When I read the 9.5 of Topping's Lectures on the Ricci flow, I have some problem.
Assume $W$ is a vector bundle over manifold $(M,g)$, and $A$ is connection on $W$. $\{e_1,...,e_l\}$ is a frame of $W_p$, and extend it to a local frame for $W$ by radial parallel translation using the connection $A$, then we have $Ae_i=0 $, and $$ A^2e_i=-\frac{1}{2} R_A(\cdot, \cdot)e_i \tag{1} $$ where $R_A$ is the curvature of the connection $A$. I don't know how to get $(1)$. In my view, for any $v_a,v_b\in T_pM$ $$ R_A(v_a,v_b)e_k= A_a A_b e_k - A_b A_a e_k $$ where $A_ae_k=A_{v_a}e_k$. How it imply $(1)$ ?
PS(2022-7-4): I use the notation of chapter 4 of S.S. Chern and W.H. Chen's 《微分几何讲义》.
Assume the connection matrix of $W$ is $$ \omega= \begin{pmatrix} \omega_1^1 ,...\omega_1^l \\ ~~\\ \omega_1^l ,...\omega_l^l \end{pmatrix} $$ where $\omega_a^b=\Gamma_{ai}^b dx^i$. And the curvature matrix is $\Omega= d\omega -\omega\wedge \omega$, namely $$ \Omega= \begin{pmatrix} d\Gamma_{ai}^b \wedge dx^i - \omega_a^r\wedge \omega_r^b \end{pmatrix} $$ Therefore, the curvature operator is $$ R_A(\cdot, \cdot) e_a = \begin{pmatrix} d\Gamma_{ai}^b \wedge dx^i - \omega_a^r\wedge \omega_r^b \end{pmatrix} \otimes e_b $$ where $e_a=\frac{\partial}{\partial x_a}$ is tangent vector. I use the definition $$ e_1\wedge e_2 = e_1 \otimes e_2 - e_2 \otimes e_1 $$ Then, I have $$ R_A(\cdot, \cdot) e_a = d\Gamma_{ai}^r\otimes dx^i \otimes e_r - dx^i \otimes d\Gamma_{ai}^r\otimes e_r \\ -\Gamma_{ai}^b \Gamma_{bj}^r dx^i\otimes dx^j\otimes e_r +\Gamma_{bi}^r \Gamma_{aj}^b dx^i\otimes dx^j\otimes e_r \\ =\left( \frac{\partial \Gamma_{aj}^r}{\partial x_i} -\frac{\partial \Gamma_{ai}^r}{\partial x_j} -\Gamma_{ai}^b\Gamma_{bj}^r +\Gamma_{bi}^r\Gamma_{aj}^b \right) dx^i\otimes dx^j \otimes e_r \tag{2} $$ On the other hand, first, I have $$ A e_a = \Gamma_{ai}^b dx^i\otimes e_b $$ Then $$ A^2 e_a = A( \Gamma_{ai}^b dx^i \otimes e_b )\\ = d\Gamma_{ai}^b \otimes dx^i \otimes e_b - \Gamma_{ai}^b \Gamma_{cd}^i dx^c \otimes dx^d\otimes e_b + \Gamma_{ai}^b \Gamma_{bk}^r dx^i \otimes dx^k \otimes e_r \\ =\left( \frac{\partial \Gamma_{aj}^r}{\partial x_i} -\Gamma_{ak}^r\Gamma_{ij}^k +\Gamma_{ai}^b\Gamma_{bj}^r \right)dx^i\otimes dx^j \otimes e_r \tag{3} $$ Then, I think I should use an analogue of Bianchi to get $(1)$ from $(2)$ and $(3)$. But now, I still don't know how to use a suitable Bianchi.
Besides, of course, if assume local trivialization, (2) and (3) will become much simple.
PS(2022-7-5): If I use the geodesic normal coordinate at $p$, then (2) and (3) become $$ R_A(\cdot, \cdot) e_a = \begin{pmatrix} d\Gamma_{ai}^b \wedge dx^i - \omega_a^r\wedge \omega_r^b \end{pmatrix} \otimes e_b $$ where $e_a=\frac{\partial}{\partial x_a}$ is tangent vector. I use the definition $$ e_1\wedge e_2 = e_1 \otimes e_2 - e_2 \otimes e_1 $$ Then, I have $$ R_A(\cdot, \cdot) e_a = \left( \frac{\partial \Gamma_{aj}^r}{\partial x_i} -\frac{\partial \Gamma_{ai}^r}{\partial x_j} \right) dx^i\otimes dx^j \otimes e_r \tag{4} $$ $$ A^2 e_a = \frac{\partial \Gamma_{aj}^r}{\partial x_i} dx^i\otimes dx^j \otimes e_r \tag{5} $$ The power series expansion of the metric in the coordinate is $$ h_{ij}(x) = \delta_{ij} -\frac{1}{3} R_{ikjl}(p) x^k x^l + O(|x|^3) \tag{6} $$ where $x$ is in the geodesic normal coordinate of $p$, $h$ is the metric of $W$, and $R$ is cuvature of connection $A$. But I want to show (1) at $p$, seemly, it has nothing to the point $x$.
PS(2022-7-10): In fact, the next calculation is finished in two or three days ago. Just since feeling boring, I stopped.
First, since $Ah=0$, we have $$ \Gamma_{ij}^m = \frac{1}{2}h^{lm} (\partial_i h_{jl}+ \partial_j h_{li} - \partial_l h_{ij}) $$ Therefore, in the geodesic normal coordinate of $p$, we have $$ \partial_k \Gamma_{ij}^m(p) = \frac{1}{2}h^{lm} (\partial_k\partial_i h_{jl}+ \partial_k\partial_j h_{li} - \partial_k\partial_l h_{ij})|_{x=p} $$ Using (6) to calculate $\partial_k\partial_i h_{jl} (p)$ and $h^{lm} =\delta^{lm}$,we have $$ \partial_k \Gamma_{ij}^m(p) = -\frac{1}{3} (R_{mijk}(p) +R_{mjik}(p)) \tag{7} $$ From (4),(5), we have $$ R_A(\cdot, \cdot) e_a =R_{raij} dx^i \otimes dx^j \otimes e_r \tag{8} $$ $$ A^2e_a = \frac{1}{3}(R_{raij} - R_{rjai}) dx^i\otimes dx^j\otimes e_r \tag{9} $$ Obviously, they are not sustaining the (1). Besides, I am sure (7) is right, since before I calculated it , I found it in Wiki. The possible error is the miscalculate of (8) and (9). But, in fact, I have done 5 or 6 times, and not sustaining the (1). Therefore, I feel boring...