371: Restriction of C^\infty Map on Open Domain and Open Codomain Is C^\infty

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

A description/proof of that restriction of $C^{\mathrm{\infty }}$ map on open domain and open 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: Description
• 2: Proof

---

Starting Context

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

---

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, the restriction of the map on any open domain and any valid open 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: Description

For any $C^{\mathrm{\infty }}$ manifolds, $M_1,M_2$, any $C^{\mathrm{\infty }}$ map, $f:M_1\to M_2$, any open subset, $U_1\subseteq M_1$, and any open subset, $U_2\subseteq M_2$, such that $f(U_1)\subseteq U_2$, $f{|}_{U_1}:U_1\to U_2$ is $C^{\mathrm{\infty }}$.

---

2: Proof

For any point, $p\in U_1$, there are charts, $(U_p\subseteq M_1,{\varphi }_p)$ and $(U_{f(p)}\subseteq M_2,{\varphi }_{f(p)})$, and ${\varphi }_{f(p)}\circ f\circ {{\varphi }_p}^{-1}$ is $C^{\mathrm{\infty }}$ at ${\varphi }_p(p)$. $(U_p\cap U_1\subseteq U_1,{\varphi }_p{|}_{U_p\cap U_1})$ and $(U_{f(p)}\cap U_2\subseteq U_2,{\varphi }_{f(p)}{|}_{U_{f(p)}\cap U_2})$ are charts on $U_1$ and $U_2$, and ${\varphi }_{f(p)}{|}_{U_{f(p)}\cap U_2}\circ f{|}_{U_1}\circ {{\varphi }_p{|}_{U_p\cap U_1}}^{-1}$ is $C^{\mathrm{\infty }}$ at ${\varphi }_p{|}_{U_p\cap U_1}(p)$, because it is a restriction of $C^{\mathrm{\infty }}$ ${\varphi }_{f(p)}\circ f\circ {{\varphi }_p}^{-1}$ on the open domain, ${\varphi }_p(U_p\cap U_1)$.

---

References

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