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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Notes
- 1.
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.
- 2.
The epithet “continuous” is used here only for emphasis, as all curves are implicitly assumed to be continuous.
- 3.
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.
- 4.
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.
- 5.
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.
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Jost, J. (2015). What Is Space?. In: Mathematical Concepts. Springer, Cham. https://doi.org/10.1007/978-3-319-20436-9_5
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DOI: https://doi.org/10.1007/978-3-319-20436-9_5
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