1034: (p,q)-Tensors Space at Point on C∞ Manifold with Boundary

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definition of $(p,q)$-tensors space at point on $C^{\mathrm{\infty }}$ 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 tangent vectors space at point on $C^{\mathrm{\infty }}$ manifold with boundary.
• The reader knows a definition of tensors space with respect to field and $k$ vectors spaces and vectors space over field.
• The reader knows a definition of tensor product of $k$ vectors spaces over field.
• The reader knows a definition of %field name% vectors space.

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Target Context

• The reader will have a definition of $(p,q)$-tensors space at point on $C^{\mathrm{\infty }}$ 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:
$M$: $\in \{\text{ the }{C}^{\mathrm{\infty }}\text{ manifolds with boundary }\}$
$m$: $\in M$
$T_{m}M$: $=\text{ the tangent vectors space at }m$
$T_m{M}^{\ast }$: $=\text{ the covectors space of }{T}_{m}M$
$p$: $\in \mathbb{N}$
$q$: $\in \mathbb{N}$
$\ast T_q^p(T_{m}M)$: $=T_{m}M\otimes ...\otimes T_{m}M\otimes T_m{M}^{\ast }\otimes ...\otimes T_m{M}^{\ast }$ where $T_{m}M$ appears $p$ times and $T_m{M}^{\ast }$ appears $q$ times when $p\ne 0$ or $q\ne 0$; $=\mathbb{R}$ when $p=q=0$, $\in \{\text{ the }\mathbb{R}\text{ vectors spaces }\}$
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Conditions:
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2: Note

$T_{m}M$ is an $\mathbb{R}$ vectors space as is shown in Note for the definition of tangent vectors space at point on $C^{\mathrm{\infty }}$ manifold with boundary, $T_m{M}^{\ast }$ is an $\mathbb{R}$ vectors space as is shown in Note for the definition of tensors space with respect to field and $k$ vectors spaces and vectors space over field, and $T_q^p(T_{m}M)$ is indeed an $\mathbb{R}$ vectors space as is shown in Note for the definition of tensor product of $k$ vectors spaces over field.

$T_q^p(T_{m}M)$ is canonically 'vectors spaces - linear morphisms' isomorphic to $L(T_m{M}^{\ast },...,T_m{M}^{\ast },T_{m}M,...,T_{m}M:\mathbb{R})$, by Note for 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, and quite often, the 2 spaces are implicitly identified by the isomorphism, which is the reason why $T_q^p(T_{m}M)$ is called "tensors space".

$T_0^1(T_{m}M)$ is the tangent vectors space.

$T_1^0(T_{m}M)$ is called "cotangent vectors space".

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References

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