Let \(M\) be a \(n\)-manifold. A smooth manifold is a pair \( (M, \mathcal{A}) \), where \( M \) is a topological manifold and 𝒜 is a smooth structure on \( M \). If it is clear out of context, we do not write \(\mathcal{A}\) explicitly.
We call \(M\) a \(C^k\)- or \(C^{k,\alpha}\)-manifold, for \(k\in \mathbb{N}\) and \(0<\alpha\le 1\), if it has a \(C^k\)- or \(C^{k,\alpha}\)-structure, respectively, and it is real analytic if there exists a \(C^\omega\)-atlas on \(M\).
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Structures on a Smooth Manifold Link to heading
- smooth coordinates
- smooth structure
- tangent space
- smooth boundary
- smooth maps on a manifold
- Riemannian metric