# Space of Alternating $k$-Tensors Notation

I will be taking a Differential Geometry class in the Fall, so I decided to get somewhat of a head start by going through Spivak's "Calculus on Manifolds." Before reading, though, I saw the Addenda at the end, which stated that his notation $\Lambda^{k}\left(V\right)$ for the space of alternating $k$-tensors was incorrect, although it is naturally isomorphic to $\Lambda^{k}\left(V^{*}\right)$ for fin. dim. $V$ (and, after a little more digging around on Wikipedia, is naturally isomorphic to $\left(\Lambda^{k}\left(V\right)\right)^{*}$ in general). Is the notation suggested by Spivak, $\Omega^{k}\left(V\right)$, standard or is there some other notation that is typically used?

EDIT: To quote Spivak: "Finally, the notation $\Lambda^{k}\left(V\right)$ appearing in this book is incorrect, since it conflicts with the standard definition of $\Lambda^{k}\left(V\right)$ (as a certain quotient of the tensor algebra of $V$). For the vector space in question (which is naturally isomorphic to $\Lambda^{k}\left(V^{*}\right)$ for finite dimensional vector spaces $V$) the notation $\Omega^{k}\left(V\right)$ is probably on the way to becoming standard."

I don't know if this is still the case, though, or if his use of $\Lambda^{k}\left(V\right)$ became the standard.

## 2 Answers

In Lee's 'Intro to Smooth Manifolds', $\Lambda^k(V)$ refers to the space of alternating $k$-tensors on a vector space $V$, as you mentioned. However, the space $\Omega^k(M)$ is the space of smooth $k$-forms on a smooth manifold $M$. That is, an element of $\omega \in \Omega^k(M)$ is a smooth map $M \to \Lambda^k(T^* M)$ (called a smooth section of of the bundle $\Lambda^k(T^* M)$), so for each point $x \in M$, we get an alternating $k$-tensor $\omega(x) \in \Lambda^k(T^* M)$. This space is often written as $\Omega^k(M) = \Gamma(\Lambda^k(T^* M))$.

Not entirely sure however what $\Omega^k(V)$ is, when $V$ is just a vector space.

• Small clarification: In ISM (2nd ed.), I use $\Lambda^k(V)$ to denote the space of contravariant alternating $k$-tensors on $V$, and $\Lambda^k(V^*)$ for the space of covariant ones. Then $\Lambda^k(T^*M)$ is the bundle of covariant alternating $k$-tensors on $M$, and as you said, $\Omega^k(M)$ is shorthand for the space of $k$-forms, which are global smooth sections of $\Lambda^k(T^*M)$. I chose these notations because they seem to be the most common in the literature. (In the first edition of the book, I used different notations, heavily influenced by Spivak.) – Jack Lee Jul 16 '14 at 0:01
• To quote Spivak: "Finally, the notation $\Lambda^{k}\left(V\right)$ appearing in this book is incorrect, since it conflicts with the standard definition of $\Lambda^{k}\left(V\right)$ (as a certain quotient of the tensor algebra of $V$). For the vector space in question (which is naturally isomorphic to $\Lambda^{k}\left(V^{*}\right)$ for finite dimensional vector spaces V) the notation $\Omega^{k}\left(V\right)$ is probably on the way to becoming standard." I don't know if this is still the case, though, or if his use of $\Lambda^{k}\left(V\right)$ became the standard. – Brian Jul 16 '14 at 0:13
• I just realized that I had forgotten to accept this answer. I have accepted it now. – Brian Sep 5 '14 at 4:33

Something like $\mathcal{A}^k(V)$ (Munkres, Analysis on Manifolds, and Loomis and Sternberg, Advanced Calculus) or $A_k(V)$ (Tu, An Introduction to Manifolds) seems to be used by several texts for alternating $k$-linear functions (tensors) $T:V^k\to \mathbb{R}$. On the other hand, $\Omega^k(U)$ seems to be standard for the space of (smooth) $k$-forms on open set $U$.

$\bigwedge^k(T^*_p M)$ is a space that is isomorphic to (and the usual notation for) $\mathcal{A}^k(T_p M)$, the space of alternating $k$-tensors on tangent space $T_pM$, so that if $\omega\in\Omega^k(M)$, $\omega(p)=\omega_p\in\bigwedge^k T^*_p M$.