1110: C∞ (p,q)-Tensors Bundle over C∞ Manifold with Boundary

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definition of $C^{\mathrm{\infty }}$ $(p,q)$-tensors bundle over $C^{\mathrm{\infty }}$ manifold with boundary

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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: Note

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Starting Context

• The reader knows a definition of $(p,q)$-tensors space at point on $C^{\mathrm{\infty }}$ manifold with boundary.
• The reader knows a definition of $C^{\mathrm{\infty }}$ vectors bundle of rank $k$.

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Target Context

• The reader will have a definition of $C^{\mathrm{\infty }}$ $(p,q)$-tensors bundle over $C^{\mathrm{\infty }}$ manifold with boundary.

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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:
$M$: $\in \{\text{ the }d\text{ -dimensional }{C}^{\mathrm{\infty }}\text{ manifolds with boundary }\}$
$p$: $\in \mathbb{N}$
$q$: $\in \mathbb{N}$
$T_q^p(TM)$: $=\underset{m\in M}{⨄}{T}_q^p(T_{m}M)$, denoted also as $TM\otimes ...\otimes TM\otimes T{M}^{\ast }\otimes ...\otimes T{M}^{\ast }$
$\pi$: $:T_q^p(TM)\to M,t\mapsto m$, where $t\in T_q^p(T_{m}M)$
$\ast (T_q^p(TM),M,\pi )$:
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Conditions:
$(T_q^p(TM),M,\pi )$ is the $C^{\mathrm{\infty }}$ vectors bundle constructed with the following to-be-set-of-local-trivializations: for any covering set of charts, $\{(U_j\subseteq M,{\varphi }_j)\}$, ${\mathrm{\Phi }}_j:{\pi }^{-1}(U_j)\to U_j\times {\mathbb{R}}^{d^{(p+q)}},t=t_{l_1,...,l_q}^{j_1,...,j_p}[((\partial /\partial x^{j_1},...,\partial /\partial x^{j_p},d{x}^{l_1},...,d{x}^{l_q}))]\mapsto (\pi (t),(t_{1,...,1}^{1,...,1},...,t_{d,...,d}^{d,...,d}))$, by the proposition that for any $C^{\mathrm{\infty }}$ manifold with boundary and any real vectors space at each point with any same dimension, $k$, any to-be-set-of-local-trivializations determines the canonical $C^{\mathrm{\infty }}$ vectors bundle of rank $k$
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$T_q^p(TM)$ is very often implicitly identified with $\underset{m\in M}{⨄}L(T_m{M}^{\ast },...,T_m{M}^{\ast },T_{m}M,...,T_{m}M:\mathbb{R})$ by the canonical bijection, $f^{\prime }:T_q^p(TM)\to \underset{m\in M}{⨄}L(T_m{M}^{\ast },...,T_m{M}^{\ast },T_{m}M,...,T_{m}M:\mathbb{R}),t\in T_q^p(T_{m}M)\mapsto f(t)$ where $f$ is the canonical 'vectors spaces - linear morphisms' isomorphism mentioned in the definition of canonical 'vectors spaces - linear morphisms' isomorphism between tensors space at point on $C^{\mathrm{\infty }}$ manifold with boundary and tensors space with respect to real numbers field and $p$ cotangent vectors spaces and $q$ tangent vectors spaces and field.

So, although each $t=t_{l_1,...,l_q}^{j_1,...,j_p}[((\partial /\partial x^{j_1},...,\partial /\partial x^{j_p},d{x}^{l_1},...,d{x}^{l_q}))]\in T_q^p(TM)$ is not really any multilinear map, $f(t)=t_{l_1,...,l_q}^{j_1,...,j_p}\widetilde{\partial /\partial x^{j_1}}\otimes ...\otimes \widetilde{\partial /\partial x^{j_p}}\otimes d{x}^{l_1}\otimes ...\otimes d{x}^{l_q}$ is very often implicitly used instead of $t$.

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2: Note

Let us see that the definition indeed conforms to the requirements for the proposition that for any $C^{\mathrm{\infty }}$ manifold with boundary and any real vectors space at each point with any same dimension, $k$, any to-be-set-of-local-trivializations determines the canonical $C^{\mathrm{\infty }}$ vectors bundle of rank $k$.

