662: C∞ Immersion

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definition of $C^{\mathrm{\infty }}$ immersion

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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 $C^k$ map between arbitrary subsets of $C^{\mathrm{\infty }}$ manifolds with boundary, where $k$ includes $\mathrm{\infty }$.
• The reader knows a definition of differential of $C^{\mathrm{\infty }}$ map between $C^{\mathrm{\infty }}$ manifolds with boundary at point.
• The reader knows a definition of injection.

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Target Context

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

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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:
$M_1$: $\in \{\text{ the }{C}^{\mathrm{\infty }}\text{ manifolds with boundary }\}$
$M_2$: $\in \{\text{ the }{C}^{\mathrm{\infty }}\text{ manifolds with boundary }\}$
$\ast f$: $:M_1\to M_2$, $\in \{\text{ the }{C}^{\mathrm{\infty }}\text{ maps }\}$
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Conditions:
$\mathrm{\forall }p\in M_1(d{f}_p:Tp{M}_1\to Tp{M}_2\in \{\text{ the injections }\})$
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2: Natural Language Description

For any $C^{\mathrm{\infty }}$ manifolds with boundary, $M_1,M_2$, any $C^{\mathrm{\infty }}$ map, $f:M_1\to M_2$, such that its differential, $d{f}_p:Tp{M}_1\to Tp{M}_2$, at any point, $p\in M_1$, is an injection

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References

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