Integrable two-dimensional gravity model is proposed (with I. V. Volovich). Conformal block method for construction of global solutions in gravity is developed for arbitrary two-dimensional metrics admitting one Killing vector field. Geometric theory of defects (dislocations and disclinations) in elastic media is proposed (with I. V. Volovich). The media with defects is shown to correspond to Riemann–Cartan manifold. Torsion and curvature tensors are interpreted as the surface densities of Burgers and Frank vectors, respectively. Full classification of global solutions to vacuum Einstein equations with cosmological constant is given under the assumptions that four-dimensional space-time is a product of two surfaces and the metric has a block diagonal form (with T. Klosch and W. Kummer). The found pseudo-riemannian manifolds include solutions describing black holes, worm holes, cosmic strings, domain walls of curvature singularities.
Main publications:
M. O. Katanaev, Geometrical methods in mathematical physics, Manuscript in Russian. Extended version of lectures delivered at the Academic Educational Center at Steklov Mathematical Institute during seven semesters, 2016 , xvi+1570 pp., arXiv: 1311.0733v3
M. O. Katanaev, “Geometric theory of defects”, Phys. Usp., 48:7 (2005), 675–701
M. O. Katanaev, “Effective action for scalar fields in two-dimensional gravity”, Ann. Physics, 296:1 (2002), 1–50 , arXiv: gr-qc/0101033
M. O. Katanaev, I. V. Volovich, “Theory of defects in solids and three-dimensional gravity”, Ann. Physics, 216:1 (1992), 1–28
M. O. Katanaev, I. V. Volovich, “String model with dynamical geometry and torsion”, Phys. Lett. B, 175:4 (1986), 413–416 , arXiv: hep-th/0209014
