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The Axiomatic Geometric Approach to Bundles

In: Quantum Field Theory III: Gauge Theory

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  • Eberhard Zeidler

    (Max Planck Institute for Mathematics in the Sciences)

Abstract

Our strategy is to define the notion of vector bundles and principal bundles in an invariant way by only using geometric properties of manifolds. In order to prove further geometric properties of these manifolds (e.g., curvature or parallel transport), we use the fact that, by definition, these properties do not depend on the choice of local bundle coordinates. Therefore, we can pass to special bundle coordinates. This is the situation of product bundles considered in Sects. 15.1 through 15.3. This way, the general results are immediate consequences of our special results about product bundles. In this chapter, we tacitly assume that all the objects are smooth, that is, they are described by smooth functions with respect to local coordinates.

Suggested Citation

  • Eberhard Zeidler, 2011. "The Axiomatic Geometric Approach to Bundles," Springer Books, in: Quantum Field Theory III: Gauge Theory, chapter 17, pages 875-903, Springer.
    • Handle: RePEc:spr:sprchp:978-3-642-22421-8_18
    DOI: 10.1007/978-3-642-22421-8_18
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