$T_q^p(T_{m}M)$ is a $d^{(p+q)}$-dimensional $\mathbb{R}$ vectors space, by the proposition that the tensor product of any $k$ finite-dimensional vectors spaces has the basis that consists of the classes induced by any basis elements: $T_{m}M$ is $d$-dimensional and $T_m{M}^{\ast }$ is $d$-dimensional, as is well-known.

For each $m\in U_j$, ${\mathrm{\Phi }}_j{|}_{T_q^p(T_{m}M)}:T_q^p(T_{m}M)\to \{m\}\times {\mathbb{R}}^{d^{(p+q)}},t=t_{l_1,...,l_q}^{j_1,...,j_p}[((\partial /\partial x^{j_1},...,\partial /\partial x^{j_p},d{x}^{l_1},...,d{x}^{l_q}))]\mapsto (\pi (t),(t_{1,...,1}^{1,...,1},...,t_{d,...,d}^{d,...,d}))$ is a 'vectors spaces - linear morphisms' isomorphism, by the proposition that between any vectors spaces, any map that maps any basis onto any basis bijectively and expands the mapping linearly is a 'vectors spaces - linear morphisms' isomorphism.

$\left(\begin{array}{c}{T}_{j,l}\end{array}\right)$ is $\left(\begin{array}{c}\partial ({\varphi }_l\circ {{\varphi }_j}^{-1}{)}^{j_1}/\partial x^{m_1}...\partial ({\varphi }_l\circ {{\varphi }_j}^{-1}{)}^{j_p}/\partial x^{m_p}\partial ({\varphi }_j\circ {{\varphi }_l}^{-1}{)}^{n_1}/\partial x^{\prime l_1}...\partial ({\varphi }_j\circ {{\varphi }_l}^{-1}{)}^{n_q}/\partial x^{\prime l_q}\end{array}\right)$ where $\{(j_1,...,j_p,l_1,...,l_q)\}$ constitutes the index for $l$ and $\{(m_1,...,m_p,n_1,...,n_q)\}$ constitutes the index for $j$, by the proposition that for any $C^{\mathrm{\infty }}$ manifold with boundary and the $(p,q)$-tensors space at any point, the transition of the components of any tensor with respect to the standard bases with respect to any charts is this.

$\left(\begin{array}{c}{T}_{j,l}\end{array}\right)$ is $C^{\mathrm{\infty }}$, because ${\varphi }_l\circ {{\varphi }_j}^{-1}$ and ${\varphi }_j\circ {{\varphi }_l}^{-1}$ are $C^{\mathrm{\infty }}$: while $\partial ({\varphi }_l\circ {{\varphi }_j}^{-1}{)}^{j_1}/\partial x^{m_1}...\partial ({\varphi }_l\circ {{\varphi }_j}^{-1}{)}^{j_p}/\partial x^{m_p}$ is $C^{\mathrm{\infty }}$ as a map from ${\varphi }_j(U_j\cap U_l)$ and $\partial ({\varphi }_j\circ {{\varphi }_l}^{-1}{)}^{n_1}/\partial x^{\prime l_1}...\partial ({\varphi }_j\circ {{\varphi }_l}^{-1}{)}^{n_q}/\partial x^{\prime l_q}$ is $C^{\mathrm{\infty }}$ as a map from ${\varphi }_l(U_j\cap U_l)$, $\partial ({\varphi }_l\circ {{\varphi }_j}^{-1}{)}^{j_1}/\partial x^{m_1}...\partial ({\varphi }_l\circ {{\varphi }_j}^{-1}{)}^{j_p}/\partial x^{m_p}\partial ({\varphi }_j\circ {{\varphi }_l}^{-1}{)}^{n_1}/\partial x^{\prime l_1}...\partial ({\varphi }_j\circ {{\varphi }_l}^{-1}{)}^{n_q}/\partial x^{\prime l_q}$ is $C^{\mathrm{\infty }}$ as a map from $U_j\cap U_l$, because ${\varphi }_j{|}_{U_j\cap U_l}:U_j\cap U_l\to {\varphi }_j(U_j\cap U_l)$ and ${\varphi }_l{|}_{U_j\cap U_l}:U_j\cap U_l\to {\varphi }_l(U_j\cap U_l)$ are $C^{\mathrm{\infty }}$.

