definition of canonical \(C^\infty\) atlas for finite-dimensional real vectors space
Topics
About: vectors space
About: \(C^\infty\) manifold
The table of contents of this article
- Starting Context
- Target Context
- Orientation
- Main Body
- 1: Structured Description
- 2: Natural Language Description
- 3: Note
Starting Context
- The reader knows a definition of %field name% vectors space.
- The reader knows a definition of canonical topology for finite-dimensional real vectors space.
- The reader knows a definition of \(C^\infty\) manifold.
Target Context
- The reader will have a definition of canonical \(C^\infty\) atlas for finite-dimensional real vectors space.
Orientation
There is a list of definitions discussed so far in this site.
There is a list of propositions discussed so far in this site.
Main Body
1: Structured Description
Here is the rules of Structured Description.
Entities:
\( V\): \(\in \{\text{ the }d\text{ -dimensional real vectors spaces }\}\)
\( \{b_1, ..., b_d\}\): \(\subseteq V\), \(\in \{\text{ the bases of } V\}\)
\( \mathbb{R}^d\): \(= \text{ the Euclidean topological space }\)
\( f\): \(: V \to \mathbb{R}^d\), \(v \mapsto (v^1, ..., v^d)\) such that \(v = v^j b_j\)
\( O\): \(= \{U \subseteq V \vert f (U) \in \text{ the topology of } \mathbb{R}^d\}\)
\( (V, f)\): \(\in \{\text{ the charts for } V\}\)
\(*A\): \(= \text{ the maximal }C^\infty \text{ atlas compatible with } (V, f)\)
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Conditions:
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\(A\) does not depend on the choice of \(\{b_1, ..., b_d\}\), because with another \(\{b'_1, ..., b'_d\}\) and the corresponding \(f': V \to \mathbb{R}^d\), the charts transition maps, \(f \circ f'^{-1}: \mathbb{R}^d \to \mathbb{R}^d\) and \(f' \circ f^{-1}: \mathbb{R}^d \to \mathbb{R}^d\) are \(C^\infty\): \(f \circ f'^{-1}\) is linear with a non-singular matrix and \(f' \circ f^{-1}\) is its inverse.
2: Natural Language Description
For any \(d\)-dimensional real vectors space, \(V\), any basis, \(\{b_1, ..., b_d\} \subseteq V\), the Euclidean topological space, \(\mathbb{R}^d\), the map, \(f: V \to \mathbb{R}^d\), \(v \mapsto (v^1, ..., v^d)\) such that \(v = v^j b_j\), and the chart, \((V, f)\), the maximal \(C^\infty\) atlas compatible with \((V, f)\)
3: Note
As \(V\) with \(O\) is obviously a Hausdorff, 2nd-countable, locally topological Euclidean topological space, \(V\) with \(O\) and \(A\) is a \(C^\infty\) manifold.
