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Let $\mathfrak{A}$, $\mathfrak{B}$, and $\mathfrak{C}$ be subcategories of the category of Banach spaces (over $\mathbb{R}$). Suppose we have a functor $\lambda:\mathfrak{A}^{op}\times\mathfrak{B}\to \mathfrak{C}$.

Let $f:E'\to E$ be a morphism belonging to $\mathfrak{A}$, and let $g:F\to F'$ be a morphism belonging to $\mathfrak{B}$. (Note: These are morphisms of topological vector spaces).

Then we have a map $$\matrix{Hom(E',E) \times Hom(F,F')\to Hom(\lambda(E,F),\lambda(E',F'))\\ (f,g)\mapsto\lambda(f,g)}$$

We say $\lambda$ is of class $C^p$ if for all manifolds $U$, and any two $C^p$ morphisms $U\to Hom(E',E)$ and $U\to Hom(F,F')$, the composition $$U\to Hom(E',E) \times Hom(F,F')\to Hom(\lambda(E,F),\lambda(E',F'))$$ is also of class $C^p$. (Note: We can replace $\mathfrak{A}$ and $\mathfrak{B}$ with categories of tuples to generalize this to several variables. In fact, this is what we do below.)

It is not hard to show that this induces a unique functor $$\lambda_X:VB(X, \mathfrak{A})^{op}\times VB(X,\mathfrak{B})\to VB(X,\mathfrak{C}).$$ on vector bundles taking values in the appropriate vector bundle categories over $X$.

We define a tensor bundle of type $\mathbf{\lambda}$ on $X$ to be $\lambda_X(TX)=\lambda_X((TX,\dots,TX),(TX,\dots,TX))$, where $TX$ is the tangent bundle.

However, this doesn't agree with the definition given on Wikipedia or anywhere else I've looked.


  • Is this terminology nonstandard?

  • Is the notion itself nonstandard?

  • If the terminology is nonstandard, but the notion is standard, does it have a different name?

  • Is this definition useful?

  • Does this include more vector bundles as tensor bundles than the standard definition?

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By the way, for anyone who'd like to look up the section in the book, it's on page 58 of Fundamentals of Differential Geometry by Serge Lang. It should also be included in Differential and Riemannian Manifolds Chapter 3, Section 4. I don't have that book on hand, so I can't give you the page number. – Harry Gindi Mar 4 '10 at 0:06
I'd also just like to note that I believe that OP4 in the statement of Theorem 4.1 appears to have a typo. (I think it should read $\lambda_Y(h^*\alpha,h^*\beta)=h^*\lambda_(\alpha,\beta)$). I say this only because the notation $\lambda_Y^*$ does not appear to be defined in the text. Thanks. – Harry Gindi Mar 4 '10 at 1:41
Please check if you're concerned about the recent disappearing comments here. – Scott Morrison Mar 4 '10 at 5:08

2 Answers 2

up vote 5 down vote accepted

Is the notion non-standard? As Emerton says, the answer is no, except perhaps in minor details.

Is the terminology non-standard (and more permissive than the usual notion of tensor)? Yes, I'd say it is.

Because Lang allows arbitrary categories of Banach spaces, his notion is very general; by taking the categories small, one needn't have much functoriality at all. For instance, in the case where $\lambda$ has just one, covariant input, we can take $\mathfrak{B}$ to be the category with one object, $\mathbb{R}^n$, and morphisms $GL(n)$. Then $\lambda(\mathbb{R}^n)$ is a representation $V$ of $GL(n)$. On an $n$-manifold $X$, the resulting bundle $\lambda_X(TX)$ is usually known as an associated bundle. Form the principal $GL(n)$-bundle $Fr_X$ of frames (consisting of pairs of $x\in X$ with an isomorphism $T_x X \to \mathbb{R}^n$). Then $\lambda_X(TX)= Fr_X \times_{GL(n)}V$.

The notion of associated bundle is one that differential geometers often find more convenient than Lang's, because it allows one to say exactly which structure groups of interest (here, $GL(n,\mathbb{R})$) and exactly which representations.

A restrictive definition of tensor bundles allows those associated bundles where $V$ is a tensor product of copies of $\mathbb{R}^n$ and its dual. A broader (and I suspect fairly standard) definition allows subquotient representations of such - in particular, symmetric tensors and differential forms. But Lang apparently allows other things, e.g. $r$-densities (take the representation of $GL(n)\to \mathbb{R}^\times$ given by $g\mapsto |\det g|^r$).

Is the greater generality (compared to the standard notion of tensor) useful? In examples (e.g. densities) yes; in the abstract, not really - the interesting geometry is attached to specific classes of structure groups and their representations.

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He calls the more specific type of tensor a "multilinear tensor field". Thanks though. This is a great answer. – Harry Gindi Mar 4 '10 at 4:20

This is a slight variation of the standard definition, as far as I can tell.

First of all, let me restrict to the finite dimensional context, since this is more standard. Then a typical example of $\lambda$ is the functor which sends $V_1,\ldots,V_p,W_1,\ldots,W_q$ to $V_1^{\*}\otimes\cdots\otimes V_p^{\*}\otimes W_1\otimes \cdots \otimes W_q.$ The corresponding tensor bundle will be $(T_X^*)^{\otimes p}\otimes T_X^{\otimes q}$ (here I mean the usual tensor product of bundles), and I can imagine sections of this being referred to as tensors of type $(p,q)$ classically. (It may be that the $p$ and $q$ would be reversed; I would check the conventions carefully of any reference that used terminology of this kind.)

You could check in Spivak, or on Wikipedia, or in any number of other sources to see the various kinds of terminology that are used for this construction, but whatever the terminology, sections of these sorts of bundles are precisely what are referred to as tensors in classical differential geometry.

In more modern treatments, you may see less of this terminology, because people will just write out explicitly the tensor products of bundles as I did above. But this terminology evolved over a long period of time, and tensors in differential geometry were being considered well before the functorial notion of tensor product of vector spaces was introduced.

As for how more general Lang's definition is, I can't think of any other $\lambda$s of the top of my head (and it may be that, if you impose some natural axioms on $\lambda$, there essentially are no further examples). As far as I can tell, he has simply abstracted the properties you need to have a functor of vector spaces give rise to a corresponding functor of vector bundles.

[Edit: As pointed out in the comment below, and in Tim Perutz's answer, there are indeed other $\lambda$'s: e.g., there are symmetric tensors and exterior tensors (the latter giving differential forms, of course), which I mysteriously neglected when I wrote the above; one should certainly single them out, and my statement about there being no other $\lambda$s is wrong as it stands --- all these Schur-type functors are certainly candidate $\lambda$s. One thing I hadn't realized, which is pointed out by Tim Perutz, is that the case of densities is also included in Lang's definition.]

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There is a discussion of such $\lambda$s in Ivan Kolar, Jan Slovak and Peter W. Michor's awsome Natural operations in differential geometry (which one can get online). There are other examples (you can mix variously exterior powers, tensor products, duality, general Schur functors, etc) – Mariano Suárez-Alvarez Mar 4 '10 at 4:14
I'll have to check it out. Thanks for the reference. – Harry Gindi Mar 4 '10 at 4:34

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