We provide a concise approach to generalized dilaton theories with and without torsion and coupling to Yang - Mills fields. Transformations on the space of fields are used to trivialize the field equations locally. In this way their solution becomes accessible within a few lines of calculation only. In this first of a series of papers we set the stage for a thorough global investigation of classical and quantum aspects of more or less all available 2D gravity - Yang - Mills models.

A set of simple rules for constructing the maximal (e.g. analytic) extensions for any metric with a Killing field in an (effectively) two-dimensional spacetime is formulated. The application of these rules is extremely straightforward, as is demonstrated at various examples and illustrated with numerous figures. Despite the resulting simplicity we also comment on some subtleties concerning the concept of Penrose diagrams. Most noteworthy among these, maybe, is that (smooth) spacetimes which have both degenerate and non-degenerate (Killing) horizons do not allow for globally smooth Penrose diagrams. Physically speaking this obstruction corresponds to an infinite relative red/blueshift between observers moving across the two horizons. -The present work provides a further step in the classification of all global solutions of the general class of two-dimensional gravity-Yang-Mills systems introduced in Part I [1], comprising, e.g., all generalized (linear and nonlinear) dilaton theories. In Part I we constructed the local solutions, which were found to always have a Killing field; in this paper we provide all universal covering solutions (the simply connected maximally extended spacetimes). A subsequent Part III [2] will treat the diffeomorphism inequivalent solutions for all other spacetime topologies.Part II is kept entirely self-contained; a prior reading of Part I is not necessary.

We provide a concise approach to generalized dilaton theories with and without torsion and coupling to Yang-Mills fields. Transformations on the space of fields are used to trivialize the field equations locally. In this way their solution becomes accessible within a few lines of calculation only. In this first of a series of papers we set the stage for a thorough global investigation of classical and quantum aspects of more or less all available 2D gravity-Yang-Mills models.

All global solutions of arbitrary topology of the most general 1+1 dimensional dilaton gravity models are obtained. We show that for a generic model there are globally smooth solutions on any non-compact 2-surface. The solution space is parametrized explicitly and the geometrical significance of continuous and discrete labels is elucidated. As a corollary we gain insight into the (in general non-trivial) topology of the reduced phase space. The classification covers basically all 2D metrics of Lorentzian signature with a (local) Killing symmetry.PACS numbers: 04.20.Gz 04.60.KzClass. Quantum Grav. 14 (1997), 1689. * e-mail: kloesch@tph.tuwien.ac.at † e-mail: tstrobl@physik.rwth-aachen.de Motivation and first resultsMuch of the interest in two-dimensional gravity models centers around their quantization. However, for any interpretation of quantum results and, even more, for a comparison and possibly an improvement of existing quantization schemes, a sound understanding of the corresponding classical theory is indispensable.Therefore, in this paper we pursue quite an ambitious goal: Given any 2D gravity Lagrangian of the form [1] we want to classify all its global, diffeomorphism inequivalent classical solutions. This shall be done without any restriction on the topology of the spacetime M . For some of the popular, but specific choices of the potentials D, V, Z, such as those of ordinary (i.e. string inspired, 'linear') dilaton gravity, of deSitter gravity, or of spherically reduced gravity, cf. [2], the possible topologies of the maximally extended solutions turn out to be restricted considerably through the field equations. In particular their first homotopy is either trivial or (at most) Z. (Allowing e.g. also for conical singularities, cf. Sec. 7, the fundamental group might become more involved.)For any 'sufficiently generic' (as specified below) smooth/analytic choice of D, V, Z, on the other hand, the field equations of L allow for maximally extended, globally smooth/ analytic solutions on all non-compact two-surfaces with an arbitrary number of handles (genus) and holes (≥ 1). 2 This shall be one of the main results of the present paper. These solutions are smooth and maximally extended, more precisely, the boundaries are either at an infinite distance (geodesically complete) or they correspond to true singularities (of the curvature R and/or the dilaton field Φ). We will call such solutions global, as there are other kinds of inextendible solutions (cf. below and Sec. 7).The existence of solutions on such non-trivial spacetimes is a qualitatively new challenge for any programme of quantizing a gravity theory. Take, e.g., a Hamiltonian approach to quantization: In any dimension D + 1 of spacetime the Hamiltonian formulation necessarily is restricted to topologies of the form Σ×IR where Σ is some (usually spacelike) D-manifold. In our two-dimensional setting Σ may be IR or S 1 only. Thus π 1 (M ) can be Z at most. According to our discussion above this is far from exhaustive in most of the models (1...

Assuming the four-dimensional space-time to be a general warped product of two surfaces we reduce the four-dimensional Einstein equations to a two-dimensional problem which can be solved. All global vacuum solutions are explicitly constructed and analysed. The classification of the solutions includes the Schwarzschild, the (anti-)de Sitter, and other well-known solutions but also many exact ones whose detailed global properties to our knowledge have not been discussed before. They have a natural physical interpretation describing single or several wormholes, domain walls of curvature singularities, cosmic strings, cosmic strings surrounded by domain walls, solutions with closed timelike curves, etc. Academic Press