Abstract
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 by observing that the tangent spaces of a manifold combine in a natural way into what is called the tangent bundle. By taking the properties of the tangent bundle as axioms, we arrive at the notion of a vector bundle. We derive elementary properties of vector bundles, including their description in terms of transition functions, and discuss sections and frames. In order to keep in touch with the physics literature, we present the local coordinate description in some detail. In particular, we discuss transformation properties, this way making contact with classical tensor analysis. Next, we show that, by fibrewise application, the algebraic constructions for vector spaces (like taking the dual, the direct sum or the tensor product) carry over to vector bundles. In the special case of the tangent bundle, this yields the whole variety of tensor bundles. The final two sections are devoted to induced (or pull-back) bundles, subbundles and quotient bundles. As special cases, this includes regular distributions, kernel and image bundles, annihilators, as well as normal and conormal bundles.
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Notes
- 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.
- 2.
Denoted by the same symbol.
- 3.
And with tangent vectors being nowhere parallel to the fibres, but this is not relevant for the argument.
- 4.
For a guide to octonions, see [29].
- 5.
Taking into account that Φ T projects to φ −1.
- 6.
Beware that there exist different conventions concerning the choice of the factor in Formula (2.4.17).
- 7.
In the definition of vector bundle, replace “\(\mathbb{K}\)-vector space” by “\(\mathbb{K}\)-algebra” and “linear mapping” by “algebra homomorphism”.
- 8.
Like for the tangent bundle we will stick to this notation (instead of writing (T∗ M) m ).
- 9.
is called a Euclidean vector bundle if \(\mathbb {K}=\mathbb{R}\) and a Hermitian vector bundle if \(\mathbb {K}=\mathbb{C}\). - 10.
Compactness of M is necessary here, see Example 3.6 in [125].
- 11.
Such local frames exist by Proposition 2.7.5.
- 12.
By construction, \(\tilde{X}\) is also the restriction of X in domain to the submanifold (M,φ) and in range to the subbundle (TM,φ′). This does not help for the argument though, because the latter need not be embedded, so that Proposition 1.6.10 does not apply here.
is called a Euclidean vector bundle if References
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Agricola, I., Friedrich, T.: Globale Analysis. Vieweg, Wiesbaden (2001), in German
Baez, J.: The octonions. Bull. Am. Math. Soc. 39, 145–205 (2001)
Hatcher, A.: Vector Bundles & K-Theory (2009). Available at Allen Hatcher’s homepage, http://www.math.cornell.edu/~hatcher/
Hirsch, M.W.: Differential Topology. Springer, Berlin (1976)
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Rudolph, G., Schmidt, M. (2013). Vector Bundles. In: Differential Geometry and Mathematical Physics. Theoretical and Mathematical Physics. Springer, Dordrecht. https://doi.org/10.1007/978-94-007-5345-7_2
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