368: Restriction of C^\infty Vectors Bundle on Regular Submanifold Base Space Is C^\infty Vectors Bundle

<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 $C^{\mathrm{\infty }}$ vectors bundle on regular submanifold base space is $C^{\mathrm{\infty }}$ vectors bundle

---

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 restriction of $C^{\mathrm{\infty }}$ vectors bundle on regular submanifold base space.
• The reader admits the proposition that for any transversal map from any $C^{\mathrm{\infty }}$ manifold to any regular submanifold, the preimage of the regular submanifold under the transversal map is a regular submanifold of the domain.
• The reader admits the proposition that for any $C^{\mathrm{\infty }}$ map between $C^{\mathrm{\infty }}$ manifolds, the restriction of the map on any regular submanifold domain and any regular submanifold codomain is $C^{\mathrm{\infty }}$.
• The reader admits the proposition that for any $C^{\mathrm{\infty }}$ manifold and its any regular submanifold, any open subset of the super manifold is canonically a $C^{\mathrm{\infty }}$ manifold, and the intersection of the open subset and the regular submanifold is a regular submanifold of the open subset manifold.
• The reader admits the proposition that for any map, the map preimage of any intersection of sets is the intersection of the map preimages of the sets.
• The reader admits the proposition that the intersection of the same-indices-set products of possibly uncountable number of sets is the product of the intersections of the subsets.
• The reader admits the proposition that for the product of any 2 $C^{\mathrm{\infty }}$ manifolds, the product for which one of the constituents is replaced with any regular submanifold is a regular submanifold of the original product.

---

Target Context

• The reader will have a description and a proof of the proposition that for any $C^{\mathrm{\infty }}$ vectors bundle, the restriction of the vectors bundle on any regular submanifold base space is a $C^{\mathrm{\infty }}$ vectors bundle.

---

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 $C^{\mathrm{\infty }}$ vectors bundle, $\pi :M_1\to M_2$, and any regular submanifold, $M_3\subseteq M_2$, the restriction of $\pi$ on $M_3$, ${\pi }_{M_3}:{\pi }^{-1}(M_3)\to M_3$, is a $C^{\mathrm{\infty }}$ vectors bundle.

---

2: Proof

$\pi$ is a submersion, $\pi$ is a transversal map to $M_3$, and ${\pi }^{-1}(M_3)$ is a regular submanifold of $M_1$ as a preimage of a regular submanifold under a transversal map, by the proposition that for any transversal map from any $C^{\mathrm{\infty }}$ manifold to any regular submanifold, the preimage of the regular submanifold under the transversal map is a regular submanifold of the domain.

${\pi }_{M_3}$ is $C^{\mathrm{\infty }}$ as a restriction of a $C^{\mathrm{\infty }}$ map on a regular submanifold domain and a regular submanifold codomain, by the proposition that for any $C^{\mathrm{\infty }}$ map between $C^{\mathrm{\infty }}$ manifolds, the restriction of the map on any regular submanifold domain and any regular submanifold codomain is $C^{\mathrm{\infty }}$. ${\pi }_{M_3}$ is obviously surjective.

For any $p\in M_3$, ${{\pi }_{M_3}}^{-1}(p)$ is a real vectors space, because ${{\pi }_{M_3}}^{-1}(p)={\pi }^{-1}(p)$.

For any $p\in M_3$, there is an open neighborhood, $U_p\subseteq M_2$, such that there is a fiber-preserving diffeomorphism, $f^{\prime }:{\pi }^{-1}(U_p)\to U_p\times {\mathbb{R}}^r$. There is the fiber-preserving bijection, $f=f^{\prime }{|}_{{\pi }^{-1}(U_p\cap M_3)}:{\pi }^{-1}(U_p\cap M_3)\to (U_p\cap M_3)\times {\mathbb{R}}^r$, because $f^{\prime }$ is fiber-preserving.

Is $f$ diffeomorphic? ${\pi }^{-1}(U_p)$ is an open subset of $M_1$, because $\pi$ is continuous, and is a $C^{\mathrm{\infty }}$ manifold, by the proposition that for any $C^{\mathrm{\infty }}$ manifold and its any regular submanifold, any open subset of the super manifold is canonically a $C^{\mathrm{\infty }}$ manifold, and the intersection of the open subset and the regular submanifold is a regular submanifold of the open subset manifold. $U_p\times {\mathbb{R}}^r$ is an open subset of $M_2\times {\mathbb{R}}^r$, and is a $C^{\mathrm{\infty }}$ manifold likewise.

${\pi }^{-1}(U_p\cap M_3)={\pi }^{-1}(U_p)\cap {\pi }^{-1}(M_3)$, by the proposition that for any map, the map preimage of any intersection of sets is the intersection of the map preimages of the sets. So, ${\pi }^{-1}(U_p\cap M_3)$ is a regular submanifold of ${\pi }^{-1}(U_p)$, by the proposition that for any $C^{\mathrm{\infty }}$ manifold and its any regular submanifold, any open subset of the super manifold is canonically a $C^{\mathrm{\infty }}$ manifold, and the intersection of the open subset and the regular submanifold is a regular submanifold of the open subset manifold. $(U_p\cap M_3)\times {\mathbb{R}}^r=(U_p\times {\mathbb{R}}^r)\cap (M_3\times {\mathbb{R}}^r)$, by the proposition that the intersection of the same-indices-set products of possibly uncountable number of sets is the product of the intersections of the subsets, and $M_3\times {\mathbb{R}}^r$ is a regular submanifold of $M_2\times {\mathbb{R}}^r$, by the proposition that for the product of any 2 $C^{\mathrm{\infty }}$ manifolds, the product for which one of the constituents is replaced with any regular submanifold is a regular submanifold of the original product, and $(U_p\cap M_3)\times {\mathbb{R}}^r$ is a regular submanifold of $U_p\times {\mathbb{R}}^r$ likewise.

So, $f=f^{\prime }{|}_{{\pi }^{-1}(U_p\cap M_3)}:{\pi }^{-1}(U_p\cap M_3)\to (U_p\cap M_3)\times {\mathbb{R}}^r$ is a restriction of the $C^{\mathrm{\infty }}$ $f^{\prime }$ on a regular submanifold domain and a regular submanifold codomain, and is $C^{\mathrm{\infty }}$, by the proposition that for any $C^{\mathrm{\infty }}$ map between $C^{\mathrm{\infty }}$ manifolds, the restriction of the map on any regular submanifold domain and any regular submanifold codomain is $C^{\mathrm{\infty }}$. As $f$ is a bijection, there is the inverse, $f^{-1}$, which is a restriction of the $C^{\mathrm{\infty }}$ $f^{\prime -1}$ on a regular submanifold domain and a regular submanifold codomain, and is $C^{\mathrm{\infty }}$, likewise.

For any point, $p^{\prime }\in U_p\cap M_3$, $f{|}_{{\pi }^{-1}(p^{\prime })}:{\pi }^{-1}(p^{\prime })\to \{p^{\prime }\}\times {\mathbb{R}}^r$ is a 'vectors spaces - linear morphisms' isomorphism, because it is the same with $f^{\prime }{|}_{{\pi }^{-1}(p^{\prime })}$.

---

References

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