It is unusual to hypothesize that the four-dimensional space-time universe of general relativity is compact (i.e., “finite”). But in such a case several interesting conclusions can be drawn. In the first place, if the mass distribution is assumed to be continuous, so that the metric tensor has no singularities, then the Euler-Poincaré characteristic
In the second place, it seems to be generally known that in a finite cosmology there must exist a closed curve in space-time whose tangent vector at every point is time-like. Professor L. Markus has indicated a proof to us. Let
4-manifold of the universe. Now, on
construct a continuous, nowhere vanishing field of time-like vectors (, pp. 6-7; cf., p. 207). By Birkhoff’s fundamental theorem
A more standard hypothesis, however, is that the universe is not compact, but is the topological product of the infinite real line (a time axis) with a 3-manifold . The manifold is often assumed to be compact, and any local (hence experimentally verifiable) condition which implies compactness is of much interest. For example, if has constant curvature then is compact if, and only if, is positive (, pp. 84 and 203), and in this case is a 3-sphere if its first Betti number vanishes, and in general admits the 3-sphere as a covering space.
We wish to point out a new method for studying the topology of manifolds such as and . This method consists of the construction of a continuous, nowhere vanishing, irrotational vector field on the manifold under consideration. Once such vector field has been constructed, we can assert that either the manifold is non-compact (i.e., open or “infinite”), or that it cannot be simply-connected.
We shall prove a slight generalization of this theorem; but first, let us note that a similar, but more restrictive and less easily applicable condition is a trivial consequence of Hodge’s well-known theorem
• Theorem 1 (Hodge):: Let be an -dimensional Riemannian manifold (with positive definite metric tensor), and let denote a non-trivial class vector field defined on . Suppose that the curl and the divergence of both vanish identically; or equivalently, suppose that the field satisfies the generalized Laplace equation for harmonic vector fields. Then, if is compact, its first Betti number is not zero.
• Corollary (Bochner-Myers):: If is orientable and has positive definite Ricci curvature throughout, then its first Betti number vanishes. (, p. 37).
Recall that the curl tensor of a vector field is independent of the metric tensor, and so is a non-metric notion. Accordingly, the following theorem applies equally well to with its indefinite hyperbolic metric as to with its positive definite Riemannian metric.
• Theorem 2:: Let be an -dimensional differentiable manifold, and let be a continuous, class vector field defined on . Suppose that F vanishes at most once and that its curl vanishes identically on . Then either is non-compact, or is compact and its first Betti number does not vanish. In either case, of course, if actually vanishes nowhere, the Euler-Poincaré characteristic of is zero.
this theorem is a consequence of a more general theorem  which applies, for example, to manifolds with boundary. In fact, by a generalization to arbitrary flows of a theorem proved by Lichnerowicz
It is possible that Theorems 1 and 2 have applications to the study of specific cosmological models. In fact, there are many ways of constructing on , or on continuous vector fields which are unique once the indefinite metric (or set of gravitational potentials) for has been specified.
Professor J.A. Wheeler
• Theorem 3:: Consider the combined Einstein-Maxwell field theory on . If the vector field
is defined everywhere and of class on , and if does not vanish more than once, then the universe cannot be compact.
The vector field
, which was defined by Dr. C. W. Misner,