665: Tangent Vectors Bundle over C∞ Manifold with Boundary

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

definition of tangent vectors bundle over $C^{\mathrm{\infty }}$ manifold with boundary

---

Topics

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

---

The table of contents of this article

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

---

Starting Context

• The reader knows a definition of $C^{\mathrm{\infty }}$ manifold with boundary.
• The reader knows a definition of tangent vector.

---

Target Context

• The reader will have a definition of tangent vectors bundle over $C^{\mathrm{\infty }}$ manifold with boundary.

---

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:
$M$: $\in \{\text{ the }d\text{ -dimensional }{C}^{\mathrm{\infty }}\text{ manifolds with boundary }\}$
$TM$: $={\cup }_{p\in M}{T}_{p}M$, $\in \{\text{ the }2d\text{ -dimensional }{C}^{\mathrm{\infty }}\text{ manifolds with boundary }\}$, with the topology and the $C^{\mathrm{\infty }}$ atlas specified below
$\pi$: $:TM\to M,v\mapsto p$, where $v\in T_{p}M$
$\ast (TM,M,\pi )$:
//

Conditions:
$\mathrm{\forall }(U_{\alpha }\subseteq M,{\varphi }_{\alpha })\in \{\text{ the charts of }M\}$
(
$\mathrm{\exists }({\pi }^{-1}(U_{\alpha })\subseteq TM,{\varphi }_{\alpha }^{\prime })\in \{\text{ the charts of }TM\}$
(
${\varphi }_{\alpha }^{\prime }:{\pi }^{-1}(U_{\alpha })\to \varphi (U_{\alpha })\times {\mathbb{R}}^d\subseteq {\mathbb{R}}^{2d}\text{ or }{\mathbb{H}}^{2d},v\mapsto ({\varphi }_{\alpha }(\pi (v)),{\varphi }_{\alpha }^{″}(v))$, where ${\varphi }_{\alpha }^{″}:{\pi }^{-1}(U_{\alpha })\to {\mathbb{R}}^d,v\mapsto (v^1,...,v^d)$, where $v=v^j\partial x^j$, where $(x^1,...,x^n)$ are the coordinates of the chart, $(U_{\alpha },{\varphi }_{\alpha })$
$B_{\alpha }:=\{{{\varphi }_{\alpha }^{\prime }}^{-1}(U_{\alpha }^{\prime })|\mathrm{\forall }{U}_{\alpha }^{\prime }\in \{\text{ the open subsets of }{\varphi }_{\alpha }(U_{\alpha })\times {\mathbb{R}}^d\}\}$
)
)
$\wedge$
$B:={\cup }_{\alpha }{B}_{\alpha }\in \{\text{ the topological bases of }TM\}$
//

As ${\varphi }_{\alpha }^{\prime }$ is obviously bijective, ${{\varphi }_{\alpha }^{\prime }}^{-1}$ is valid.

Usually, it is called "tangent bundle", but the author prefers "tangent vectors bundle", because the author prizes expressing explicitly: it is the bundle of the tangent vectors, while "tangent vectors bundle" will be immediately and successfully guessed what it means by anyone who knows what "tangent bundle" is.

---

2: Natural Language Description

For any $d$-dimensional $C^{\mathrm{\infty }}$ manifold with boundary, $M$, $(TM,M,\pi )$, where $TM$ is the $2d$-dimensional $C^{\mathrm{\infty }}$ manifold with boundary, ${\cup }_{p\in M}{T}_{p}M$, with the topology and the $C^{\mathrm{\infty }}$ atlas specified below and $\pi :TM\to M$ is $v\mapsto p$, where $v\in T_{p}M$: the topology and the $C^{\mathrm{\infty }}$ atlas of $TM$ is defined as this: for each chart, $(U_{\alpha }\subseteq M,{\varphi }_{\alpha })$, define the chart, $({\pi }^{-1}(U_{\alpha })\subseteq TM,{\varphi }_{\alpha }^{\prime })$, where ${\varphi }_{\alpha }^{\prime }:{\pi }^{-1}(U_{\alpha })\to \varphi (U_{\alpha })\times {\mathbb{R}}^d\subseteq {\mathbb{R}}^{2d}\text{ or }{\mathbb{H}}^{2d}$, $v\mapsto ({\varphi }_{\alpha }(\pi (v)),{\varphi }_{\alpha }^{″}(v))$, where ${\varphi }_{\alpha }^{″}:{\pi }^{-1}(U_{\alpha })\to {\mathbb{R}}^d$, $v\mapsto (v^1,...,v^d)$, where $v=v^j\partial x^j$, where $(x^1,...,x^d)$ are the coordinates of the chart, $(U_{\alpha },{\varphi }_{\alpha })$, and take $B_{\alpha }:=\{{{\varphi }_{\alpha }^{\prime }}^{-1}(U_{\alpha }^{\prime })|\mathrm{\forall }{U}_{\alpha }^{\prime }\in \{\text{ the open subsets of }{\varphi }_{\alpha }(U_{\alpha })\times {\mathbb{R}}^d\}\}$, and take $B:={\cup }_{\alpha }{B}_{\alpha }$ as the topological basis

---

3: Note

The basis is valid according to Description 2 of some criteria for any set of open sets to be a basis, as is shown subsequently.

