In my Global Analysis course at university we saw the notion of Euclidean connection on vector bundle and I do not quite understand its definition. Let me recall the settings.
Let $\pi: E \to M$ be a smooth vector bundle. We define a connection on $E$ as an $\mathbb R$-linear map $$\nabla: \Gamma(E) \to \Omega^1(E) = \Gamma(T^*M \otimes E) = \Omega^1(M)\otimes \Gamma(E),$$ such that $$\nabla (fs) = df \otimes s + f \nabla s$$ for all $f \in C^\infty(M)$ and $s \in \Gamma(E)$. Now we say that $E$ has a Euclidean structure if there exists $\langle \cdot, \cdot \rangle \in \Gamma(E^*\otimes E^*)$ such that $$\langle v, w \rangle = \langle w, v \rangle, \quad \text{for } v \neq 0, ~\langle v, v \rangle > 0$$ for $v, w \in \Gamma(E)$. Afterwards, my teacher said that a connection $\nabla$ was called Euclidean if $$d\langle v, w \rangle = \langle \nabla v, w \rangle + \langle v, \nabla w \rangle $$ for $v, w \in \Gamma(E)$. However, I am not sure to see what he meant by this, because $\nabla v \in \Omega^1(M)\otimes \Gamma(E)$ so that $\langle \nabla v, w \rangle$ doesn't really make sense. If we write $$\nabla v = \sum_{j = 1}^n v_j \otimes e_j,$$ can we say that $\langle \nabla v, w \rangle$ is in fact equal to $$\sum_{j = 1}^n v_j \otimes \langle e_j, w \rangle \in \Omega^1(M)?$$ This makes sense because $\langle v, w \rangle \in C^\infty(M)$ so that $d\langle v, w \rangle \in \Omega^1(M)$. But still, I am not totally sure. Could one of you enlighten me with this ? Maybe there is a reference where all of this is well-explained ?
It could help to see $\Gamma(T^{*}M \otimes E)$ simply as $\text{Hom}(TM,E)$. This is the case because if $\sigma\in\Gamma(T^{*}M \otimes E)$, then for every $p\in M$ we have $$\sigma_{p}\in \left(T_{p}M\right)^{*}\otimes E_p\simeq \text{Hom}(T_p M,E_p).$$ This explains why we sometimes call $\sigma\in\Gamma(T^{*}M \otimes E)$ a differential $1$-form with values in $E$. Now, if $v,w\in\Gamma(E)$, then $\nabla v\in \text{Hom}(TM,E)$ and therefore the differential $1$-form $\langle \nabla v,w\rangle\in\Omega^{1}(M)$ is defined simply as $$\langle \nabla v,w\rangle_{p}(u):=\langle \nabla v\vert_{p}(u), w\vert_{p}\rangle_{p}\in\mathbb{R}$$ for all $p\in M$ and $u\in T_{p}M$.