Realisation of holonomy algebras on pseudo-Riemannian manifolds by means of Manakov operators

In the present thesis we construct a new class of holonomy algebras in pseudo-Riemannian geometry. Starting from a smooth connected manifold M, we consider its (1;1)-tensor fields acting on the tangent spaces. We then prove that there exists a class of pseudo- Riemannian metrics g on M such that the (1;1)-tensor fields are g-self adjoint and their centralisers in the Lie algebra so(g) are holonomy algebras for the Levi-Civita connection of g. Our construction is elaborated with the aid of Manakov operators and holds for any signature of the metric g

On a new class of holonomy groups in pseudo-Riemannian geometry

On a new class of holonomy groups in pseudo-Riemannian geometr

Finite-dimensional integrable systems: a collection of research problems

This article suggests a series of problems related to various algebraic and geometric aspects of integrability. They reflect some recent developments in the theory of finite-dimensional integrable systems such as bi-Poisson linear algebra, Jordan-Kronecker invariants of finite dimensional Lie algebras, the interplay between singularities of Lagrangian fibrations and compatible Poisson brackets, and new techniques in projective geometry