homogeneous manifolds; Lie groups; generalized symmetric spaces (homogeneous $\Phi$-spaces, in particular, $k$-symmetric spaces); geometric structures on manifolds (almost complex, almost product structures, $f$-structures, etc.); pseudo-Riemannian and almost symplectic manifolds; almost Hermitian structures; generalized Hermitian geometry; nearly Kahler manifolds and their generalizations; twistor and spinor structures on manifolds; geometric structures in physics.
Subject:
The problem of describing all canonical affinor structures of classical type (almost complex, almost product, $f$-structures) on regular $\Phi$-spaces was completely solved (jointly N.A.Stepanov). In the case of homogeneous $\Phi$-spaces of arbitrary finite order $k$ ($k$-symmetric spaces) precise computational formulae for the above structures were indicated. It was proved that all classical canonical structures on $k$-symmetric spaces are compatible with natural pseudo-Riemannian metrics. Wide classes of invariant examples for generalized Hermitian geometry were presented on the base of canonical $f$-structures on $k$-symmetric spaces. In particular, a concept of nearly K\"ahler $f$-structures was introduced. It was proved that canonical $f$-structures on naturally reductive $\Phi$-spaces of orders 4 and 5 belong to these structures. As a result, the analogy with well-known homogeneous nearly K\"ahler manifolds and 3-symmetric spaces was obtained. The problem of classifying regular $\Phi$-spaces with respect to the commutative algebra of all canonical affinor structures was solved.
Biography
Graduated from Faculty of Mathematics of the Belarusian State University in 1973 (department of geometry). Ph.D. thesis was defended in 1980. A list of my works contains about 60 titles. Since 1991 I have been a member of the Bureau of Minsk Geometric seminar at the Belarusian State University.
In 1981 I was awarded the first prize in the Competition of young scientists at the Belarusian State University.
Main publications:
Balashchenko V. V. Invariant nearly Kahler $f$-structures on homogeneous spaces // Global Differential Geometry: The Mathematical Legacy of Alfred Gray, Contemporary Mathematics, 2001, vol. 288, 263–267.
V. V. Balashchenko, “Generalized symmetric spaces, Yu. P. Solovyov's formula, and the generalized Hermitian geometry”, Fundam. Prikl. Mat., 13:8 (2007), 43–60; J. Math. Sci., 159:6 (2009), 777–789
V. V. Balashchenko, “Canonical $f$-structures of hyperbolic type on regular $\Phi$-spaces”, Uspekhi Mat. Nauk, 53:4(322) (1998), 213–214; Russian Math. Surveys, 53:4 (1998), 861–863
V. V. Balashchenko, S. V. Vedernikov, N. A. Stepanov, A. S. Fedenko, “The scientific heritage of Vasilii Ivanovich Vedernikov (Feb. 11, 1919–March 16, 1991)”, Itogi Nauki i Tekhniki. Sovrem. Mat. Pril. Temat. Obz., 8 (1995), 37–58; J. Math. Sci., 74:3 (1995), 961–976
8.
V. V. Balashchenko, N. A. Stepanov, “Canonical affinor structures of classical type on regular $\Phi$-spaces”, Mat. Sb., 186:11 (1995), 3–34; Sb. Math., 186:11 (1995), 1551–1580
V. V. Balashchenko, O. V. Dashevich, “Geometry of canonical structures on homogeneous $\Phi$-spaces of order 4”, Uspekhi Mat. Nauk, 49:4(298) (1994), 153–154; Russian Math. Surveys, 49:4 (1994), 149–150
V. V. Balashchenko, N. A. Stepanov, “Canonical affinor structures on regular $\Phi$-spaces”, Uspekhi Mat. Nauk, 46:1(277) (1991), 205–206; Russian Math. Surveys, 46:1 (1991), 247–248
V. V. Balashchenko, Yu. D. Churbanov, “Invariant structures on homogeneous $\Phi$-spaces of order 5”, Uspekhi Mat. Nauk, 45:1(271) (1990), 169–170; Russian Math. Surveys, 45:1 (1990), 195–197
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