2.1. Manifolds¶

2.1.1. Charts¶

A manifold \(\mathcal{M}\) is a locally flat (Euclidean) open subspace of \(\mathbb{R}^n\), which is covered by a collection (union) of open subsets \(\mathscr{U}\), sometimes referred to as neighbourhoods.

Definition: Chart

A chart \(\left( \mathscr{U}, \phi_\alpha \right)\) on a manifold \(\mathcal{M}\) is defined by the open subset \(\mathscr{U} \subset \mathcal{M}\) and the associated homeomorphism

\[\begin{split}\phi_\alpha : \mathscr{U} &\rightarrow \mathbb{R}^n, \\ x &\mapsto \left( x^1, ..., x^n \right),\end{split}\]

where \(x^i\) are the coordinates of \(\mathscr{U}\).

For two intersecting charts, \(\left( \mathscr{U}, \phi_\alpha \right)\) and \(\left( \mathscr{V}, \psi_\alpha \right)\), the composition

\[\begin{split}\phi_\alpha \circ \psi_\alpha^{-1} : \mathbb{R}^n &\rightarrow \mathbb{R}^n, \\ \psi_\alpha (x) & \mapsto \phi_\alpha (x)\end{split}\]

takes points defined in the coordinates of \(\mathscr{V}\) to those of \(\mathscr{U}\), i.e. from \(\left( \tilde{x}^1, ..., \tilde{x}^n \right) \mapsto \left(x^1, ..., x^n \right)\).

In the case of points lying in an intersection of open subsets, \(x \in \mathscr{U} \cap \mathscr{V}\), then, from the definition of the manifold, it is required that \(x^i\) are smooth functions of \(\tilde{x}^i\).

Definition: Cartesian product of manifolds

Two manifolds, \(\mathcal{M} \subset \mathbb{R}^n\) and \(\mathcal{N} \subset \mathbb{R}^k\), admit a cartesian product defined by the set of pairs of points

\[\mathcal{M} \times \mathcal{N} = \left\{ (x, y) \ : \ x \in \mathcal{M}, y \in \mathcal{N} \right\}.\]

As a corollary, if \(\left( \mathscr{U}, \phi_\alpha \right)\) and \(\left( \mathscr{V}, \psi_\alpha \right)\) are distinct charts of \(\mathcal{M}\) and \(\mathcal{N}\) respectively, and

\[\begin{split}\phi_\alpha(x) &= \left(x^1, ..., x^n \right), \\ \psi_\alpha(y) &= \left(y^1, ..., y^m \right),\end{split}\]

then the map,

\[\left( \phi_\alpha \times \psi_\alpha \right) (x, y) = \left( x^1, ..., x^n, y^1, ..., y^m \right),\]

is sufficient to define \(\mathcal{M} \times \mathcal{N}\) as a manifold of \(n+k\) dimensions.

2.1.2. Functions on manifolds¶

Definition: Function on a manifold

A function \(f\) on an open subset \(\mathscr{U}\) of a manifold \(\mathcal{M}\) is defined as a map

\[f: \mathscr{U} \rightarrow \mathbb{R}.\]

Functions on manifolds may be composed with the homeomorphisms of charts as \(f \circ \phi_\alpha^-1\), taking \(\mathbb{R}^n\) into \(\mathbb{R}\); such compositions are smooth functions of the coordinates \(x^i\).

We are also permitted to express a smooth curve over an interval \(Q : T \rightarrow \mathscr{U_\alpha}\), \(T \subset \mathbb{R}\), parameterized by a real variable \(\tau \in \left[ a, b \right]\):

We write

\[\left( \phi_\alpha \circ Q \right) (\tau) = \left( x^1(\tau), ..., x^n (\tau) \right),\]

requiring \(x^i(\tau)\) to be smooth functions of \(\tau\).

We may also consider the composition \(f \circ Q\), along with \(\phi_\alpha \circ Q\), to describe the action of the function on \(\tau\):

\[\begin{split}\left( f \circ Q \right)(t) =& f(Q(t)) \\ =& f\left( x^1(\tau), ..., x^n (\tau) \right)\end{split}\]

where \(p\) is the point imaged by \(Q(\tau)\) in the open subset \(\mathscr{U}\).

Mathematics Reference

2.1. Manifolds¶

2.1.1. Charts¶

2.1.2. Functions on manifolds¶