I am trying to get through the proof of the index theorem.
The background: I have been stuck for quite a while on the following point which Milnor says is evident:
Let $\gamma: [0,1]\rightarrow M$ be a geodesic in a Riemannian manifold. Let $(t_0=0, t_1,…,t_k=1)$ be a partition of $[0,1]$ so that $\gamma$ sends $[t_i, t_{i+1}]$ into an open set U with the property that any two points in U can be connected by a distance minimizing geodesic which depends smoothly on the two endpoints. If $\tau$ is between $t_j$ and $t_{j+1}$ then the space of "broken" Jacobi fields along $\gamma|_{[0,\tau]}$ (i.e. those piecewise smooth $V$ which are Jacobi fields along each piece of the partition of $[0, \tau]$) which vanish at $t=0$ and $t=\tau$ is isomorphic as a real vector space to the direct sum $T_{\gamma (t_1)}M\oplus...\oplus T_{\gamma (t_j)}M$ . Call this latter sum $\Sigma$. Then the Hessian of the energy function associated with $\gamma|_{[0,\tau]}$ (call it $E_\tau$) can be viewed as a bilinear form on $\Sigma$.
My question: I want to know why this bilinear form should vary continuously with $(\tau, V, W) \in (t_j,t_{j+1})\times \Sigma \times \Sigma$. I.e. if $V_\tau$ and $W_\tau$ are the broken Jacobi fields along $\gamma|_{[0,\tau]}$ associated with $V, W\in \Sigma$, why is $(t_j,t_{j+1})\times \Sigma \times \Sigma \rightarrow \mathbb{R}, (\tau, V, W) \mapsto E_\tau (V_\tau , W_\tau )$ continuous?
Where I am struck: From the second variation formula it seems as though I should start by proving that $D_t(V_\tau|_{[t_j,\tau]} )|_{t_j}$ varies continuously with $(\tau, V)$. I'm having trouble showing this though.