The book explores the profound impact of small domains on the entirety of Riemann surfaces and analytic manifolds, emphasizing Riemann's insights into meromorphic functions and their singularities. It discusses the ongoing relevance of Euclidean space, challenging the notion that it is the only "real" space, while acknowledging its comforting familiarity. The text highlights Riemann space as a unique geometrical framework that is locally Euclidean, making it significant in both mathematics and physics, particularly in applications like mechanics involving submanifolds.
A very small domain influences the entire Riemann surface or analytic manifold through analytic continuation. Riemann excelled in applying this principle and was the first to highlight that a meromorphic function is determined by its singularities, earning him recognition as the father of the rapidly evolving theory of singularities, which holds immense significance, particularly in physics. The role of Euclidean space remains a fascinating and complex topic. Many philosophers, following Kant, once believed that 'real space' is Euclidean, dismissing other spaces as mere abstract constructs. However, this view is outdated, as modern physics contradicts it. Nonetheless, there is some truth to the notion that three-dimensional Euclidean space (E) feels special and familiar, providing a sense of confidence and safety compared to non-Euclidean spaces. This may explain why Riemann space (M) is prominent among various geometries; it is locally Euclidean, meaning M is a differentiable manifold with tangent spaces equipped with a Euclidean metric. Additionally, every submanifold of Euclidean space inherits a natural Riemann metric, which is frequently utilized in mechanics, such as in the case of the spherical pendulum.
