I'm reading John Lee's Introduction to Smooth Manifolds, and I got stuck in the definition of the bundle $\Lambda^kT^*M$.
Let $M$ be a $n$ dimensional smooth manifold, $\Lambda^k(T^*_pM)$ be the $\frac{n!}{k!(n-k)!}$ dimensional subspace of tensor product $T^k(T^*_pM)$, then one defines $\Lambda^kT^*M:=\bigsqcup_{p\in M}\Lambda^k(T^*_pM)$. Due to John Lee's Lemma 10.32, I need to find $\frac{n!}{k!(n-k)!}$ local sections $\sigma_i:U\to T^kT^*M$ of the tensor bundle $T^kT^*M$ on some open neighbourhood $U\subset M$ of $p$, such that $\{\sigma_i(q)\}\subset\Lambda^k(T^*_qM)$ is a basis of the vector space, for any $q\in U$. Then $\Lambda^kT^*M$ is a subbundle of $T^kT^*M$ by the lemma.
Here is my question: How do I find these local sections?Can I just claim that they are those who map a point of the manifold to an alternating tensor?
Best Answer
It's necessary to argue that such sections actually exist, such as by constructing them.
Choosing $U$ contained in a single coordinate chart, the standard choice of local frame with indices $1\le i_1<i_2<\cdots<i_k\le n$ is the antisymmetrized product of coordinate $1$-forms. $$ dx^{i_1}\wedge\dots\wedge dx^{i_k}:=\sum_{\pi\in S_k}\operatorname{sgn}(\pi)dx^{i_{\pi(1)}}\otimes\dots\otimes dx^{i_{\pi(k)}} $$ Where $S_k$ is the set of permutations of $\{1,\cdots,k\}$ and $\operatorname{sgn}$ denotes the sign of a permutation. One can verify that these are smooth in $T^kT^*M$ and span $\Lambda^kT^*M$.