In [BT1,BT2] we describe a new phenomenon (the socalled
"integrable chaos") in dynamical systems. Namely, we have
constructed a series of examples of integrable geodesic flows with
positive topological entropy. These systems, in particular, are
counterexamples to the Paternain conjecture on nonintegrability of
geodesic flows on compact manifolds with fundamental groups of exponential
growth. In [BDV] we have studied the spectral problem for these examples.
In particular, we have described the spectrum ot the Laplace operator, its
eigenfunctions and the quantum monodromy phenomenon.
[BT1] Bolsinov A. V., Taimanov I. A.
Integrable geodesic flows with positive topological entropy,
// Invent. Math., 140(2000), pp. 639650.
[BT2] Bolsinov A. V., Taimanov I. A.
Integrable geodesic flows on suspensions of automorphisms of
tori // Proc. of the Steklov Institute of Mathematics, 231(2000),
pp.4258.
[BDV] Bolsinov A., Dullin H., Veselov A. Spectra of
SOLmanifolds: arithmetic and quantum monodromy // Comm. Math.
Phys., 2006, V.264, pp.583611.
The following question is classical in modern Riemannian
geometry: what are smooth compact manifolds
that admit integrable geodesic flows? In [BJ1, BJ2] we generalize a series
of results obtained by Mishchenko, Thimm, Paternain, Spatzier and Bazaikin
by proving that all homogeneous spaces G/H as well as biquotients
K\G/H of a compact Lie group G admit integrable geodesic
flows.
[BJ1] Bolsinov A., Jovanovic B. Noncommutative integrability, moment
map and geodesic flows// Annals of Glob. Anal. and Geom., 2003, Vol. 23,
pp. 305322.
[BJ2] Bolsinov A., Jovanovic B. Complete involutive algebras of
functions on cotangent bundles of homogeneous spaces// Math. Zeit.,
2004, Vol. 246, pp. 213236.
Two Riemannian metrics g and g' on a manifold M are said to
be projectively equivalent if they have the same geodesics (considered as
unparameterized curves). In [BM1] we have found a new condition for two
metrics to be projectively equivalent and show that the class of
projectively equivalent geodesic flows coincide with the socalled Benenti
systems. In particular, such geodesic flows are always bihamiltonian.
[BM1] Bolsinov A.V., Matveev V.S.
Geometrical interpretation of Benenti systems //
Jour. Geometry and Physics, 2003, Vol. 44, pp. 489506.
In [B1] we study nondegenerate singularities of Lagrangian foliations
on fourdimensional symplectic manifolds.
The main result of this paper is the topological
classification of saddlesaddle singularities of complexity two
(i.e., with 2 critical points on the singular leaf). We have shown that
there are exactly 39 different topological types of such singularities.
This classification turns out to be quite useful for the topological
analysis of integrable Hamiltonian systems with two degrees of freedom. In
particular, in [BRF] we use the results and ideas of [B1] to obtain the
complete topological description of the Kovalevskaya top in rigid body
dynamics.
[B1] Bolsinov A.V. Methods of calculation of FomenkoZieschang
topological invariant// In book : Advances in Sov. Math., 1991, Vol. 6,
AMS, Providence, pp. 147183.
[BRF] Bolsinov A.V., Richter P.H., Fomenko A.T.
Loop molecule method and the topology of the Kovalevskaya top//
Sbornik: Mathematics, 191(2000) No. 2, pp. 151188.
In [BF1], [B2] we have constructed
an invariant (more precisely, a family of invariants) that allows one to
classify integrable Hamiltonian systems with two degrees of freedom up to
orbital equivalence. By applying this technics, in [BF2] we have
found a new nontrivial orbital isomorphism between two classical systems:
Euler top in rigid body dynamics and the geodesic flow on the ellipsoid
(Jacobi problem). Then, using smooth orbital invariants, in [BD] we have
shown that this isomorphism is only continious: in the smooth sense the
two systems are not equivalent.
[BF1] Bolsinov A.V., Fomenko A.T. Orbital equivalence of
integrable systems with two degrees of freedom. The classification
theorem. Part I, II// Russian Acad. Sci. Sb.
Math. 81(1995), No. 2, pp. 421465 and 82(1995), No. 1, pp. 2163.
[B2] Bolsinov A.V. A smooth trajectory classification of
integrable Hamiltonian systems with two degrees of freedom //
Sbornik: Mathematics 186(1995) No. 1, pp. 127.
[BF2] Bolsinov A.V., Fomenko A.T.
Orbital classification of geodesic flows on twodimensional ellipsoids.
The Jacobi problem is orbitally equivalent to the integrable Euler case in
rigid body dynamics // Functional Analysis and
its Appl., 29(1995) No. 3, pp. 149160.
[BD] Bolsinov A.V., Dullin H. On the Euler case in rigid body
dynamics and the Jacobi problem // Regular and Chaotic Dynamics, 2(1997)
No. 2, pp. 1325 (in Russian).
It is wellknown that a pair of compatible Poisson
brackets on a manifold M always generates a commutative
subalgebra A in the Poisson algebra of all smooth functions on M. A natural
problem appearing very often in applications is to find out whether A is complete (i.e., guarantees the complete Liouville
integrability). In [B3, B4, B5] I have obtained a completeness criterion which
allows one to answer efficiently this question. A particular case of this
theorem gives necessary and sufficient conditions for completeness of some
wellknown commutative subalgebras in the PoissonLie algebra P(g) of a finitedimensional Lie algebra g
(see [B3, B4, B5]) . These results are closely related to the
MischenkoFomenko conjecture (1978) saying that P(g) always
admits a complete commutative subalgebra.
In [B6] we give a constructive proof of this conjecture following the
method suggested recently by S. Sadetov.
[B3] Bolsinov A.V. A completeness criterion for a family of
function in involution constructed by the argument shift method//
Sov. Math. Dokl., 38(1989) No. 1, pp. 161165.
[B4] Bolsinov A.V. Compatible Poisson brackets on Lie algebras and
completeness of families of functions in involution//
Math. USSR Izvestiya, 38(1992) No. 1, pp. 6989.
[B5] Bolsinov A.V. Commutative families of functions related to
consistent Poisson brackets// Acta Appl. Math., 24(1991), pp. 253274.
[B6] Bolsinov A. Complete commutative families of polynomials
in PoissonLie algebras: A proof of the MischenkoFomenko conjecture
// In book: Tensor and Vector Analysis, Vol. 26, Moscow State
University, 2005, pp. 87109. (in Russian).
In [BM2] we study symplectic invariants of singular Lagrangian
foliations on simplectic 4manifolds and then apply them to
the classical rigidity problem in Riemannian geometry. Namely, we
construct new nontrivial examples of Riemannian metrics with
conjugate geodesic flows on 2surfaces
of arbitrary genus.
In [BV] we study semilocal symplectic invariants of nondegenerate
singularities of Lagrangian foliations of arbitrary dimension.
In particular, we obtain a symplectic
classification of nondegenerate singularities of complexity one.
[BM2] Bolsinov A., Matveev V. Symplectic invariants of Liouville
foliations and conjugacy of geodesic flows (in preparation).
[BV] Bolsinov A., Vu Ngoc San, Symplectic equivalence for
integrable systems with common action integrals (in prepration).