1) $TM$ is obviously the union of all the elements of the basis.

2) for any ${{\varphi }_{\alpha }^{\prime }}^{-1}(U_{\alpha }^{\prime })$ and ${{\varphi }_{\beta }^{\prime }}^{-1}(U_{\beta }^{\prime })$ such that ${{\varphi }_{\alpha }^{\prime }}^{-1}(U_{\alpha }^{\prime })\cap {{\varphi }_{\beta }^{\prime }}^{-1}(U_{\beta }^{\prime })\ne \mathrm{\varnothing }$, $U_{\alpha }^{\prime }={\cup }_{\gamma \in A_{\alpha }}{U}_{\alpha ,\gamma }^{″}\times U_{\alpha ,\gamma }^{‴}$ where $A_{\alpha }$ is a possibly uncountable index set and $U_{\beta }^{\prime }={\cup }_{{\gamma }^{\prime }\in A_{\beta }}{U}_{\beta ,{\gamma }^{\prime }}^{″}\times U_{\beta ,{\gamma }^{\prime }}^{‴}$ where $A_{\beta }$ is a possibly uncountable index set, by Note for the definition of product topology, and ${{\varphi }_{\alpha }^{\prime }}^{-1}(U_{\alpha }^{\prime })\cap {{\varphi }_{\beta }^{\prime }}^{-1}(U_{\beta }^{\prime })=({\cup }_{\gamma \in A_{\alpha }}{{\varphi }_{\alpha }^{\prime }}^{-1}(U_{\alpha ,\gamma }^{″}\times U_{\alpha ,\gamma }^{‴}))\cap ({\cup }_{{\gamma }^{\prime }\in A_{\beta }}{{\varphi }_{\beta }^{\prime }}^{-1}(U_{\beta ,{\gamma }^{\prime }}^{″}\times U_{\beta ,{\gamma }^{\prime }}^{‴}))$, by the proposition that for any map, the map preimage of any union of sets is the union of the map preimages of the sets.

For any $p\in {{\varphi }_{\alpha }^{\prime }}^{-1}(U_{\alpha }^{\prime })\cap {{\varphi }_{\beta }^{\prime }}^{-1}(U_{\beta }^{\prime })$, $p\in {{\varphi }_{\alpha }^{\prime }}^{-1}(U_{\alpha ,\gamma }^{″}\times U_{\alpha ,\gamma }^{‴})\cap {{\varphi }_{\beta }^{\prime }}^{-1}(U_{\beta ,{\gamma }^{\prime }}^{″}\times U_{\beta ,{\gamma }^{\prime }}^{‴})$ for a $\gamma \in A_{\alpha }$ and a ${\gamma }^{\prime }\in A_{\beta }$. We are going to show that $S:={{\varphi }_{\alpha }^{\prime }}^{-1}(U_{\alpha ,\gamma }^{″}\times U_{\alpha ,\gamma }^{‴})\cap {{\varphi }_{\beta }^{\prime }}^{-1}(U_{\beta ,{\gamma }^{\prime }}^{″}\times U_{\beta ,{\gamma }^{\prime }}^{‴})$ is an element of the basis. In order for that, we are going to show that ${\varphi }_{\alpha }^{\prime }(S)$ is open on ${\varphi }_{\alpha }^{\prime }({\pi }^{-1}(U_{\alpha }))$, because then, $S$ will be the preimage of the open subset under ${\varphi }_{\alpha }^{\prime }$, which exists in the basis by definition.

