Geometric interpretation and classification of global solutions in generalized dilaton gravity

Two-dimensional gravity with torsion is proved to be equivalent to special types of generalized 2D dilaton gravity. For example, in one version, the dilaton field is shown to be expressible by the extra scalar curvature, constructed for an independent Lorentz connection corresponding to a nontrivial torsion. Elimination of that dilaton field yields an equivalent torsionless theory, nonpolynomial in curvature. These theories, although locally equivalent, exhibit quite different global properties of the general solution. We discuss the example of a (torsionless) dilaton theory equivalent to the $R^2+T^2$ model. Each global solution of this model is shown to split into a set of global solutions of generalized dilaton gravity. In contrast to the theory with torsion, the equivalent dilaton one exhibits solutions which are asymptotically flat in special ranges of the parameters. In the simplest case of ordinary dilaton gravity we clarify the well-known problem of removing the Schwarzschild singularity by a field redefinition.