World manifold

In gravitation theory, a world manifold endowed with some Lorentzian pseudo-Riemannian metric and an associated space-time structure is a space-time. Gravitation theory is formulated as classical field theory on natural bundles over a world manifold. A world manifold is a four-dimensional orientable real smooth manifold. It is assumed to be a Hausdorff and second countable topological space. Consequently, it is a locally compact space which is a union of a countable number of compact subsets, a separable space, a paracompact and completely regular space. Being paracompact, a world manifold admits a partition of unity by smooth functions. Paracompactness is an essential characteristic of a world manifold. It is necessary and sufficient in order that a world manifold admits a Riemannian metric and necessary for the existence of a pseudo-Riemannian metric. A world manifold is assumed to be connected and, consequently, it is arcwise connected. The tangent bundle of a world manifold and the associated principal frame bundle of linear tangent frames in possess a general linear group structure group . A world manifold is said to be parallelizable if the tangent bundle and, accordingly, the frame bundle are trivial, i.e., there exists a global section (a frame field) of . It is essential that the tangent and associated bundles over a world manifold admit a bundle atlas of finite number of trivialization charts. Tangent and frame bundles over a world manifold are natural bundles characterized by general covariant transformations. These transformations are gauge symmetries of gravitation theory on a world manifold. By virtue of the well-known theorem on structure group reduction, a structure group of a frame bundle over a world manifold is always reducible to its maximal compact subgroup . The corresponding global section of the quotient bundle is a Riemannian metric on . Thus, a world manifold always admits a Riemannian metric which makes a metric topological space. In accordance with the geometric Equivalence Principle, a world manifold possesses a Lorentzian structure, i.