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2 - Manifolds

Published online by Cambridge University Press:  31 January 2011

Amnon Neeman
Affiliation:
Australian National University, Canberra
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Summary

This chapter is intended as a gentle introduction to Chapter 3. In Chapter 3 we will define schemes; by way of preparation, before we begin the technicalities, it might be helpful to take a close look at a related concept that could already be somewhat familiar, that of a manifold. The reader might be outraged, and complain that differential geometry was not listed among the prerequisites for this book. How dare I presume that the reader will find manifolds familiar?

My answer is twofold: first, this chapter was written in such a way that it should be readable, even by the reader who has never before met a manifold. And second, we do only a very minimal amount of differential geometry. In this chapter we will not go beyond the definition of a manifold, and the definition of Ck–functions on Ck–manifolds. Even a reader without much background in differential geometry might have seen this much.

The idea of a scheme, which will occupy us from Chapter 3 on, mimics that of a manifold; but to make the parallel transparent it helps to start with the right definition of a manifold. With the right definitions the formalisms really are precisely the same, not just similar. In this chapter we treat manifolds. We will start with the traditional definition of a manifold, then modify it slightly.

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Publisher: Cambridge University Press
Print publication year: 2007

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  • Manifolds
  • Amnon Neeman, Australian National University, Canberra
  • Book: Algebraic and Analytic Geometry
  • Online publication: 31 January 2011
  • Chapter DOI: https://doi.org/10.1017/CBO9780511800443.003
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  • Manifolds
  • Amnon Neeman, Australian National University, Canberra
  • Book: Algebraic and Analytic Geometry
  • Online publication: 31 January 2011
  • Chapter DOI: https://doi.org/10.1017/CBO9780511800443.003
Available formats
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Save book to Google Drive

To save content items to your account, please confirm that you agree to abide by our usage policies. If this is the first time you use this feature, you will be asked to authorise Cambridge Core to connect with your account. Find out more about saving content to Google Drive.

  • Manifolds
  • Amnon Neeman, Australian National University, Canberra
  • Book: Algebraic and Analytic Geometry
  • Online publication: 31 January 2011
  • Chapter DOI: https://doi.org/10.1017/CBO9780511800443.003
Available formats
×