References
V. I. Arnold, “Sur la géométrie différentielle des groupes de Lie de dimension infinite et ses application à l'hydrodynamique des fluides parfaits,” Ann. de l'Institut Fourier,16, No. 1, 319–361 (1966).
V. I. Arnold, Mathematical Methods in Classical Mechanics [in Russian], Nauka, Moscow (1974).
A. M. Lukatskiî, “The curvature of the group of diffeomorphisms preserving the measure of then-dimensional torus,’ Sib. Mat. Zh.,25, No. 6, 76–88 (1984).
N. K. Smolentsev, “A bi-invariant metric on the group of diffeomorphisms of a three-dimensional manifold,” Sib. Mat. Zh.,24, No. 1, 152–159 (1983).
N. K. Smolentsev, “A bi-invariant metric on a group of symmetric diffeomorphisms and the equation\(\frac{\partial }{{\partial t}}\Delta F = \{ \Delta F,F\}\),” Sib. Mat. Zh.,27, No. 1, 150–156 (1986).
N. K. Smolentsev, “The Maupertuis principle,” Sib. Mat. Zh.,20, No. 5, 1092–1098 (1979).
H. Omori, Infinite Dimensional Lie Transformation Groups, Springer, Berlin etc. (1974).
D. Ebin and J. Marsden, “Groups of diffeomorphisms and the motion of an incompressible fluid,” Annals of Math.,92, No. 1, 102–163 (1970).
L. I. Sedov, Continuum Mechanics [in Russian], Mir, Moscow (1972).
D. Gromoll, W. Klingenberg, and W. Meyer, Riemannian Geometry in the Large [Russian translation], Mir, Moscow (1972).
A. F. Solov'ev, “Curvature of a distribution,” Mat. Zametki,35, No. 1, 11–124 (1984).
N. K. Smolentsev, “Geometrical properties of flows of ideal barotropic fluid,” Tr. Tomsk Univ., No. 21, 68–78 (1980).
N. K. Smolentsev, “Integrals of flows of an ideal barotropic fluid,” Sib. Mat. Zh.,23, No. 1, 205–207 (1982).
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Kemerovo. Translated fromSibirskiî Matematicheskiî Zhurnal, Vol. 33, No. 4, pp. 135–141, July–August, 1992.
Translated by N. S. Dairbekov
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Smolentsev, N.K. Curvatures of the diffeomorphism group and the space of volume elements. Sib Math J 33, 669–674 (1992). https://doi.org/10.1007/BF00971132
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DOI: https://doi.org/10.1007/BF00971132