I am trying to work out Lemma 4.14 of the book "Heat Kernels and Dirac Operators" by Berline and Getzler. I am stuck with the proof. For the sake of brevity I am uploading a picture from the book before showing what I have tried.
Here's what I have proved.
Let $M$ be a Riemannian manifold and $V\rightarrow M$ be a complex vector bundle. Then we fix a point $q_0$ and trivialize $E$ on a geodesic-ally convex neighbourhood using normal co-ordinates and parallel transport as has been done in this section. Let $\nabla$ be a connection on $V$ and we can write on the trivializing neighbourhood $\nabla s_\alpha=\sum_\beta \omega_{\beta \alpha}\otimes s_\beta$ where $d$ is the trivial connection in terms of the local frame field $\{s_\alpha \}_\alpha$ obtained by parallel transporting a frame at the point $0$. Then we have
$\omega(0)=0$
Let $F_{\nabla}=\displaystyle{\sum_{i<j}}F_{ij}dx_i\wedge dx_j$ where $F_{ij}(x)=F_\nabla(\partial^i,\partial^j)(x)$.
Then we get around $0$, $$\omega(x)=-\frac{1}{2}\sum_{i,j}F_{ij}(0)x_jdx_i+O(|x|^2)$$ $\nabla_{\partial^i}=\partial^i-\displaystyle{\frac{1}{2}\sum_{j=1}^n}{F_\nabla(\partial^i,\partial^j)_0}x_j+O(|x|^2)$
Using this I am trying to get the more precise asymptotic result for a Dirac bundle $\mathcal E$ on $M$ with a compatible Clifford connection $\nabla^\mathcal E$. It will be great if someone can explain the proof to me or help me complete the proof. Thanks a lot in advance.