The Unapologetic Mathematician

Mathematics for the interested outsider

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.

About these ads

July 6, 2011 - Posted by | Differential Topology, Topology

6 Comments »

  1. [...] as for vector fields, we need a good condition to identify tensor fields in the wild. And the condition we will use is similar: if is a smooth tensor field of type , then [...]

    Pingback by Identifying Tensor Fields « The Unapologetic Mathematician | July 7, 2011 | Reply

  2. [...] we build up the coordinates on the tensor bundles as the canonical ones induced on the tensor spaces by the coordinate bases on and , we immediately [...]

    Pingback by Change of Variables for Tensor Fields « The Unapologetic Mathematician | July 8, 2011 | Reply

  3. [...] tensor field over a manifold gives us a tensor at each point $latex . And we know that can be considered as [...]

    Pingback by Tensor Fields and Multilinear Maps « The Unapologetic Mathematician | July 9, 2011 | Reply

  4. [...] John Armstrong: Distributions, Integral Submanifolds, The Hopf Fibration, Tensor Bundles [...]

    Pingback by Ninth Linkfest | July 9, 2011 | Reply

  5. [...] defined the exterior bundle over a manifold . Given any open we’ve also defined a -form over to be a section of this [...]

    Pingback by The Algebra of Differential Forms « The Unapologetic Mathematician | July 12, 2011 | Reply

  6. [...] a Riemannian metric on a manifold is nothing more than a certain kind of tensor field of type on . At each point , the field gives us a [...]

    Pingback by (Pseudo-)Riemannian Metrics « The Unapologetic Mathematician | September 20, 2011 | Reply


Leave a Reply

Fill in your details below or click an icon to log in:

WordPress.com Logo

You are commenting using your WordPress.com account. Log Out / Change )

Twitter picture

You are commenting using your Twitter account. Log Out / Change )

Facebook photo

You are commenting using your Facebook account. Log Out / Change )

Google+ photo

You are commenting using your Google+ account. Log Out / Change )

Connecting to %s

Follow

Get every new post delivered to your Inbox.

Join 366 other followers

%d bloggers like this: