dynamical systems on homogeneous spaces of Lie groups; integrability questions of invariant Hamiltonian systems; critical point of sectional curvatures; invariant metrics on homogeneous spaces.
Subject:
By using the soliton theory methods, the solution of the problem of the integration of the equations of the geodesic lines for the important class of left-invariant metrics on simple Lie groups has been obtained in terms of special functions. It is shown that there are no self-intersections on extremals of the invariant Lagrangians on the homogeneous spaces. Using symplectic geometry, it is found a characteristic property of the one class of the left-invariant metrics on the semisimple Lie groups, and the definition of perfect quadratic Lie algebras has been given. A number of papers were devoted to the critical points and critical values of the sectional curvatures (pseudo) Riemannian manifolds. For the symmetric Riemannian manifolds the problem of finding of the critical points and values is solved completely. The embedding of the Grassmannian manifolds, which linearize the sectional curvature, is pointed out. With this linearization the estimates of the curvature reduce to the study of the geometry of the convex hull of the orbits of some Lie groups.
Main publications:
Mescheryakov M. V. Integrirovanie uravnenii geodezicheskikh levoinvariantnykh metrik na prostykh gruppakh Li s pomoschyu spetsialnykh funktsii // Matem. sbornik, 1982, 117(4), 481–493.
Mescheryakov M. V. O kharakteristicheskom svoistve tenzora inertsii mnogomernogo tverdogo tela // UMN, 1983, 38(5), 201–202.
Mescheryakov M. V. Neskolko zamechanii o gamiltonovykh potokakh na odnorodnykh prostranstvakh // UMN, 1985, 40(3), 215–216.
Mescheryakov M. V. Otsenki razmernosti prostranstva invariantnykh affinnykh svyaznostei, soglasovannykh s prisoedinennym predstavleniem poluprostykh grupp Li // Matematika. Izv. vuzov, 1984, 3, 41–43.
Mescheryakov M. V. Otsenki sektsionnykh krivizn rimanovykh simmetricheskikh prostranstv // UMN, 1996, 51(1), 157–158.
M. V. Meshcheryakov, “Reach of orbits of the isotropy representations of Riemannian symmetric spaces”, Algebra i Analiz, 36:6 (2024), 112–121
2021
2.
M. V. Meshcheryakov, “Upper estimates for the Morse numbers of the matrix elements of real linear irreducible representations for connected compact simple Lie groups”, Algebra i Analiz, 33:6 (2021), 107–120; St. Petersburg Math. J., 33:6 (2022), 971–980
2020
3.
M. V. Meshcheryakov, “Classification of taut irreducible real linear representations of compact connected Lie groups”, Algebra i Analiz, 32:1 (2020), 40–50; St. Petersburg Math. J., 32:1 (2021), 31–38
M. V. Meshcheryakov, “Estimates of the sectional curvatures of Riemannian symmetric spaces”, Uspekhi Mat. Nauk, 51:1(307) (1996), 157–158; Russian Math. Surveys, 51:1 (1996), 152–154
1994
5.
M. V. Meshcheryakov, “Estimates of the dimension of the space of invariant affine connections that are compatible with the adjoint representation of semisimple Lie groups”, Izv. Vyssh. Uchebn. Zaved. Mat., 1994, no. 3, 41–23; Russian Math. (Iz. VUZ), 38:3 (1994), 39–41
1985
6.
M. V. Meshcheryakov, “Some remarks on Hamiltonian flows on homogeneous spaces”, Uspekhi Mat. Nauk, 40:3(243) (1985), 215–216; Russian Math. Surveys, 40:3 (1985), 243–244
1983
7.
M. V. Meshcheryakov, “A characteristic property of the inertial tensor of a multidimensional solid body”, Uspekhi Mat. Nauk, 38:5(233) (1983), 201–202; Russian Math. Surveys, 38:5 (1983), 156–157
M. V. Meshcheryakov, “The integration of the equations for geodesics of left-invariant metrics on simple Lie groups using special functions”, Mat. Sb. (N.S.), 117(159):4 (1982), 481–493; Math. USSR-Sb., 45:4 (1983), 473–485
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