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

Question

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 $T_{gamma (t_1)}Moplus...oplus T_{gamma (t_j)}M$ .  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 $V, Win Sigma$, 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.

doggerel · Answer

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

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

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

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 $V_0in mathbb{R}^n$ 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 $Wmapsto theta(tau_0, V_0, W)$ 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 $tauin (tau_0-delta, tau_0+delta)$ and whose value at $t=0$ is $Vin B(V_0, delta)$.

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

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