# The Unapologetic Mathematician

## Tensor Fields and Multilinear Maps

A tensor field $T$ over a manifold $M$ gives us a tensor $T_p$ at each point $p\in M$. And we know that $T_p$ can be considered as a multilinear map. Specifically, if $T$ is a tensor field of type $(r,s)$, then we find

$\displaystyle T_p\in\mathcal{T}_pM^{\otimes r}\otimes\mathcal{T}^*_pM^{\otimes s}$

which we can interpret as a multilinear map:

$\displaystyle T_p:\mathcal{T}_pM^{\times r}\times\mathcal{T}^*_pM^{\times s}\to\mathbb{R}$

where multilinearity means that $T_p$ is linear in each variable separately.

As we let $p$ vary over $M$, we can interpret $T$ as defining a function which takes $r$ vector fields and $s$ covector fields and gives a function. Explicitly:

$\displaystyle\left[T(X_1,\dots,X_r,\alpha^1,\dots,\alpha^s)\right](p)=T_p(X_1(p),\dots,X_r(p),\alpha^1(p),\dots,\alpha^s(p))$

And, in particular, this function is multilinear over $\mathcal{O}M$. That is,

\displaystyle\begin{aligned}\left[T(fX_1,\dots,X_r,\alpha^1,\dots,\alpha^s)\right](p)&=T_p(\left[fX_1\right](p),\dots,X_r(p),\alpha^1(p),\dots,\alpha^s(p))\\&=T_p(f(p)X_1(p),\dots,X_r(p),\alpha^1(p),\dots,\alpha^s(p))\\&=f(p)T_p(X_1(p),\dots,X_r(p),\alpha^1(p),\dots,\alpha^s(p))\\&=\left[fT(X_1,\dots,X_r,\alpha^1,\dots,\alpha^s)\right](p)\end{aligned}

And a similar calculation holds for any of the other variables, vector or covector.

So each tensor field gives us a multilinear function $T:\mathfrak{X}M^{\times r}\times\mathfrak{X}^*M^{\times s}\to\mathcal{O}M$, and this multilinearity is not only true over $\mathbb{R}$ but over $\mathcal{O}M$ as well.

Conversely, let $T:\mathfrak{X}M^{\times r}\times\mathfrak{X}^*M^{\times s}\to\mathcal{O}M$ be an $\mathbb{R}$-multilinear function. If it’s also linear over $\mathcal{O}M$ in each variable, then it “lives locally”. That is, if $X_i(p)=Y_i(p)$ and $\alpha^j(p)=\beta^j(p)$ then

$\displaystyle\left[T(X_1,\dots,X_r,\alpha^1,\dots,\alpha^s)\right](p)=\left[T(Y_1,\dots,Y_r,\beta^1,\dots,\beta^s)\right](p)$

and so at each $p$ there is some tensor $T_p\in T^r_s\left(\mathcal{T}_pM\right)$ so that $T$ is a tensor field.

This is as distinguished from things like differential operators — $X\to L_Y(X)^1$, for instance — which fail both sides. On the one side, we can calculate

\displaystyle\begin{aligned}L_Y(fX)^1&=[Y,fX]^1\\&=f[X,Y]^1+Y(f)X^1\\&=fL_Y(X)^1+Y(f)X^1\end{aligned}

which picks up an extra term. It’s $\mathbb{R}$-linear but not $\mathcal{O}M$-linear. On the other side, the value of this function at $p$ doesn’t just depend on the value of $X$ at $p$, but on how $X$ changes through $p$. That is, this operator does not “live locally”, and is not a tensor field.

To prove this assertion, it will suffice to deal with the case where $T$ takes a single vector variable $X$, and we only need to verify that if $X_p=0$ then $\left[T(X)\right](p)=0$. Let $(U,x)$ be a chart around $p$, and write

$\displaystyle X=\sum\limits_{i=1}^nf^i\frac{\partial}{\partial x^i}$

where by assumption each $f^i(p)=0$. We let $V$ be a neighborhood of $p$ whose closure is contained in $U$. We know we can find a smooth bump function $\phi$ supported in $U$ and with $\phi(q)=1$ on $\bar{V}$.

Now we define vector fields $X_i=\phi\frac{\partial}{\partial x^i}$ on $U$ and $0$ on $M\setminus U$. Similarly we define $g^i=\phi f^i$ on $U$ and $0$ on $M\setminus U$. Then we can write

$\displaystyle X=\phi^2X+(1-\phi^2)X=\sum\limits_{i=1}^ng^iX_i+(1-\phi^2)X$

and thus

\displaystyle\begin{aligned}\left[T(X)\right](p)&=\sum\limits_{i=1}^ng^i(p)\left[T(X_i)\right](p)+(1-\phi(p)^2)\left[T(X)\right](p)\\&=\sum\limits_{i=1}^n\phi(p)f^i(p)\left[T(X_i)\right](p)+(1-\phi(p)^2)\left[T(X)\right](p)\\&=\sum\limits_{i=1}^n1\cdot0\cdot\left[T(X_i)\right](p)+(1-1^2)\left[T(X)\right](p)=0\end{aligned}

as asserted.