So, the definition indeed conforms to the requirements for the proposition that for any $C^{\mathrm{\infty }}$ manifold with boundary and any real vectors space at each point with any same dimension, $k$, any to-be-set-of-local-trivializations determines the canonical $C^{\mathrm{\infty }}$ vectors bundle of rank $k$.

Let us see that the topology and the $C^{\mathrm{\infty }}$ atlas of $T_q^p(TM)$ do not depend on the choice of $\{(U_j\subseteq M,{\varphi }_j)\}$, by the proposition that for any set and any 2 topology-atlas pairs, if and only if there is a common chart domains open cover and the transition for each common chart is a diffeomorphism, the pairs are the same.

A set of covering charts by $\{(U_j\subseteq M,{\varphi }_j)\}$ is $\{({\pi }^{-1}(U_m)\subseteq T_q^p(TM),\widetilde{{\varphi }_m})\}$, where $m\in U_m\subseteq U_j$.

Let another choice be $\{(U_{j^{\prime }}^{\prime }\subseteq M,{\varphi }_{j^{\prime }}^{\prime })\}$.

A set of covering charts by $\{(U_{j^{\prime }}^{\prime }\subseteq M,{\varphi }_{j^{\prime }}^{\prime })\}$ is $\{({\pi }^{-1}(U_{m^{\prime }}^{\prime })\subseteq T_q^p(TM),\widetilde{{\varphi }_{m^{\prime }}^{\prime }})\}$, where $m^{\prime }\in U_{m^{\prime }}^{\prime }\subseteq U_{j^{\prime }}^{\prime }$.

$\{{\pi }^{-1}(U_m)\cap {\pi }^{-1}(U_{m^{\prime }}^{\prime })\}$ is a common chart domains open cover, because $({\pi }^{-1}(U_m)\cap {\pi }^{-1}(U_{m^{\prime }}^{\prime })\subseteq T_q^p(TM),\widetilde{{\varphi }_m}{|}_{{\pi }^{-1}(U_m)\cap {\pi }^{-1}(U_{m^{\prime }}^{\prime })})$ and $({\pi }^{-1}(U_m)\cap {\pi }^{-1}(U_{m^{\prime }}^{\prime })\subseteq T_q^p(TM),\widetilde{{\varphi }_{m^{\prime }}^{\prime }}{|}_{{\pi }^{-1}(U_m)\cap {\pi }^{-1}(U_{m^{\prime }}^{\prime })})$ are some charts.

The transition is $\widetilde{{\varphi }_{m^{\prime }}^{\prime }}{|}_{{\pi }^{-1}(U_m)\cap {\pi }^{-1}(U_{m^{\prime }}^{\prime })}\circ (\widetilde{{\varphi }_m}{|}_{{\pi }^{-1}(U_m)\cap {\pi }^{-1}(U_{m^{\prime }}^{\prime })}{)}^{-1}=\lambda \circ ({\varphi }_{m^{\prime }}^{\prime },id)\circ {\mathrm{\Phi }}_{j^{\prime }}^{\prime }\circ {{\mathrm{\Phi }}_j}^{-1}\circ ({{\varphi }_m}^{-1},i{d}^{-1})\circ {\lambda }^{-1}$: strictly speaking, each constituent needs to be appropriately restricted, but let us omit the marks of the restrictions here.

${\mathrm{\Phi }}_{j^{\prime }}^{\prime }\circ {{\mathrm{\Phi }}_j}^{-1}$ is a diffeomorphism, because it is essentially no different from ${\mathrm{\Phi }}_l\circ {{\mathrm{\Phi }}_j}^{-1}$.

The transition is a diffeomorphism, because $\lambda$, $({\varphi }_{m^{\prime }}^{\prime },id)$, ${\mathrm{\Phi }}_{j^{\prime }}^{\prime }\circ {{\mathrm{\Phi }}_j}^{-1}$, $({{\varphi }_m}^{-1},i{d}^{-1})$, and ${\lambda }^{-1}$ are some diffeomorphisms.

So, the topology and the $C^{\mathrm{\infty }}$ atlas of $T_q^p(TM)$ do not depend on the choice of $\{(U_j\subseteq M,{\varphi }_j)\}$.

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References

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