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Charts, atlases, smooth structures, and smooth manifolds.
A reference for this material is Chapter 1 of John M. Lee. Introduction to smooth
manifolds. Second edition. Grad. Texts in Math., 218. Springer, New York, 2013.
xvi+708pp. ISBN: 978-1-4419-9981-8.
Topological manifold Let \(n \in \mathbb {N}\). An \(n\)-dimensional topological manifold is a second-countable
Hausdorff topological space \(M\) such that each point \(p \in M\) has an open neighborhood
homeomorphic to an open set in \(\mathbb {R}^n\).
Charts and parametrizations Let \(M\) be an \(n\)-dimensional topological manifold, \(U \subset M\) an open
set, and \(\varphi : U \to V\) a homeomorphism with \(V \subset \mathbb {R}^n\) open. The pair \((U, \varphi )\) is called a chart and the pair \((V ,\varphi ^{-1} )\) is
called a parametrization. By an abuse of notation, we often call \(\varphi \) a chart.
Compatible charts Let \(M\) be a topological manifold. We say two charts \((U , \varphi )\), \((V, \psi )\) are compatible
if the map
\[ \psi \circ \varphi ^{-1} : \varphi (U \cap V) \to \psi (U \cap V) \]
is a diffeomorphism.
Being compatible is an equivalence relation in the set of charts.
Homework.
Smooth atlas Let \(M\) be a topological manifold. A smooth atlas is a collection of
compatible charts \(\mathcal {A} = \{ (U_i , \varphi _i ) \} _{i \in I } \) with
\[ M \subset \bigcup _{i \in I} U_i . \]
Smooth structure Let \(M\) be a topological manifold. A smooth structure is a maximal
smooth atlas \(\mathcal {A}\). In this definition, maximal means that any smooth atlas containing \(\mathcal {A}\)
equals \(\mathcal {A}\).
Any smooth atlas \(\mathcal {A}\) is contained in a unique smooth structure \(\mathcal {S}\). Moreover, \(\mathcal {S}\) consists
precisely of the charts compatible with all charts in \(\mathcal {A}\).
Homework.
Smooth manifold A smooth manifold is a pair \((M, \mathcal {S})\) with \(M\) a topological manifold and \(\mathcal {S}\) a
smooth structure on \(M\).
Smooth function to \(\mathbb {R}^m\) Let \((M, \mathcal {S})\) be a smooth manifold and \(f : M \to \mathbb {R}^m\) a function. We say \(f\) is smooth if
for any chart \((U , \varphi )\) in \( \mathcal {S}\), the composition
\[ f \circ \varphi ^{-1} : \varphi (U ) \to \mathbb {R}^m \]
is smooth. We denote by \(C^{\infty }(M)\) the set of smooth
functions \(f : M \to \mathbb {R}\).
Smoothness is local Let \((M, \mathcal {S})\) be a smooth manifold, \(\mathcal {A} \subset \mathcal {S}\) a smooth atlas, and \(f : M \to \mathbb {R}^m\) a function.
Assume that for any chart \((U, \varphi )\) in \(\mathcal {A}\), the composition
\[ f \circ \varphi ^{-1} : \varphi (U ) \to \mathbb {R}^m \]
is smooth. Then \(f\) is smooth.
Exercise. Use that the change of coordinates is smooth the chain rule.
Given a smooth manifold \((M, \mathcal {S})\), by an abuse of noation, whenever we say “chart” we mean
an element of \(\mathcal {S}\).
While a smooth manifold is technically a pair \((M , \mathcal {S})\), it is often simply denoted by \(M\).
Smooth manifolds