Differential Geom., 19 (1984), 241-256.
10) S. Kobayashi and K. Nomizu, Foundations of Differential Geometry, Vol. I, John Wiley, New York, 1963.
11) R. Kuwabara, On spectra of the Laplacian on vector bundles, J. Math. Tokushima Univ., 16 (1982), 1-23.
12) R. Kuwabara, Isospectral connections on line bundles, Math. Z., 204 (1990), 465-473.
13) F. Marhuenda, Microlocal analysis of some Isospectral deformations, 1991, preprint.
14) J. Marsden and A. Weinstein, Reduction of symplectic manifolds with symmetry, Rep. Math. Phys., 5 (1974), 121-130.
15) J. Milnor, Eigenvalues of the Laplace operator on certain manifolds, Proc. Nat. Acad. Sci. U. S. A., 51 (1964), 542.
16) R. Montgomery, Canonical formulations of a classical particle in a Yang-Mills field and Wong's equations, Lett. Math. Phys., 8 (1984), 59-67.
17) T. Sunada, Riemannian coverings and isospectral manifolds, Ann. of Math., 121 (1985), 169-186.
18) S. Zelditch, Isospectrality in the FIO category, J. Differential Geom., 35 (1992), 689-710.

Right : [1] R. Abraham and J. E. Marsden, Foundations of Mechanics, 2nd ed., Benjamin/Cummings, Reading MA, 1978.
[2] P. Bérard, Variétés riemanniennes isospectrales non isométriques, Séminaire BOURBAKI, 1988-89, 705, 1989.
[3] L. Corwin and F. P. Greenleaf, Representations of nilpotent Lie groups and their applications. Part I: Basic theory and examples, Cambridge Univ. Press, 1989.
[4] D. M. DeTurck and C. S. Gordon, Isospectral deformation I: Riemannian structures on two-step nilspaces, Comm. Pure Appl. Math., 40 (1987), 367-387.
[5] D. M. DeTurck and C. S. Gordon, lsospectral deformation II: trace formulas, metrics, and potentials, Comm. Pure Appl. Math., 42 (1989), 1067-1095.
[6] D. DeTurck, H. Gluck, C. Gordon and D. Webb, How can a drum change shape, while sounding the same? Part II, to appear in Mechanics, Analysis and Geometry 200 years after Lagrange.
[7] C. S. Gordon, The Laplace spectra versus the length spectra of Riemannian manifolds, Contemp. Math., 51 (1986), 63-80.
[8] C. S. Gordon, You can't hear the geodesic flow, preprint.
[9] C. S. Gordon and E. N. Wilson, lsospectral deformations of compact solvmanifolds, J. Differential Geom., 19 (1984), 241-256.
[10] S. Kobayashi and K. Nomizu, Foundations of Differential Geometry, Vol. I, John Wiley, New York, 1963.
[11] R. Kuwabara, On spectra of the Laplacian on vector bundles, J. Math. Tokushima Univ., 16 (1982), 1-23.
[12] R. Kuwabara, Isospectral connections on line bundles, Math. Z., 204 (1990), 465-473.
[13] F. Marhuenda, Microlocal analysis of some Isospectral deformations, 1991, preprint.
[14] J. Marsden and A. Weinstein, Reduction of symplectic manifolds with symmetry, Rep. Math. Phys., 5 (1974), 121-130.
[15] J. Milnor, Eigenvalues of the Laplace operator on certain manifolds, Proc. Nat. Acad. Sci. U. S. A., 51 (1964), 542.
[16] R. Montgomery, Canonical formulations of a classical particle in a Yang-Mills field and Wong's equations, Lett. Math. Phys., 8 (1984), 59-67.
[17] T. Sunada, Riemannian coverings and isospectral manifolds, Ann. of Math., 121 (1985), 169-186.
[18] S. Zelditch, Isospectrality in the FIO category, J. Differential Geom., 35 (1992), 689-710.

訂正日: 2006/10/20 訂正理由: - 訂正箇所: PDFファイル訂正内容: -

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