Speciality:
01.01.04 (Geometry and topology)
Birth date:
15.05.1951
E-mail: ,
Keywords: 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.
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.
