366: For C∞ Map Between C∞ Manifolds with Boundary, Restriction of Map on Embedded Submanifold with Boundary Domain and Embedded Submanifold with Boundary Codomain Is C∞

<The previous article in this series | The table of contents of this series | The next article in this series>

description/proof of that for $C^{\mathrm{\infty }}$ map between $C^{\mathrm{\infty }}$ manifolds with boundary, restriction of map on embedded submanifold with boundary domain and embedded submanifold with boundary codomain is $C^{\mathrm{\infty }}$

---

Topics

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

---

The table of contents of this article

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

---

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 embedded submanifold with boundary of $C^{\mathrm{\infty }}$ manifold with boundary.
• The reader admits the proposition that for any maps between any arbitrary subsets of any $C^{\mathrm{\infty }}$ manifolds with boundary $C^k$ at corresponding points, where $k$ includes $\mathrm{\infty }$, the composition is $C^k$ at the point.
• The reader admits the proposition that for any map between any arbitrary subsets of any $C^{\mathrm{\infty }}$ manifolds with boundary $C^k$ at any point, where $k$ includes $\mathrm{\infty }$, the restriction or expansion on any codomain that contains the range is $C^k$ at the point.
• The reader admits the proposition that for any $C^{\mathrm{\infty }}$ manifold with boundary and any embedded submanifold with boundary, the inverse of the codomain restricted inclusion is $C^{\mathrm{\infty }}$.

---

Target Context

• The reader will have a description and a proof of the proposition that for any $C^{\mathrm{\infty }}$ map between any $C^{\mathrm{\infty }}$ manifolds with boundary, the restriction of the map on any embedded submanifold with boundary domain and any embedded submanifold with boundary codomain is $C^{\mathrm{\infty }}$.

---

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:
$M_1^{\prime }$: $\in \{\text{ the }{C}^{\mathrm{\infty }}\text{ manifolds with boundary }\}$
$M_2^{\prime }$: $\in \{\text{ the }{C}^{\mathrm{\infty }}\text{ manifolds with boundary }\}$
$M_1$: $\in \{\text{ the embedded submanifolds with boundary of }{M}_1^{\prime }\}$
$M_2$: $\in \{\text{ the embedded submanifolds with boundary of }{M}_2^{\prime }\}$
${\iota }_1$: $:M_1\to M_1^{\prime }$, $=\text{ the inclusion }$
${\iota }_2$: $:M_2\to M_2^{\prime }$, $=\text{ the inclusion }$
$f^{\prime }$: $:M_1^{\prime }\to M_2^{\prime }$, $\in \{\text{ the }{C}^{\mathrm{\infty }}\text{ maps }\}$, such that $f^{\prime }({\iota }_1(M_1))\subseteq {\iota }_2(M_2)$
$f$: $=f^{\prime }{|}_{M_1}:M_1\to M_2$
//

Statements:
$f\in \{\text{ the }{C}^{\mathrm{\infty }}\text{ maps }\}$
//

---

2: Proof

Whole Strategy: Step 1: see that $f^{″}:=f^{\prime }\circ {\iota }_1:M_1\to M_2^{\prime }$ is $C^{\mathrm{\infty }}$; Step 2: take the codomain restriction of $f^{″}$, $f^{‴}:M_1\to f^{″}(M_1)\subseteq M_2^{\prime }$, the codomain restriction of ${\iota }_2$, ${{\iota }_2}^{\prime }:M_2\to {\iota }_2(M_2)\subseteq M_2^{\prime }$, and its inverse, ${{\iota }_2}^{\prime -1}:{\iota }_2(M_2)\to M_2$, and see that $f={{\iota }_2}^{\prime -1}\circ f^{‴}$ and $f$ is $C^{\mathrm{\infty }}$.

Step 1:

${\iota }_1$ is $C^{\mathrm{\infty }}$, by the definition of embedded submanifold with boundary of $C^{\mathrm{\infty }}$ manifold with boundary

$f^{″}:=f^{\prime }\circ {\iota }_1:M_1\to M_2^{\prime }$ is $C^{\mathrm{\infty }}$, by the proposition that for any maps between any arbitrary subsets of any $C^{\mathrm{\infty }}$ manifolds with boundary $C^k$ at corresponding points, where $k$ includes $\mathrm{\infty }$, the composition is $C^k$ at the point.

Step 2:

Let $f^{‴}:M_1\to f^{″}(M_1)\subseteq M_2^{\prime }$ be the codomain restriction of $f^{″}$.

$f^{‴}$ is $C^{\mathrm{\infty }}$, by the proposition that for any map between any arbitrary subsets of any $C^{\mathrm{\infty }}$ manifolds with boundary $C^k$ at any point, where $k$ includes $\mathrm{\infty }$, the restriction or expansion on any codomain that contains the range is $C^k$ at the point.

Let ${{\iota }_2}^{\prime }:M_2\to {\iota }_2(M_2)\subseteq M_2^{\prime }$ be the codomain restriction of ${\iota }_2$.

Let its inverse be ${{\iota }_2}^{\prime -1}:{\iota }_2(M_2)\to M_2$, which is valid and $C^{\mathrm{\infty }}$, by the proposition that for any $C^{\mathrm{\infty }}$ manifold with boundary and any embedded submanifold with boundary, the inverse of the codomain restricted inclusion is $C^{\mathrm{\infty }}$.

$f^{″}(M_1)\subseteq {\iota }_2(M_2)$, because $f^{″}(M_1)=f^{\prime }\circ {\iota }_1(M_1)=f^{\prime }({\iota }_1(M_1))\subseteq {\iota }_2(M_2)$.

So, ${{\iota }_2}^{\prime -1}\circ f^{‴}:M_1\to M_2$ is valid, and $=f$, because ${{\iota }_2}^{\prime -1}\circ f^{‴}={{\iota }_2}^{\prime -1}\circ f^{\prime }\circ {\iota }_1$.

${{\iota }_2}^{\prime -1}\circ f^{‴}$ is $C^{\mathrm{\infty }}$, by the proposition that for any maps between any arbitrary subsets of any $C^{\mathrm{\infty }}$ manifolds with boundary $C^k$ at corresponding points, where $k$ includes $\mathrm{\infty }$, the composition is $C^k$ at the point.

So, $f$ is $C^{\mathrm{\infty }}$.

---

References

<The previous article in this series | The table of contents of this series | The next article in this series>