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2015 | OriginalPaper | Chapter

5. What Is Space?

Author : Jürgen Jost

Published in: Mathematical Concepts

Publisher: Springer International Publishing

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Abstract

How can the notion of space be conceptualized and formalized? What is the space that we are living in? In this chapter, we discuss the concepts of a manifold (including an introduction to Riemannian geometry), a simplicial complex and a scheme as possible answers.

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previous chapter Spaces

next chapter Spaces of Relations

Let us recall the definition of a holomorphic map. On \({\mathbb C}^d\), we use complex coordinates \(z^1=x^1+ iy^1,\dots ,z^d=x^d + iy^d\). A complex function \(h:U \rightarrow {\mathbb C}\) where U is an open subset of \({\mathbb C}^d\), is called holomorphic if \(\frac{\bar{\partial } h}{\partial \bar{z}^{k}}:=\frac{1}{2}(\frac{{\partial } h}{\partial x^{k}} +i \frac{{\partial } h}{\partial y^{k}})=0\) for \(k=1,\dots ,d\), and a map \(H:U \rightarrow {\mathbb C}^m\) is then holomorphic if all its components are. We do not want to go into the details of complex analysis here, and we refer the reader to [58] for more details about complex manifolds.

The epithet “continuous” is used here only for emphasis, as all curves are implicitly assumed to be continuous.

This theorem says that every homogeneous quadratic polynomial is reducible by real orthogonal substitutions to the form of a sum of positive and negative squares. See [75], p.577.

For the present discussion, other rings would do as well, but we stick to \({\mathbb R}\) here for concreteness. \({\mathbb C}\) will become important below.

In complex and algebraic geometry, there is a way to circumvent this problem, namely, to look at meromorphic functions, that is, also for functions assuming the value \(\infty \) in a controlled manner.

Title: What Is Space?
Author: Jürgen Jost
Publisher: Springer International Publishing
Book: Mathematical Concepts
Print ISBN: 978-3-319-20435-2

Electronic ISBN: 978-3-319-20436-9

Copyright Year: 2015
DOI: https://doi.org/10.1007/978-3-319-20436-9_5

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