global-analysis-and-the-banach-manifold-of-class-h1-curvers

diff --git a/sections/applications.tex b/sections/applications.tex
@@ -13,10 +13,10 @@ Morse index theorem and the Jacobi-Darboux theorem. We start by defining\dots
   variational vector field of \(\{ \gamma_t \}_t\)}.
 \end{definition}
 
-We should note that the previous definition encompasses the classical definition
-of a variation of a curve, as defined in \cite[ch.~5]{gorodski} for instance:
-any piece-wise smooth function \(H : I \times (-\epsilon, \epsilon) \to M\)
-determines a variation \(\{ \gamma_t \}_t\) as in
+We should note that the previous definition encompasses the classical
+definition of a variation of a curve, as defined in \cite[ch.~5]{gorodski} for
+instance: any piece-wise smooth function \(H : I \times (-\epsilon, \epsilon)
+\to M\) determines a variation \(\{ \gamma_t \}_t\) as in
 definition~\ref{def:variation} given by \(\gamma_t(s) = H(s, t)\). This is
 representative of the theory that lies ahead, in the sense that most of the
 results we'll discuss in the following are minor refinements to the classical
@@ -482,7 +482,7 @@ point with the notion of focal points of \(N\) -- see theorem 7.5.4 of
 \cite{gorodski} for the classical approach. What we are really interested in,
 however, is the following consequence of Morse's theorem.
 
-\begin{theorem}[Jacobi-Darboux]
+\begin{theorem}[Jacobi-Darboux]\label{thm:jacobi-darboux}
   Let \(\gamma \in \Omega_{p q} M\) be a critical point of \(E\).
   \begin{enumerate}
     \item If there are no conjugate points of \(\gamma\) then there exists a
@@ -539,3 +539,20 @@ however, is the following consequence of Morse's theorem.
   we find that \(E(i(v)) < E(\gamma)\) for sufficiently small \(\delta\). In
   particular, \(\ell(i(v)) \le E(i(v))^2 < E(\gamma)^2 = \ell(\gamma)\).
 \end{proof}
+
+We should point out that part \textbf{(i)} of theorem~\ref{thm:jacobi-darboux}
+is weaker than the classical formulation of the Jacobi-Darboux theorem -- such
+as in theorem 5.5.3 of \cite{gorodski} for example -- in two aspects. First, we
+do not compare the length of curves in question. This could be amended by
+showing that the length functional \(\ell : H^1(I, M) \to \RR\) is smooth and
+that its Hessian \(d^2 \ell_\gamma\) is given by \(C \cdot d^2 E_\gamma\) for
+some \(C > 0\). 
+
+Secondly, inlike the classical formulation we only consider curves in an
+\(H^1\)-neighborhood of \(\gamma\) -- instead of a neighborhood of \(\gamma\)
+in \(\Omega_{p q} M\) in the uniform topology. On the other hand, part
+\textbf{(ii)} is definitevely an improvement of the classical formulation: we
+can find curves \(\eta = i(v)\) shorter than \(\gamma\) already in an
+\(H^1\)-neighborhood of \(\gamma\).
+
+We hope that short notes could provide the reader