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Vector Bundles

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Topology and Condensed Matter Physics

Part of the book series: Texts and Readings in Physical Sciences ((TRiPS,volume 19))

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Abstract

The aim in the first part of this chapter is to understand the basics of Vector bundles and K-theory. We will define vector bundles and give several examples of vector bundles that arise naturally in geometry. We will give constructions of important natural operations on vector bundles, and we will show how deformation of spaces controls the structure of vector bundles. In section 6.5 we will define K-theory, an important abelian invariant of a space. We will show that this invariant can be used to answer non-trivial questions about the geometry of a space.

The second part gives a brief overview of the Chern-Weil theory in the context of vector bundles. The Chern-Weil theory is a vast topic which has been studied from various aspects. In this short note we take the differential geometric approach. We start with smooth manifolds and affine connections and generalize the notion of connections and curvature to vector bundles with an aim to produce global invariants of vector bundles in terms of characteristic classes. We conclude with a few simple examples.

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References

  • [1] A. Hatcher, Vector Bundles and K-Theory (Cambridge University Press, New York, 2002).

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  • [2] S. Gallot, D. Hulin and J. Lafontaine, Riemannian Geometry, Third edition, (Springer-Verlag, Berlin, 2004).

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  • [3] J. W. Milnor and J. D. Stasheff, Characteristic Classes, (Princeton University Press, 1976).

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  • [5] S. Morita, Geometry of Differential Forms (American Mathematical Society, Providence, RI, 2001).

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  • [6] S. Morita, Geometry of Characteristic Classes (American Mathematical Society, Providence, RI, 2001).

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Correspondence to Utsav Choudhury .

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© 2017 Springer Nature Singapore Pte Ltd. and Hindustan Book Agency

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Choudhury, U., Bhattacharya, A. (2017). Vector Bundles. In: Bhattacharjee, S., Mj, M., Bandyopadhyay, A. (eds) Topology and Condensed Matter Physics. Texts and Readings in Physical Sciences, vol 19. Springer, Singapore. https://doi.org/10.1007/978-981-10-6841-6_6

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