I with differential geometry computing connection forms. Please respond

[jdinatale]

I need help with Part (b). I finished part (a) and attached it as well. My issue comes from how to apply the definition of connection forms to compute them. The definition states: Let E_1, E_2, E_3 be a frame field on R^3. For each tangent vector v at R^3 at the point p let $\omega_{ij}$(v )= $\nabla_v$ E_i $\cdot$E_j (p), $(i \leq i, j \leq 3)$.

 

[lippyka]

(a) Show that \omega_{ij} is a connection form, i.e., for each p in R^3 and for each v,w in T_pR^3, \omega_{ij}(v + w) = \omega_{ij}(v ) + \omega_{ij}(w ).Let E_1, E_2, E_3 be a frame field on R^3. For each tangent vector v at R^3 at the point p let \omega_{ij}(v )= \nabla_v E_i \cdotE_j (p), (i \leq i, j \leq 3).We must show that \omega_{ij}(v + w) = \omega_{ij}(v ) + \omega_{ij}(w ) for all vectors v,w in T_pR^3.We can rewrite \omega_{ij}(v + w) as: \begin{align*}\omega_{ij}(v + w) &= \nabla_{v + w} E_i \cdotE_j (p) \\&= \nabla_v E_i \cdotE_j (p) + \nabla_w E_i \cdotE_j (p) \\&= \omega_{ij}(v ) + \omega_{ij}(w )\end{align*}Therefore, \omega_{ij} is a connection form.(b) Compute the connection forms for the flat Euclidean metric. The flat Euclidean metric is given by \delta_{ij} = \langle E_i, E_j \rangle. By definition of connection forms, we have \omega_{ij}(v )= \nabla_v E_i \cdotE_j (p), (i \leq i, j \leq 3). To compute the connection forms, we need to find \nabla_v E_i. We know that \nabla_v E_i = \partial_v E_i + \Gam