${\varphi }_{\alpha }^{\prime }(S)={\varphi }_{\alpha }^{\prime }({{\varphi }_{\alpha }^{\prime }}^{-1}(U_{\alpha ,\gamma }^{″}\times U_{\alpha ,\gamma }^{‴}))\cap {\varphi }_{\alpha }^{\prime }({{\varphi }_{\beta }^{\prime }}^{-1}(U_{\beta ,{\gamma }^{\prime }}^{″}\times U_{\beta ,{\gamma }^{\prime }}^{‴})\cap {\pi }^{-1}(U_{\alpha }))$, by the proposition that for any injective map, the map image of the intersection of any sets is the intersection of the map images of the sets, $=(U_{\alpha ,\gamma }^{″}\times U_{\alpha ,\gamma }^{‴})\cap {\varphi }_{\alpha }^{\prime }({{\varphi }_{\beta }^{\prime }}^{-1}(U_{\beta ,{\gamma }^{\prime }}^{″}\times U_{\beta ,{\gamma }^{\prime }}^{‴})\cap {\pi }^{-1}(U_{\alpha }))$.

${\varphi }_{\beta }^{\prime }({{\varphi }_{\beta }^{\prime }}^{-1}(U_{\beta ,{\gamma }^{\prime }}^{″}\times U_{\beta ,{\gamma }^{\prime }}^{‴})\cap {\pi }^{-1}(U_{\alpha }))={\varphi }_{\beta }^{\prime }({{\varphi }_{\beta }^{\prime }}^{-1}(U_{\beta ,{\gamma }^{\prime }}^{″}\times U_{\beta ,{\gamma }^{\prime }}^{‴}))\cap {\varphi }_{\beta }^{\prime }({\pi }^{-1}(U_{\alpha })\cap {\pi }^{-1}(U_{\beta }))$, by the proposition that for any injective map, the map image of the intersection of any sets is the intersection of the map images of the sets, $=(U_{\beta ,{\gamma }^{\prime }}^{″}\times U_{\beta ,{\gamma }^{\prime }}^{‴})\cap {\varphi }_{\beta }^{\prime }({\pi }^{-1}(U_{\alpha })\cap {\pi }^{-1}(U_{\beta }))$.

In fact, ${\varphi }_{\beta }^{\prime }({\pi }^{-1}(U_{\alpha })\cap {\pi }^{-1}(U_{\beta }))={\varphi }_{\beta }(U_{\alpha }\cap U_{\beta })\times {\mathbb{R}}^d$, and is open on ${\mathbb{R}}^{2d}$ or ${\mathbb{H}}^{2d}$. Likewise, ${\varphi }_{\alpha }^{\prime }({\pi }^{-1}(U_{\alpha })\cap {\pi }^{-1}(U_{\beta }))$ is open on ${\mathbb{R}}^{2d}$ or ${\mathbb{H}}^{2d}$, by the symmetry.

So, ${\varphi }_{\alpha }^{\prime }(S)=(U_{\alpha ,\gamma }^{″}\times U_{\alpha ,\gamma }^{‴})\cap {\varphi }_{\alpha }^{\prime }\circ {{\varphi }_{\beta }^{\prime }}^{-1}((U_{\beta ,{\gamma }^{\prime }}^{″}\times U_{\beta ,{\gamma }^{\prime }}^{‴})\cap {\varphi }_{\beta }^{\prime }({\pi }^{-1}(U_{\alpha })\cap {\pi }^{-1}(U_{\beta })))$.

Let us think of ${\varphi }_{\alpha }^{\prime }\circ {{\varphi }_{\beta }^{\prime }}^{-1}:{\varphi }_{\beta }^{\prime }({\pi }^{-1}(U_{\alpha })\cap {\pi }^{-1}(U_{\beta }))\to {\varphi }_{\alpha }^{\prime }({\pi }^{-1}(U_{\alpha })\cap {\pi }^{-1}(U_{\beta }))$, obviously bijective.

Let us prove that ${\varphi }_{\alpha }^{\prime }\circ {{\varphi }_{\beta }^{\prime }}^{-1}$ is diffeomorphic. ${\varphi }_{\alpha }^{\prime }\circ {{\varphi }_{\beta }^{\prime }}^{-1}:(x^1,...,x^d,v^1,...,v^d)\mapsto ({\varphi }_{\alpha }({{\varphi }_{\beta }}^{-1})(x),\partial {\varphi }_{\alpha }^1(x)/\partial x^j{v}^j,...,\partial {\varphi }_{\alpha }^d(x)/\partial x^j{v}^j)$, which is $C^{\mathrm{\infty }}$. The inverse, $({\varphi }_{\alpha }^{\prime }\circ {{\varphi }_{\beta }^{\prime }}^{-1}{)}^{-1}={\varphi }_{\beta }^{\prime }\circ {{\varphi }_{\alpha }^{\prime }}^{-1}$, is likewise $C^{\mathrm{\infty }}$, by the symmetry. So, ${\varphi }_{\alpha }^{\prime }\circ {{\varphi }_{\beta }^{\prime }}^{-1}$ is indeed diffeomorphic.

