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Differential manifolds

In the following sections of differential geometry we make use of the Einstein summation convention and \(\mathbb{K} = \mathbb{R}\) or \(\mathbb{K} = \mathbb{C}\).

Definition

Differential geometry is concerned with differential manifolds, smooth continua that are locally Euclidean.

Definition 1: let \(n \in \mathbb{N}\), a \(n\)-dimensional differential manifold is a Hausdorff (T2) space \(M\) furnished with a family of smooth diffeomorphisms \(\phi_\alpha: \mathscr{D}(\phi_\alpha) \to \mathscr{R}(\phi_\alpha)\) with \(\mathscr{D}(\phi_\alpha) \subset\mathrm{M}\) and \(\mathscr{R}(\phi_\alpha) \subset E\), with the following axioms

  1. \(\mathscr{D}(\phi_\alpha)\) is open and \(\bigcup_{\alpha \in \mathbb{N}} \mathscr{D}(\phi_\alpha) =\mathrm{M}\),
  2. if \(\Omega = \mathscr{D}(\phi_\alpha) \cap \mathscr{D}(\phi_\beta) \neq \empty\) then \(\phi_\alpha(\Omega), \phi_\beta(\Omega) \subset E\) are open sets and \(\phi_\alpha \circ \phi_\beta^{-1}, \phi_\beta \circ \phi_\alpha\) are diffeomorphisms,
  3. the atlas \(\mathscr{A} = \{(\mathscr{D}(\phi_\alpha), \phi_\alpha)\}\) is maximal.

with \(E\) a \(n\)-dimensional Euclidean space.

The last axiom ensures that any chart is tacitly assumed to be already contained in the atlas.

Coordinate transformations

Definition 2: let \(p,q \in \mathrm{M}\) be points on the differential manifold and let \(\psi: \mathscr{D}(\psi) \to\mathrm{M}: p \mapsto \psi(p) \overset{\text{def}}{=} q\) be a transformation from \(p\) to \(q\) on the manifold, we define two diffeomorphisms

\[ \phi_\alpha: \mathscr{D}(\phi_\alpha) \to \mathscr{R}(\phi_\alpha): p \mapsto \phi_\alpha(p) \overset{\text{def}}{=} x, \]
\[ \phi_\beta: \mathscr{D}(\phi_\beta) \to \mathscr{R}(\phi_\beta): q \mapsto \phi_\beta(q) \overset{\text{def}}{=} y, \]

with \(\mathscr{D}(\phi_{\alpha,\beta}) \subset\mathrm{M}\) and \(\mathscr{R}(\phi_{\alpha,\beta}) \subset E\). Then we have a coordinate transformation given by

\[ \phi_{\alpha \beta}^\psi = \phi_\beta \circ \psi \circ \phi_\alpha^{-1}: x \mapsto y, \]

then \(\phi_{\alpha \beta}^\psi\) is an active transformation if \(p \neq q\) and \(\phi_{\alpha \beta}^\psi\) is a passive transformation if \(p = q\).

To clarify the definitions, a passive transformation corresponds only to a descriptive transformation. Whereas an active transformation corresponds to a transformation on the manifold \(M\).

A passive transformation may also be given directly by \(\phi_\beta \circ \phi_\alpha: x \mapsto y\) since \(\psi = \mathrm{id}\) in this case. Note that the definitions could also have been given by the inverse as the transformations are all diffeomorphisms.