105 research outputs found
Argument shift method and sectional operators: applications to differential geometry
This paper does not contain any new results, it is just an attempt to
present, in a systematic way, one construction which establishes an interesting
relationship between some ideas and notions well-known in the theory of
integrable systems on Lie algebras and a rather different area of mathematics
studying projectively equivalent Riemannian and pseudo-Riemannian metrics
Some remarks about Mishchenko-Fomenko subalgebras
We discuss and compare two different approaches to the notion of
Mishchenko--Fomenko subalgebras in Poisson-Lie algebras of finite-dimensional
Lie algebras. One of them, commonly accepted by the algebraic community, uses
polynomial \Ad^*-invariants. The other is based on formal \Ad^*-invariants
and allows one to deal with arbitrary Lie algebras, not necessarily algebraic.
In this sense, the latter is more universal
On one class of holonomy groups in pseudo-Riemannian geometry
We describe a new class of holonomy groups on pseudo-Riemannian manifolds.
Namely, we prove the following theorem. Let g be a nondegenerate bilinear form
on a vector space V, and L:V -> V a g-symmetric operator. Then the identity
component of the centraliser of L in SO(g) is a holonomy group for a suitable
Levi-Civita connection.Comment: (revised version
Smooth invariants of focus-focus singularities and obstructions to product decomposition
We study focus-focus singularities (also known as nodal singularities, or
pinched tori) of Lagrangian fibrations on symplectic -manifolds. We show
that, in contrast to elliptic and hyperbolic singularities, there exist
homeomorphic focus-focus singularities which are not diffeomorphic.
Furthermore, we obtain an algebraic description of the moduli space of
focus-focus singularities up to smooth equivalence, and show that for double
pinched tori this space is one-dimensional. Finally, we apply our construction
to disprove Zung's conjecture which says that any non-degenerate singularity
can be smoothly decomposed into an almost direct product of standard
singularities.Comment: Final version accepted to Journal of Symplectic Geometry; 25 pages, 2
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Jordan-Kronecker invariants of finite-dimensional Lie algebras
For any finite-dimensional Lie algebra we introduce the notion of
Jordan-Kronecker invariants, study their properties and discuss examples. These
invariants naturally appear in the framework of the bi-Hamiltonian approach to
integrable systems on Lie algebras and are closely related to
Mischenko-Fomenko's argument shift method
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