Meaning of $B^2$ for $B\in \Omega^*(M,\operatorname{End}(E))$

Question

Let $E\rightarrow M$ a vector bundle over a smooth manifold and let $B\in \Omega^*(M,,\operatorname{End}(E))$ be an $,\operatorname{End}$-valued form. Is there a natural meaning to $B^2$?

I guess that if there is we would like to have $B^2\in \Omega^*(M,,\operatorname{End}(E))$ as well so if we write $B$ as a sum of $\omega\otimes f$ with $\omega\in\Omega^*(M)$ and $f\in\Gamma(,\operatorname{End}(E))$ we would want something like a sum of $$\omega\wedge \eta\otimes f\circ g$$ Is there something I am missing?

Answer 1

You are not missing anything, except this operation is usually denoted by $B \wedge B$ and not $B \circ B$ or $B^2$ (so that one won't be confused with the pointwise composition) although this notation also has its drawbacks. This is an instance of the following general construction:

Let's assume that $V,W$ are vector spaces over $k$ and consider the space $\Lambda(V) \otimes W$. You can always define the product of two elements in $\Lambda(V) \otimes W$ to be an element of $\Lambda(V) \otimes W \otimes W$ by the formula $$ (\alpha \otimes w) \wedge (\beta \otimes u) := (\alpha \wedge \beta) \otimes (w \otimes u). $$ If, in addition, $W$ has the structure of a $k$-algebra so that you have a map $m : W \otimes W \rightarrow W$, you can apply it to the $W \otimes W$ factor and get a multiplication map $\wedge_m \colon (\Lambda(V) \otimes W) \times (\Lambda(V) \otimes W) \rightarrow \Lambda(V) \otimes W$ defined by: $$ (\alpha \otimes w) \wedge_m (\beta \otimes u) := (\alpha \wedge \beta) \otimes m(w,u). $$ This resulting algebra $\Lambda(V) \otimes W$ is precisely the tensor product of the algebras $\Lambda(V)$ and $W$ as algebras.

The two most important cases that appear in Differential Geometry are where:

1. $W = \operatorname{End}(E)$ with the multiplication being composition. In this case, the corresponding product on $\Omega^{*}(M;\operatorname{End}(E))$ is usually denoted by $\wedge$ for the lack of better notation.
2. $W = \mathfrak{g}$ is a Lie algebra with the "multiplication" being the Lie bracket. In this case, the corresponding product on $\Omega^{*}(M;\mathfrak{g})$ is denoted by $\mu \wedge \nu$ or $[\mu \wedge \nu]$ (the latter is used to remind us that the product involves both $\wedge$ and $[,]$).

An example of a formula which uses this notation is the following "structure equation". If $\nabla$ is a connection on $E$ and you choose a local trivializing frame of $E$ over $U$, then $\nabla$ can be written on $U$ as $d + \omega$ where $d$ is the standard flat connection induced from the trivialization and $\omega \in \Omega^1(U;\operatorname{End}(E))$. Then $$ d\omega + \omega \wedge \omega = d \omega + \frac{1}{2} [\omega \wedge \omega] = R_{\nabla} $$ where $R_{\nabla} \in \Omega^2(U;\operatorname{End}(E))$ is the curvature and $\omega \wedge \omega$ is defined using the wedge product and the composition while $[\omega \wedge \omega]$ is defined using the wedge product and the Lie algebra structure on $\operatorname{End}(E)$.

One can also write this formula globally if you choose some connection $\nabla_0$ and write $\nabla = \nabla_0 + \omega$. Then $$ d_{\nabla_0}(\omega) + \omega \wedge \omega = R_{\nabla}. $$ The previous instance is obtained if you apply it locally with $\nabla_0 = d$ the standard flat connection associated to a specific trivialization.