---
slug: thoughts/Tensor-field
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tags:
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title: "Tensor field"
date: 2024-11-27
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---
> a function assign a tensor to each point of a region of a mathematical space (typically a _Euclidean space_ or a [manifold](https://aarnphm.xyz/thoughts/Tensor-field/../../thoughts/manifold))

> [!math] Definition
>
> Let $M$ be a manifold, for instance the Euclidean plane $\mathbb{R}^n$
>
> Then a tensor field of type $(p, q)$ is a section
>
> $$
> T \in \Gamma(M, V^{\otimes p} \otimes (V^{*})^{\otimes q})
> $$
>
> where $V$ is a [vector bundle](https://aarnphm.xyz/thoughts/Tensor-field/../../thoughts/Tensor-field#vector-bundle) on $M$, $V^{*}$ is its [dual](https://aarnphm.xyz/thoughts/Tensor-field/../../thoughts/Tensor-field#dual) and $\otimes$ is the tensor product of vector bundles

## via coordinate transitions

See also ([McConnell, 2014](#bib-mcconnell2014applications); [Schouten, 1951](#bib-schouten1951tensor))

---

## appendix

_a few math definitions_

### metric tensors

> A tangent space is a $n$-dimensional differentiable manifold $M$ associated with each point $p$.

a non-degenerate, smooth, symmetric bilinear map that assigns a real number to pairs of tangent vectors at each tangent space of the manifold.

> [!math] metric tensor
>
> $$
> g: T_p M \times T_p M \to \mathbb{R}
> $$

The map is symmetric and bilinear, so if $X, Y, Z \in T_p M$ are tangent vectors at point $p$ to the manifold $M$ then we have:

$$
\begin{aligned} g(X,Y) &= g(Y,X) \\ g(aX + Y, Z) &= ag(X,Z) + g(Y,Z) \end{aligned}
$$

for any real number $a \in \mathbb{R}$

> $g$ is _non-degenerate_ means there is no non-zero $X \in T_p M$ such that $g(X,Y)=0 \forall \space Y \in T_p M$

### vector bundle

a topological construction that makes precise the idea of of a family of vector space parameterised by another space $X$

ex: $X$ could be a topological space, a [manifold](https://aarnphm.xyz/thoughts/Tensor-field/../../thoughts/manifold)

[](https://aarnphm.xyz/thoughts/Tensor-field/../../thoughts/images/MobiusStrip.mp4)

_Möbius strip_

> to every point $x$ of the space $X$ we “attach” a vector space $V(x)$ in such a way that these vector space fits together to form another space of the same kind as $X$

> [!math] definition
>
> A **real vector bundle** consists of
>
> - topological spaces $X$ (base space) and $E$ (total space)
> - a continuous surjection $\pi: E \rightarrow X$ (bundle projection)
> - For every $x$ in $X$ the structure of a _finite-dimensional real vector space_ on the [fiber](https://aarnphm.xyz/thoughts/Tensor-field/../../thoughts/Tensor-field#fiber) $\pi^{-1}(\{x\})$

> [!tip] compatibility condition
>
> For every point $p$ in $X$, there is an _open neighborhood_ $U \subseteq X$ of $p$ and a **[homeomorphism](https://aarnphm.xyz/thoughts/Tensor-field/../../thoughts/homeomorphism)**
>
> $$
> \varphi : U \times \mathbb{R}^k \rightarrow \pi^{-1}(U)
> $$
>
> such that for all $x$ in $U$:
>
> - $(\pi \circ \varphi)(x,v)=x$ for all vectors $v$ in $\mathbb{R}^k$
> - the map $v \mapsto \varphi(x,v)$ is a _linear isomorphism_ between vector spaces $\mathbb{R}^k$ and $\pi^{-1}(\{x\})$

#### properties

- open neighborhood $U$ together with the hoemomorphism $\varphi$ is called a _local trivialisation_ of the vector bundle

  [^local-trivial]

* every fiber $\pi^{-1}(\{x\})$ is a finite-dimensional real vector space and hence has a _dimension_ $k_x$

* function $x \to k_x$ is locally constant, and therefore constant on each _connected component_ of $X$

> [!note] rank of the vector bundle
>
> if $k_x$ is equal to constant $k$ on all of $X$, then $k$ is the rank of the vector bundle, and $E$ is a **vector bundle of rank** $k$

> [!math] trivial bundle
>
> The Cartesian product $X \times \mathbb{R}^k$ equipped with the projection $X \times \mathbb{R}^k \to X$ is considered as the _trivial bundle_ of rank $k$ over $X$

### dual

operations on vector bundle extending the operation of duality for vector space.

