notes/docs/mathematics/differential-geometry/differential-manifolds.md

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.