group of diffeomorphisms; space of Riemannian metrics; space of almost complex structures; Riemannian metrics and almost complex structures on a symplectic manifold; critical metrics.
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.
Biography
Graduated from Faculty of Mathematics and Mechanics of Tomsk State University in 1973 (Department of geometry). Ph. D. thesis was defended in 1980. D. Sci. thesis was defended in 1995. The list of my works contains more than 50 titles.
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.
N. K. Smolentsev, “Left-invariant para-Kähler structures on six-dimensional nilpotent Lie groups”, Vestn. Tomsk. Gos. Univ. Mat. Mekh., 2022, no. 78, 38–48
2021
2.
N. K. Smolentsev, “Classification of left-invariant para-Sasakian structures on five-dimensional Lie groups”, Itogi Nauki i Tekhniki. Sovrem. Mat. Pril. Temat. Obz., 203 (2021), 100–115
3.
N. K. Smolentsev, I. Y. Shagabudinova, “On parasasakian structures on five-dimensional Lie algebras”, Vestn. Tomsk. Gos. Univ. Mat. Mekh., 2021, no. 69, 37–52
N. K. Smolentsev, “Left-invariant almost para-Hermitian structures on some six-dimensional nilpotent Lie groups”, Vestn. Tomsk. Gos. Univ. Mat. Mekh., 2019, no. 58, 41–55
2018
6.
N. A. Daurtseva, N. K. Smolentsev, “On Almost Complex Structures on Six-Dimensional Products of Spheres”, Itogi Nauki i Tekhniki. Sovrem. Mat. Pril. Temat. Obz., 146 (2018), 17–47; Journal of Mathematical Sciences, 245:5 (2020), 568–600
N. K. Smolentsev, “On almost (para)complex Cayley structures on spheres $\mathbf{S}^{2,4}$ and $\mathbf{S}^{3,3}$”, Vestn. Tomsk. Gos. Univ. Mat. Mekh., 2018, no. 53, 22–38
N. K. Smolentsev, “On Almost Complex Structures on 6-dimensional Products of Spheres”, Kazan. Gos. Univ. Uchen. Zap. Ser. Fiz.-Mat. Nauki, 151:4 (2009), 116–135
N. K. Smolentsev, “On the space of Riemannian metrics on symplectic and contact manifolds”, Sibirsk. Mat. Zh., 42:6 (2001), 1402–1407; Siberian Math. J., 42:6 (2001), 1165–1169
N. K. Smolentsev, “A space of associated metrics on a two-dimensional torus”, Izv. Vyssh. Uchebn. Zaved. Mat., 1995, no. 5, 68–77; Russian Math. (Iz. VUZ), 39:5 (1995), 63–72
11.
N. K. Smolentsev, “Critical associated metrics on a symplectic manifold”, Sibirsk. Mat. Zh., 36:2 (1995), 409–418; Siberian Math. J., 36:2 (1995), 359–367
N. K. Smolentsev, “Two weak Riemannian structures on the space of Riemannian metrics”, Izv. Vyssh. Uchebn. Zaved. Mat., 1994, no. 2, 82–83; Russian Math. (Iz. VUZ), 38:2 (1994), 80–81
13.
N. K. Smolentsev, “Natural weak Riemannian structures on the space of Riemannian metrics”, Sibirsk. Mat. Zh., 35:2 (1994), 439–445; Siberian Math. J., 35:2 (1994), 396–402
N. K. Smolentsev, “On the curvature of the space of associated metrics on A contact manifold”, Sibirsk. Mat. Zh., 33:6 (1992), 188–194; Siberian Math. J., 33:6 (1992), 1119–1125
N. K. Smolentsev, “Curvature of the group of diffeomorphisms and of the space of volume elements”, Sibirsk. Mat. Zh., 33:4 (1992), 135–141; Siberian Math. J., 33:4 (1992), 669–674
N. K. Smolentsev, “On the curvature of a space of associated metrics on a symplectic manifold”, Sibirsk. Mat. Zh., 33:1 (1992), 132–139; Siberian Math. J., 33:1 (1992), 111–117
N. K. Smolentsev, “Space of associated metrics on a regular contact manifold”, Sibirsk. Mat. Zh., 31:3 (1990), 176–185; Siberian Math. J., 31:3 (1990), 505–513
1989
19.
N. K. Smolentsev, “Orthogonal decompositions of the space of symmetric tensors on an almost Kählerian manifold”, Sibirsk. Mat. Zh., 30:3 (1989), 131–139; Siberian Math. J., 30:3 (1989), 451–457
1987
20.
N. K. Smolentsev, “The space of $K$-contact metrics of a three-dimensional manifold”, Sibirsk. Mat. Zh., 28:6 (1987), 119–125; Siberian Math. J., 28:6 (1987), 960–965
N. K. Smolentsev, “A bi-invariant metric on a group of symplectic diffeomorphisms and the equation $\frac\partial{\partial t}\Delta F=\{\Delta F,F\}$”, Sibirsk. Mat. Zh., 27:1 (1986), 150–156; Siberian Math. J., 27:1 (1986), 120–126
N. K. Smolentsev, “The vector cross product on a seven-dimensional manifold”, Sibirsk. Mat. Zh., 25:5 (1984), 157–167; Siberian Math. J., 25:5 (1984), 807–815
23.
N. K. Smolentsev, “A group of diffeomorphisms that leaves a vector field fixed”, Sibirsk. Mat. Zh., 25:2 (1984), 180–185; Siberian Math. J., 25:2 (1984), 313–317
N. K. Smolentsev, “A bi-invariant metric on the group of diffeomorphisms of a three-dimensional manifold”, Sibirsk. Mat. Zh., 24:1 (1983), 152–159; Siberian Math. J., 24:1 (1983), 124–130
N. N. Danilov, N. N. Kozik, N. K. Smolentsev, “All-Union School “Optimal Control. Geometry and Analysis””, Uspekhi Mat. Nauk, 44:3(267) (1989), 199–200
1987
29.
N. N. Kozik, N. K. Smolentsev, V. A. Shalaumov, “All-Union School “Optimal Control. Geometry and Analysis””, Uspekhi Mat. Nauk, 42:5(257) (1987), 235
1985
30.
N. N. Kozik, N. K. Smolentsev, V. A. Shalaumov, “All-Union function theory school in celebration of the 100th birthday of academician N. N. Luzin”, Uspekhi Mat. Nauk, 40:3(243) (1985), 226–227
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