Question about the proof of the index theorem appearing in Milnor's Morse Theory

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.

Solution 1:

After Sleeping for 12 Hours: This seems like a possible solution. I'd be grateful for any corroboration because I wouldnt expect Milnor to dismiss something as "evident" if this were the easiest way to prove it:

Using the second variation formula it seems as though the problem reduces to the following: Let $\gamma: [0,1]\rightarrow U$ be a geodesic s.t. any two points in $U$ are connected by a minimizing geodesic which depends smoothly on the endpoints. Show that $(0,1)\times T_{\gamma(0)}\rightarrow T_{\gamma(0)}, (\tau, V)\mapsto D_t(V_\tau)|_{t=0}$ is continuous (where $V_\tau$ is the unique Jacobi field along $\gamma$ with $V_\tau(0)= V, V_\tau(\tau)=0$).

$proof$: After choosing a parallel orthonormal frame along $\gamma$, Jacobi fields are just projections of integral curves to a vector field on $[0,1]\times \mathbb{R}^{2n}$. If $0<\tau_0<1$ then by the ODE theorem and the linearity of the Jacobi equation, there exists $\epsilon>0$ such that

$\Theta: (\tau_0-\epsilon,\tau_0+\epsilon)\times \mathbb{R}^{2n}\rightarrow (\tau_0-\epsilon,\tau_0+\epsilon)\times\mathbb{R}^{2n}$

$(\tau, V,W) \mapsto (\tau, X_{t=\tau}, D_t(X)|_{t=\tau})$

is a smooth map. Where $X$ is the unique Jacobi field along $\gamma$ with $X_{t=0}=V, D_t(X)|_{t=0}=W$ (All of these vectors are coordinates w.r.t. the parallel frame). Now postcompose $\Theta$ with projection onto $X_{t=\tau}$ and call this composition $\theta$. Then I claim that the implicit function theorem applies for the zero vector in the image of $\theta$. Specifically for any $V_0\in \mathbb{R}^n$ let $W_0$ such that $\theta(\tau_0,V_0,W_0)=0$ (that such a vector $W_0$ exists is a fact about Jacobi fields along geodesics in open sets like $U$). The hypotheses of the implicit function theorem are met because $W\mapsto \theta(\tau_0, V_0, W)$ is a linear isomorphism. So we get a smooth map $(\tau_0-\delta, \tau_0+\delta)\times B(V_0, \delta)\rightarrow \mathbb{R}^n$ which gives the (coordinates of the) unique vector in $T_{\gamma(0)}M$ which is the covariant derivative of the Jacobi field vanishing at $\tau\in (\tau_0-\delta, \tau_0+\delta)$ and whose value at $t=0$ is $V\in B(V_0, \delta)$.