> [!math] definition
>
> a _dual bundle_ of a vector bundle $\pi : E \rightarrow X$ is the vector bundle $\pi^{*}: E^{*} \rightarrow X$ whose fiber are the dual spaces to fibers of $E$

Equivalently, $E^{*}$ can be defined as the Hom bundle $\text{Hom}(E, \mathbb{R} \times X)$, the vector bundle of morphisms from $E$ to the trivial line bundle $\mathbb{R} \times X \rightarrow X$

### fiber

_a space that is _locally_ a product space, but _globally_ may have different topological structure_

> [!math] definition
>
> A fiber bundle is a structure $(E, B, \pi, F)$ where:
>
> - $E, B, F$ are topological space
> - $\pi: E \rightarrow B$ is a _continuous surjection_ satisfying _local triviality_ condition

$B$ is considered as _base space_, $E$ is **total space**, and $F$ is the _fiber space_

the map $\pi$ is called the **projection map**

> [!abstract] consequences
>
> we require that for every $x \in B$, there is an open neighborhood $U \subseteq B$ of $x$ such that there is a [homeomorphism](https://aarnphm.xyz/thoughts/Tensor-field/../../thoughts/homeomorphism) $\varphi: \pi^{-1}(U) \rightarrow U \times F$ such that a way $\pi$ agrees with the projection onto the first factor. [^annotation]

```tikz
\usepackage{tikz-cd}
\begin{document}
\begin{tikzcd}
\pi^{-1}(U) \arrow[r, "\varphi"] \arrow[d, "\pi"'] & U \times F \arrow[ld, "proj_1"] \\
U &
\end{tikzcd}
\end{document}
```

where $\text{proj}_1: U \times F \rightarrow U$ is the natural projection and $\varphi : \pi^{-1}(U) \rightarrow U \times F$ is a homeomorphism.

> The set of all $\{(U_i, \varphi_i)\}$ is called a **local trivialization** of the bundle

Therefore, for any $p \in B$, the _preimage_ $\pi^{-1}(\{p\})$ is _homeomorphic_ to $F$ [^true] and is called the _fiber over_ p

> [!note] annotation
>
> a fiber bundle $(E, B, \pi, F)$ is often denoted as
>
> $$
> F \to E \xrightarrow{\pi} B
> $$

#### bundle map

Suppose that $M$ and $N$ are base space, and $\pi_E: E \to M$ and $\pi_F: F \to N$ are fiber bundles over $M$ and $N$ respectively.

> [!math] definition
>
> **bundle map/morphism** consists of a pair of continuous functions
>
> $$
> \varphi: E \to F, f: M \to N
> $$
>
> such that $\pi_F \circ \varphi = f \circ \pi_E$. That is the following is commutative:
>
> ```tikz
> \usepackage{tikz-cd}
> \begin{document}
> \begin{tikzcd}
> E \arrow[r, "\varphi"] \arrow[d, "\pi_E"'] & F \arrow[d, "\pi_F"] \\
> M \arrow[r, "f"'] & N
> \end{tikzcd}
> \end{document}
> ```

## Bibliographie

- McConnell, A. J. (2014). _Applications of Tensor Analysis_. Dover Publications. <https://books.google.ca/books?id=ZCP0AwAAQBAJ>
- Schouten, J. A. (1951). _Tensor Analysis for Physicists_. Oxford University Press.

[^local-trivial]: shows that _locally_ the map $\pi$ “looks like” the projection of $U \times \mathbb{R}^k$ on $U$ 

[^annotation]: $\pi^{-1}(U)$ is the given subspace topology, and $U \times F$ is the product space 

[^true]: since this is true of $\text{proj}_1^{-1}(\{p\})$ 
