MG11 
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Three-dimensional gravity and geometric theory of defects

A description of dislocations and disclinations defects in terms of Riemann--Cartan geometry is given, with the curvature and torsion tensors being interpreted as the surface densities of the Frank and Burgers vectors, respectively. A new free energy expression describing the static distribution of defects is presented, and equations of nonlinear elasticity theory are used to specify the coordinate system. Application of the Lorentz gauge leads to equations for the principal chiral SO(3)-field. In the defect-free case, the geometric model reduces to elasticity theory for the displacement vector field and to a principal chiral SO(3)-field model for the spin structure. As illustrated by the example of a wedge dislocation, elasticity theory reproduces only the linear approximation of the geometric theory of defects.

Two-dimensional gravity and global solutions in General Relativity

Two-dimensional Gravity is proved to be an integrable model. A general solution of the equations of motion is explicitly found without gauge fixing. It consists of two classes of solutions: (i) constant curvature and zero torsion surfaces (the Liouville theory) and (ii) nonconstant curvature and nonzero torsion surfaces (new solutions). Solutions are found locally and then extended along geodesics providing global structure of surfaces. As an example, we consider General Relativity assuming the space-time to be a warped product of two surfaces. In this case, General Relativity reduces to two-dimensional gravity. The vacuum solutions contain solutions describing many wormholes, cosmic strings, domain walls of curvature singularities, cosmic strings surrounded by domain walls.