Differentiable Manifolds
In this first chapter of the purely mathematical part on the most important tools of differential geometry needed for GR, we introduce several basic concepts connected with the notion of a differentiable manifold. We give two definitions of a differentiab
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Differentiable Manifolds
A manifold is a topological space which locally looks like the space Rn with the usual topology. Definition 11.1 An n-dimensional topological manifold M is a topological Hausdorff space with a countable base, which is locally homeomorphic to Rn . This means that for every point p ∈ M there is an open neighborhood U of p and a homeomorphism h : U −→ U which maps U onto an open set U ⊂ Rn . As an aside, we note that a topological manifold M also has the following properties: (a) M is σ -compact; (b) M is paracompact and the number of connected components is at most denumerable. The second of these properties is particularly important for the theory of integration. For a proof, see e.g. [43], Chap. II, Sect. 15. Definition 11.2 If M is a topological manifold and h : U −→ U is a homeomorphism which maps an open subset U ⊂ M onto an open subset U ⊂ Rn , then h is a chart of M and U is called the domain of the chart or local coordinate neighborhood (Fig. 11.1). The coordinates (x 1 , . . . , x n ) of the image h(p) ∈ Rn of a point p ∈ U are called the coordinates of p in the chart. A set of charts {hα |α ∈ I } with domains Uα is called an atlas of M, if α∈I Uα = M. If hα and hβ are two charts, then both define homeomorphisms on the intersection of their domains Uαβ := Uα ∩ Uβ ; one thus obtains a homeomorphism hαβ N. Straumann, General Relativity, Graduate Texts in Physics, 579 DOI 10.1007/978-94-007-5410-2_11, © Springer Science+Business Media Dordrecht 2013
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Differentiable Manifolds
Fig. 11.1 Chart
Fig. 11.2 Change of coordinates
between two open sets in Rn (Fig. 11.2) via the commutative diagram:
Thus hαβ = hβ ◦ h−1 α on the domain where the mapping is defined. This mapping gives a relation between the coordinates in the two charts and is called a change of coordinates or coordinate transformation (see Fig. 11.2). Sometimes, particularly in the case of charts, it is useful to include the domain of a mapping in the notation; thus, we write (h, U ) for the mapping h : U −→ U . Definition 11.3 An atlas defined on a manifold is said to be differentiable if all of its coordinate changes are differentiable mappings. For simplicity, unless otherwise stated, we shall always mean differentiable mappings of class C ∞ on Rn (the derivatives of all orders exist and are continuous). Obviously, for all coordinate transformations one has (on the domains for which the mappings are defined) hαα = Id and hβγ ◦ hαβ = hαγ , so that h−1 αβ = hβα , and hence the inverses of the coordinate transformations are also differentiable. They are thus diffeomorphisms.
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Differentiable Manifolds
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If A is a differentiable atlas defined on manifold M, then D(A) denotes the atlas that contains exactly those charts for which each coordinate change with every chart of A is differentiable. The atlas D(A) is then also differentiable since, locally, a coordinate change hβγ in D(A) can be written as a composition hβγ = hαγ ◦ hβα of two other coordinate changes with a chart hα ∈ A, and differentiabi
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