Some Unusual Dimensional Reductions of Gravity: Geometric Potentials, Separation of Variables, and Static - Cosmological Duality

We discuss some problems related to dimensional reductions of gravity theories to two-dimensional and one-dimensional dilaton gravity models. We first consider the most general cylindrical reductions of the four-dimensional gravity and derive the corresponding (1+1)-dimensional dilaton gravity, paying a special attention to a possibility of producing nontrivial cosmological potentials from pure geometric variables (so to speak, from `nothing). Then we discuss further reductions of two-dimensional theories to the dimension one by a general procedure of separating the space and time variables. We illustrate this by the example of the spherically reduced gravity coupled to scalar matter. This procedure is more general than the usual `naive reduction and apparently more general than the reductions using group theoretical methods. We also explain in more detail the earlier proposed `static-cosmological duality (SC-duality) and discuss some unusual cosmologies and static states which can be obtained by using the method of separating the space and time variables.