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. It should be emphasized that 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 ${\displaystyle TX}$ of a world manifold ${\displaystyle X}$ and the associated principal frame bundle ${\displaystyle FX}$ of linear tangent frames in ${\displaystyle TX}$ possess a general linear group structure group ${\displaystyle GL^{+}(4,\mathbb {R} )}$. A world manifold ${\displaystyle X}$ is said to be parallelizable if the tangent bundle ${\displaystyle TX}$ and, accordingly, the frame bundle ${\displaystyle FX}$ are trivial, i.e., there exists a global section (a frame field) of ${\displaystyle FX}$. It is essential that the tangent and associated bundles over a world manifold admit a bundle atlas of finite number of trivialization charts.

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