As ${\varphi }_{\beta }^{\prime }({\pi }^{-1}(U_{\alpha })\cap {\pi }^{-1}(U_{\beta }))$ is open on ${\mathbb{R}}^{2d}$ or ${\mathbb{H}}^{2d}$, $(U_{\beta ,{\gamma }^{\prime }}^{″}\times U_{\beta ,{\gamma }^{\prime }}^{‴})\cap {\varphi }_{\beta }^{\prime }({\pi }^{-1}(U_{\alpha })\cap {\pi }^{-1}(U_{\beta }))$ is open on ${\mathbb{R}}^{2d}$ or ${\mathbb{H}}^{2d}$, and so, is open on ${\varphi }_{\beta }^{\prime }({\pi }^{-1}(U_{\alpha })\cap {\pi }^{-1}(U_{\beta }))$, and ${\varphi }_{\alpha }^{\prime }\circ {{\varphi }_{\beta }^{\prime }}^{-1}((U_{\beta ,{\gamma }^{\prime }}^{″}\times U_{\beta ,{\gamma }^{\prime }}^{‴})\cap {\varphi }_{\beta }^{\prime }({\pi }^{-1}(U_{\alpha })\cap {\pi }^{-1}(U_{\beta })))$ is open on ${\varphi }_{\alpha }^{\prime }({\pi }^{-1}(U_{\alpha })\cap {\pi }^{-1}(U_{\beta }))$, because ${\varphi }_{\alpha }^{\prime }\circ {{\varphi }_{\beta }^{\prime }}^{-1}$ is diffeomorphic, and so, $(U_{\alpha ,\gamma }^{″}\times U_{\alpha ,\gamma }^{‴})\cap {\varphi }_{\alpha }^{\prime }\circ {{\varphi }_{\beta }^{\prime }}^{-1}((U_{\beta ,{\gamma }^{\prime }}^{″}\times U_{\beta ,{\gamma }^{\prime }}^{‴})\cap {\varphi }_{\beta }^{\prime }({\pi }^{-1}(U_{\alpha })\cap {\pi }^{-1}(U_{\beta })))$ is open on ${\varphi }_{\alpha }^{\prime }({\pi }^{-1}(U_{\alpha })\cap {\pi }^{-1}(U_{\beta }))$, and so, is open on ${\varphi }_{\alpha }^{\prime }({\pi }^{-1}(U_{\alpha }))$.

So, $S$ is an element of the basis.

So, the criterion, 2), is satisfied.

The topological space is Hausdorff, because for any distinct points on $TM$, if they are on some distinct fibers, there are some disjoint chart open subsets, $U_{\alpha },U_{\beta }$, and ${\pi }^{-1}(U_{\alpha })\cap {\pi }^{-1}(U_{\beta })=\mathrm{\varnothing }$; if they are on a same fiber, there are some open subsets of ${\varphi }_{\alpha }^{\prime }({\pi }^{-1}(U_{\alpha }))$, $(U_{\alpha }\times U^{\prime })\cap (U_{\alpha }\times U^{″})=\mathrm{\varnothing }$, because ${\mathbb{R}}^d$ is Hausdorff, and ${{\varphi }_{\alpha }^{\prime }}^{-1}(U_{\alpha }\times U^{\prime })\cap {{\varphi }_{\alpha }^{\prime }}^{-1}(U_{\alpha }\times U^{″})=\mathrm{\varnothing }$.

The topological space is 2nd-countable, because the charts on $M$ can be taken countably and the open subsets of ${\varphi }_{\alpha }^{\prime }({\pi }^{-1}(U_{\alpha }))$ can be taken countably, because ${\mathbb{R}}^{2d}$ or ${\mathbb{H}}^{2d}$ is 2nd-countable.

The atlas is $C^{\mathrm{\infty }}$, as has been proved above.

$\pi$ is $C^{\mathrm{\infty }}$, because with respect to the charts, $({\pi }^{-1}(U_{\alpha })\subseteq TM,{\varphi }_{\alpha }^{\prime })$ and $(U_{\alpha }\subseteq M,\varphi )$, the coordinates function is $({\varphi }_{\alpha }(\pi (v)),{\varphi }_{\alpha }^{″}(v))\mapsto {\varphi }_{\alpha }(\pi (v))$, $C^{\mathrm{\infty }}$.

Any tangent vectors bundle is a kind of $C^{\mathrm{\infty }}$ vectors bundle: $\pi$ is a locally trivial surjection of rank $d$, as can be confirmed straightforwardly.

---

References

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