Speciality:
01.01.04 (Geometry and topology)
Birth date:
21.06.1951
Phone: +7 (3842) 37 47 30
E-mail: ,
Keywords: group of diffeomorphisms; space of Riemannian metrics; space of almost complex structures; Riemannian metrics and almost complex structures on a symplectic manifold; critical metrics.
UDC: 514.76, 514 MSC: 58d05, 58d17, 58e11, 57s05, 58d27, 58d10, 53c15, 53c55
Subject:
It is known (V. I. Arnold, L. Ebin, J. Marsden, H. Omori), that the motion of an ideal incompressible fluid can be interpreted as geodesic on group of volume preserving diffeomorphisms. The given approach is advanced by the author for a case of a barotropic fluid. It is shown that configuration space of an ideal barotropic fluid is the group of all diffeomorphisms of a manifold $M$. The biinvariant metric on group of volume-preserving diffeomorphisms of a three-dimensional manifold is found. It is shown that its signature is equal to $\eta$-invariant of $M$. The biinvariant metrics on group of symplectic diffeomorphisms and group of contact diffeomorphisms are obtained. The properties of a curvature of groups of diffeomorphisms are investigated. Other series of the articles is devoted to study of spaces of the Riemannian metrics on a compact manifold. In case of a symplectic manifold the spaces of the associated metrics and associated almost complex structures are investigated.
Main publications:
Smolentsev N. K. Biinvariantnaya metrika na gruppe diffeomorfizmov trekhmernogo mnogoobraziya // Sib. matem. zhurn., 1983, 24, 1, 152–159.
Smolentsev N. K. Biinvariantnaya metrika na gruppe simplekticheskikh diffeomorfizmov i uravnenie $\frac{\partial}{\partial t}\Delta F = \{\Delta F,F \}$ // Sib. matem. zhurn., 1986, 27, 1, 150–156.
Smolentsev N. K. O krivizne prostranstva assotsiirovannykh metrik na simplekticheskom mnogoobrazii // Sib. matem. zhurn., 1992, 33, 1, 132–139.
Smolentsev N. K. Kriticheskie assotsiirovannye metriki na simplekticheskom mnogoobrazii // Sib. matem. zhurn., 1995, 36, 2, 409–418.
