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Smooth functions, diffeomorphisms, bump functions, and partitions of unity.
A reference for this material is Chapter 2 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.
Smooth function to a manifold Let \(M,\) \(N\) be smooth manifolds and \(f : M \to N\) a continuous function.
We say \(f\) is smooth if for any chart \((U, \varphi )\) of \(M\) and any chart \((V, \psi )\) of \(N\), the composition
Let \(M_1\), \(M_2\), \(M_3\) be smooth manifolds and \(f:M_1 \to M_2 \), \(g : M_2 \to M_3\) smooth functions. Then \(g \circ f : M_1 \to M_3\) is smooth.
Exercise. Use the definition and the chain rule.
Diffeomorphism Let \(M\), \(N\) be smooth manifolds. A smooth homeomorphism \(f : M \to N\) is called a
diffeomorphism if its inverse \(f^{-1} : N \to M\) is smooth. If there is a diffeomorphism \(f : M \to N\), we say that \(M\) is
diffeomorphic to \(N\).
Support Let \(M\) be a topological space and \(f : M \to \mathbb {R}\) a continuous function. The support of \(f\),
denoted by supp\((f)\) is defined as the topological closure of the set
\[ \{ x \in M \, \vert \, f(x) \neq 0 \} . \]
Bump functions Let \(M\) be a smooth manifold, \(U \subset M\) an open set, and \(p\in U\). Then there is \(f \in C^{\infty }(M)\) with
supp\((f) \subset U\) and \(f(p) \equiv 1\) in a neighborhood of \(p\).
Homework.
Let \(M\) be a smooth manifold, \(U \subset M\) an open set, \(p \in U\), and \(f \in \mathbb {C}^{\infty }(U)\). Then there is \(g \in \mathbb {C}^{\infty }(M)\) with \(g \equiv f\) in a
neighborhood of \(p\).
Locally finite Let \(M\) be a topological space and \( \mathcal {X} = \{ X_i \} _{i \in I}\) a collection of subsets of \(M\). We say \(\mathcal {X}\) is
locally finite if each \(p \in M\) admits an open neighborhood \(U \subset M\) such that \(U \cap X_i \neq \emptyset \) only for finitely many
\(i\)’s.
Partitions of unity Let \(M\) be a smooth manifold and \(\{ U_i \} _{i \in I} \) an open cover of \(M\). Then there is a
collection \(\{ \rho _i \} _{i \in I}\) of functions \(\rho _i \in C^{\infty }(M)\) such that
For each \(i \in I\), one has supp\((\rho _i) \subset U_i\).
\(\rho _i (M)\subset [0,1]\)
The collection \(\{ \)supp\((\rho _i)\}_{i \in I}\) is locally finite.
For each \(x \in M\), one has
\[ \sum _{i \in I} \rho _i (x) = 1 . \]
Note that the sum in the last item of Theorem thm:pou makes sense since only finitely many
summands are finite.
Smooth manifolds