On the topology of spherically symmetric space-times

Spherically symmetric space-times have attained considerable attention ever since the early beginnings of the theory of general relativity. In fact, they have appeared already in the papers of K. Schwarzschild 12] and W. De Sitter 5] which were published in 1916 and 1917 respectively soon after Einstein's epoch-making work 7] in 1915. The present survey is concerned mainly with recent results pertainig to the toplogy of spherically symmetric space-times. Definition. By space-time a connected time-oriented 4-dimensional Lorentz manifold is meant. If (M,<,>) is a space-time, and Φ: SO(3)×M→M an isometric action such that the maximal dimension of its orbits is equal to 2, then the action Φ is said to be spherical and the space-time is said to be spherically symmetric 8]; 11]. Likewise, isometric actions Ψ: O(3)×M→M are also considered (10], p. 365; 4]) which will be called quasi-spherical if the maximal dimension of its orbits is 2 and then the space-time is said to be quasi-spherically symmetric here. Each quasi-spherical action yields a spherical one by restricting it to the action of SO(3); the converse of this statement will be considered elsewhere. The main results concerning spherically symmetric space-times are generally either of local character or pertaining to topologically restricted simple situations 14], and earlier results of global character are scarce 1], 4], 6], 13]. A report on recent results concerning the global geometry of spherically symmetric space-times 16] is presented below.