452: Chart on C∞ Manifold

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definition of chart on $C^{\mathrm{\infty }}$ manifold

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Topics

About: $C^{\mathrm{\infty }}$ manifold

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The table of contents of this article

• Starting Context
• Target Context
• Orientation
• Main Body
• 1: Structured Description
• 2: Natural Language Description

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Starting Context

• The reader knows a definition of $C^{\mathrm{\infty }}$ manifold.
• The reader knows a definition of chart on topological manifold.

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Target Context

• The reader will have a definition of chart on $C^{\mathrm{\infty }}$ manifold.

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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.

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Main Body

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1: Structured Description

Here is the rules of Structured Description.

Entities:
$M$: $\in \{\text{ the }{C}^{\mathrm{\infty }}\text{ manifolds }\}$
$\ast (U\subseteq M,\varphi )$: $\in \{\text{ the charts on the topological manifold, }M\}$
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Conditions:
$(U\subseteq M,\varphi )$: $\in \text{ the maximal atlas of }M$
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2: Natural Language Description

For any $C^{\mathrm{\infty }}$ manifold, $M$, any chart on the topological manifold that (the chart) is an element of the maximal atlas, $(U\subseteq M,\varphi )$

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References

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