Definition: Differentiable Manifold, Atlas

Let \(n, k \in \mathbb {N} \) be natural numbers with \(k\ge 1\). A topological Hausdorff space \(M\), together with an open cover \(M=\bigcup _{i\in I}U_{i}\) and the charts. \[\alpha _{i}\colon U_{i}\longrightarrow V_{i}\,\]

with \(V_{i}\subseteq \mathbb {R} ^{n}\) being open subsets of the \(n\)-dimensional metric space of real numbers such that the transition maps. \[\alpha _{j}\circ (\alpha _{i})^{-1}\colon V_{i}\cap \alpha _{i}(U_{i}\cap U_{j})\longrightarrow V_{j}\cap \alpha _{j}(U_{i}\cap U_{j})\,\]

are \(C^{k}\)-diffeomorphisms for all \(i,j\in I\), is called a \(C^{k}\)-manifold or \(k\) times differentiable manifold of the dimension \(n\).

The set of the charts \((U_{i},\alpha _{i})\), \(i\in I\), is called the \(C^{k}\)-atlas of the manifold.

Definitions: 1 2 3 4

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Adapted from CC BY-SA 3.0 Sources:

  1. Brenner, Prof. Dr. rer. nat., Holger: Various courses at the University of Osnabrück