# Two connections on a smooth $SO(3)$-vector bundle

Let $$E\to M$$ be a smooth oriented real vector bundle of rank 3, so that its structure group can be reduced to $$SO(3)$$. Suppose $$E\to M$$ is given a Riemannian metric and a connection $$\nabla:\Omega^0(E)\to \Omega^1(E)$$ compatible to it.

Consider its endomorphism bundle $$\text{End}(E)$$. $$\nabla$$ induces a connection (also denoted by $$\nabla$$) on $$\text{End}(E)$$ satisfying $$(\nabla \phi)(\sigma)=\nabla(\phi(\sigma))-\phi(\nabla\sigma)$$ (cf. exterior covariant derivative of $\operatorname{End}(E)$-valued $p$-form).

Now consider the adjoint bundle $$\text{Ad}(E)$$ of $$E$$. The Lie algebra $$\mathfrak{g}\subset \text{End}(\Bbb R^3)$$ of $$SO(3)$$ is isomorphic to $$\Bbb R^3$$, via the isomorphism $$\Bbb R^3\to \mathfrak{g}$$, $$v\mapsto (v\times -)$$, where $$\times$$ is the cross product of $$\Bbb R^3$$. This gives an isomorphism $$E\to \textrm{Ad}(E)$$. Also the connection $$\nabla$$ on $$\text{End}(E)$$ defined above restricts to a connection on $$\text{Ad}(E)$$.

So now we have two connections on $$E\cong \text{Ad}(E)$$ (denoted by the same symbol). But are these two the same? It seems quite natural to be the same, but I'm not sure

More formally, the section $$c\in\Gamma\mathrm{Hom}(E,\mathrm{Ad}(E))$$ defined by $$c(\sigma)(\tau)=\sigma\times\tau$$ is parallel with respect to the induced connection, which means that for all $$\sigma\in\Gamma E$$ $$\nabla \,c(\sigma)=c(\nabla\sigma)$$ This is equivalent to the fact that $$\nabla(\sigma\times\tau)=\nabla\sigma\,\times\tau+\sigma\times\nabla\tau$$ for all $$\sigma,\tau\in\Gamma E$$. To show this, fix $$p\in M$$ and a local positively oriented orthonormal frame $$e_0,e_1,e_2$$ with $${\nabla e_i}_{|p}=0$$ and write $$\sigma=\sigma^ie_i$$, $$\tau=\tau^ie_i$$ using the double summation convention. Now consider indices modulo $$3$$, so that $$e_{i}\times e_{i+1}=e_{i+2}$$. Then at $$p$$
$$\nabla(\sigma\times\tau) =\nabla(\sigma^i\tau^{i+1}e_{i+2}-\sigma^i\tau^{i+2}e_{i+1})\\ =(d\sigma^i)\tau^{i+1}e_{i+2}-(d\sigma^i)\tau^{i+2}e_{i+1}+\sigma^i(d\tau^{i+1})e_{i+2}-\sigma^i(d\tau^{i+2})e_{i+1}\\ =\nabla\sigma\,\times\tau+\sigma\times\nabla\tau$$