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Vector Bundles

Vector bundles constitute a special class of manifolds, which is of great importance in physics. In particular, all sorts of tensor fields occurring in physical models may be viewed in a coordinate-free manner as sections of certain vector bundles. We start with the observation that the tangent spaces of a manifold combine in a natural way into a bundle, which is called tangent bundle. Next, by taking its typical properties as axioms, we arrive at the general notion of vector bundle. In Sect. 2.2, we discuss elementary aspects of this notion, including the proof that—up to isomorphy—vector bundles are completely determined by families of transition functions. In Sect. 2.3 we discuss sections and frames,1 and in Sect. 2.4 we present the tool kit for vector bundle operations. We will see that, given some vector bundles over the same base manifold, by applying fibrewise the standard algebraic operations of taking the dual vector space, of building the direct sum and of taking the tensor product, we obtain a universal construction recipe for building new vector bundles. In Sect. 2.5, by applying these operations to the tangent bundle of a manifold, we get the whole variety of tensor bundles over this manifold. The remaining two sections contain further operations, which will be frequently used in this book. In Sect. 2.6, we discuss the notion of induced bundle and Sect. 2.7 is devoted to subbundles and quotient bundles. There is a variety of special cases occurring in applications: regular distributions, kernel and image bundles, annihilators, normal and conormal bundles.

2.1 The Tangent Bundle Let M be a C k -manifold, let I ⊂ R be an open interval and let γ : I → M be a C k -curve. According to Example 1.5.6, for every t ∈ I , the tangent vector γ˙ (t) of γ at t is an element of the tangent space Tγ (t) M. Hence, while t runs through I , γ˙ (t) runs through the tangent spaces along γ , see Fig. 2.1. 1 Here, as well as in Sect. 2.5, in order to keep in touch with the physics literature, the local description is presented in some detail. In particular, we discuss transformation properties. This way, we make contact with classical tensor analysis.

G. Rudolph, M. Schmidt, Differential Geometry and Mathematical Physics, Theoretical and Mathematical Physics, DOI 10.1007/978-94-007-5345-7_2, © Springer Science+Business Media Dordrecht 2013

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2 Vector Bundles

Fig. 2.1 Tangent vectors along a curve γ in M

To follow the tangent vectors along γ it is convenient to consider the totality of all tangent spaces of M. This leads to the notion of tangent bundle of a manifold M, denoted by TM. As a set, TM is given by the disjoint union of the tangent spaces at all points of M, that is,  TM := Tm M. (2.1.1) m∈M

Let π : TM → M be the canonical projection which assigns to an element of Tm M the point m for every m ∈ M. TM can be equipped with a manifold structure as follows. Denote n = dim M. Choose a countable atlas {(Uα , κα ) : α ∈ A} on M and define the mappings   (2.1.2) καT (Xm ) := κα (m), Xκmα . κ