Topological Analysis of the Liouville Foliation for the Kovalevskaya Integrable Case on the Lie Algebra so(4)

Singularities of integrable Liouville systems, reduction of integrals to lower degree and topological billiards: Recent results

“…Let us know, that -mark on the loop molecule of any such singularity is equal to zero. In the work by V. A. Kibkalo [19] admissible bases (admissible coordinate systems) for two intersecting arcs of the bifurcation diagram are expressed via uniquely defined -cycles similarly to [18]. It has already been applied for the calculation of Fomenko-Zieschang invariants of Liouville foliation on isoenergy manifolds.…”

“…As a result we can now calculate -marks. The problem of the calculation of gluing matrices for the points of center-center type was also investigated in the work by V. A. Kibkalo [19]. In this work, the orientation of the tangent basis ( , ) to the torus 2 was defined by the condition ( ∧ )(grad , , , ) > 0, where is the outer normal vector to the 3-atom in 4 .…”

“…In this work, the orientation of the tangent basis ( , ) to the torus 2 was defined by the condition ( ∧ )(grad , , , ) > 0, where is the outer normal vector to the 3-atom in 4 . But the gluing matrix was not found explicitly in [19]. Nevertheless, theorem 3.2 determines the relations between admissible bases.…”

Topological classification of Liouville foliations for the Kovalevskaya integrable case on the Lie algebra so(3, 1)

“…, γ 11 : (λ γ 11 , λ β 4 ) (7) (λ γ 11 , −λ β 3 ) (2,5) , δ 3 : (λ δ 3 , λ β 4 ) (7) , ξ 6 : (λ ξ 6 , λ γ 5 ) (6) .…”

Topology of isoenergy surfaces of Kovalevskaya integrable case on the Lie algebra so(4)

“…In [6] Fomenko-Zieschang invariants for the classical Kovalevskays case were calculated by A.Bolsinov, A. Fomenko and P.Richter using the loop molecule approach for expression of some bases through uniquely defined cycles. In [7] by similar way V. Kibkalo expressed admissible coordinate systems that appear only in the case of so(4).…”