# Topology and Smooth Structure on the Bundle of Covariant $k$-Tensors

(All vector spaces are finite dimensional and real).

Given a vector space $V$, let $T^k(V^*)$ denote the vector space of all the covariant $k$ tensors on $V$.

Following Lee's Introduction to Smooth manifolds, 2nd edition, pg. 316:

Given a smooth manifold $M$, we define the bundle of covariant $k$-tensors on $M$ as $$T^k(T^*M)=\bigsqcup_{p\in M}T^k(T^*_pM)$$

This just defines a set.

I am wondering how to give a smooth structure to this set. Lee doesn't explicitly discuss what the smooth structure is on this.

Can somebody help me with this?

• He has a lemma somewhere in the first few chapters about constructing vector bundles as a disjoint union of vector spaces parametrized by points in a manifold. It's called something like "bundle construction theorem". I don't have the reference at hand. In any case, the construction is as follows: you topologize the subset where $p$ varies inside a coordinate patch $U$ as a product $U\times\mathbb R^N$ for the appropriate $N$ using pure tensors $\otimes d\,x^i$ as a basis for the linear part, and you check that the transition functions are smooth. – Olivier Bégassat Mar 4 '15 at 0:45
• Possibly called the "smooth bundle construction lemma"? – Olivier Bégassat Mar 4 '15 at 0:51
• @Olivier is right -- but the lemma in question is called the "vector bundle chart lemma" (Lemma 10.6). You might also look at the proof of Proposition 11.9 for inspiration. – Jack Lee Mar 4 '15 at 1:17
• @JackLee Thank you Professor Lee. Your comment really helped. – caffeinemachine Mar 5 '15 at 20:50