## Background

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.

## Questions

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?

Fundamentals of Differential Geometryby Serge Lang. It should also be included inDifferential and Riemannian ManifoldsChapter 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 0:06thinkit 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