Tangent Vectors, Vector and Tensor Fields
Three equivalent definitions of the tangent space at a point of a manifold are given. It is important that one is able to switch freely among these definitions. On the basis this notion vector fields are introduced, together with their Lie algebra structu
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Tangent Vectors, Vector and Tensor Fields
At every point p of a differentiable manifold M, one can introduce a linear space, called the tangent space Tp (M). A tensor field is a (smooth) map which assigns to each point p ∈ M a tensor of a given type on Tp (M).
12.1 The Tangent Space Before defining the tangent space, let us introduce some basic concepts. Consider two differentiable manifolds M and N , and the set of differentiable mappings {φ | φ : U −→ N for a neighborhood U of p ∈ M}. Two such mappings φ, ψ are called equivalent, φ ∼ ψ, if and only if there is a neighborhood V of p such that φ|V = ψ|V . Here φ|V denotes the restriction of φ to the domain V . In other words, φ and ψ are equivalent if they coincide on some neighborhood V of p. Definition 12.1 An equivalence class of this relation is called a germ of smooth mappings M −→ N at the point p ∈ M. A germ with representative φ is denoted by φ¯ : (M, p) −→ N
or
φ¯ : (M, p) −→ (N, q) if q = φ(p).
Compositions of germs are defined naturally via their representatives. A germ of a function is a germ (M, p) −→ R. The set of all germs of functions at a point p ∈ M is denoted by F(p). The set F(p) has the structure of a real algebra, provided the operations are defined using representatives. A differentiable germ φ¯ : (M, p) −→ (N, q) defines, through composition, the following homomorphism of algebras: φ ∗ : F(q) −→ F(p), ¯ f¯ −→ f¯ ◦ φ.
(12.1)
N. Straumann, General Relativity, Graduate Texts in Physics, 585 DOI 10.1007/978-94-007-5410-2_12, © Springer Science+Business Media Dordrecht 2013
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Tangent Vectors, Vector and Tensor Fields
(We omit the bar symbol on φ in φ ∗ .) Obviously Id∗ = Id,
(ψ ◦ φ)∗ = φ ∗ ◦ ψ ∗ .
(12.2)
If φ¯ is a germ having an inverse φ¯ −1 , then φ ∗ ◦ (φ −1 )∗ = Id and φ ∗ is thus an isomorphism. For every point p ∈ M of a n-dimensional differentiable manifold, a chart h having a neighborhood of p as its domain defines an invertible germ h¯ : (M, p) −→ (Rn , 0) and hence an isomorphism h∗ : Fn −→ F(p), where Fn is the set of germs (Rn , 0) −→ R. We now give three equivalent definitions of the tangent space at a point p ∈ M. One should be able to switch freely among these definitions. The “algebraic definition” is particularly handy. Definition 12.2 (Algebraic definition of the tangent space) The tangent space Tp (M) of a differentiable manifold M at a point p is the set of derivations of F(p). A derivation of F(p) is a linear mapping X : F(p) −→ R which satisfies the Leibniz rule (product rule) X(f¯g) ¯ = X(f¯)g(p) ¯ + f¯(p)X(g). ¯
(12.3)
A differentiable germ φ¯ : (M, p) −→ (N, q) (and thus a differentiable mapping φ : M −→ N ) induces an algebra homomorphism φ ∗ : F(q) −→ F(p) and hence also a linear mapping Tp φ¯ : Tp (M) −→ Tq (N ),
(12.4)
X −→ X ◦ φ ∗ .
Definition 12.3 The linear mapping Tp φ¯ is called the differential (or tangent map) of φ¯ at p. The set of derivations obviously forms a vector space. The Leibnitz rule gives X(1) = X(1 · 1) = X(1) + X(1),
i.e. X(1) = 0,
where 1 is the germ defined by the co
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