Let \(M\) denote a manifold . Two smooth atlases \( \mathcal{A}_1 \) and \( \mathcal{A}_2 \) determine the same smooth structure on \( M \) if and only if \( \mathcal{A}_1 \cup \mathcal{A}_2 \) is a smooth atlas on \( M \) [1, Proposition 1.17].
- This Proposition justifies another way to determine a smooth structure on \(M\) by defining an equivalence relation for smooth atlases.
If \( \mathcal{A}_1 \) and \( \mathcal{A}_2 \) determine the same smooth structure \( \mathcal{A} \), then \( \mathcal{A}_1 \cup \mathcal{A}_2 \subseteq \mathcal{A} \), and thus \( \mathcal{A}_1 \cup \mathcal{A}_2 \) is a smooth atlas.
On the other hand, let \( \bar{\mathcal{A}}_1 \) and \( \bar{\mathcal{A}}_2 \) denote the smooth structures determined by \( \mathcal{A}_1 \) and \( \mathcal{A}_2 \), respectively. Since \( \mathcal{A}_1 \cup \mathcal{A}_2 \) is a smooth atlas, every chart in \( \mathcal{A}_2 \) is compatible with every chart in \( \mathcal{A}_1 \). Thus, \( \mathcal{A}_2 \subseteq \bar{\mathcal{A}}_1 \). Then \( \bar{\mathcal{A}}_1 \) is a maximal smooth structure containing \( \mathcal{A}_2 \), and by maximality, \( \bar{\mathcal{A}}_1 = \bar{\mathcal{A}}_2 \).
References Link to heading
- J. Lee, Introduction to Smooth Manifolds. New York ; London: Springer, 2013.