# The Unapologetic Mathematician

## Tensor Bundles

We have a number of other constructions similar to the tangent bundle that will come in handy. These are all sort of analogues of certain constructions we already know about on vector spaces. Let’s review these first.

Taking the tensor product of vector spaces is old hat by now, as is using the dual space $V^*$. We’ll put them together by defining the space of “tensors of type $(r,s)$” as $\displaystyle T^r_s(V)=V\otimes\dots\otimes V\otimes V^*\otimes\dots\otimes V^*$

where we have $r$ copies of the vector space $V$ and $s$ copies of the dual space $V^*$. Vectors in $V$, then, are tensors of type $(1,0)$, while vectors in the dual space are tensors of type $(0,1)$.

We also know about the space of antisymmetric tensors of rank $k$ over a vector space. In particular, we’re most interested in carrying this construction out over the dual space: $\Lambda^*_k(V)=\Lambda_k(V^*)$. And of course we can take the direct sum of these spaces over all $k$ to get the exterior algebra $\Lambda^*(V)$.

Now, we will take these constructions and apply them to the tangent spaces to a manifold. We define the bundle of tensors of type $(r,s)$ over $M$: $\displaystyle T^r_s(M)=\bigcup\limits_{p\in M}T^r_s(\mathcal{T}_pM)$

the “exterior $k$-bundle” over $M$: $\displaystyle \Lambda^*_k(M)=\bigcup\limits_{p\in M}\Lambda_k(\mathcal{T}^*_pM)$

and the exterior algebra bundle over $M$: $\displaystyle \Lambda^*(M)=\bigcup\limits_{p\in M}\Lambda(\mathcal{T}^*_pM)$

The trick here is that for each of these constructions, if we have a basis of $V$ we automatically get a basis of each space $T^r_s(V)$, $\Lambda^*_k(V)$, and $\Lambda(V)$. If we start with a coordinate patch $(U,x)$ on $M$, we get a basis $\frac{\partial}{\partial x^i}$ of $\mathcal{T}_pM$ for each $p\in U$. Then, just as we did with the tangent bundle and the cotangent bundle we can come up with a coordinate patch “induced by $(U,x)$” on each of our new bundles. In this way, we can turn them from disjoint unions of vector spaces into manifolds of their own right, each with a smooth projection down to $M$.

Now we can define a “tensor field of type $(r,s)$” on an open region $U\subseteq M$ as a section of the projection $\pi:T^r_s(U)\to U$. That is, it’s a smooth map $t:U\to T^r_s(U)$ such that $\pi(t(p))=p$. Similarly, we define a “differential $k$-form” over $U$ to be a section of the projection $\pi:\Lambda^*_k(U)\to U$.

July 6, 2011