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

diff --git a/sections/applications.tex b/sections/applications.tex
@@ -487,8 +487,7 @@ however, is the following consequence of Morse's theorem.
   \begin{enumerate}
     \item If there are no conjugate points of \(\gamma\) then there exists a
       neighborhood \(U \subset \Omega_{p q} M\) of \(\gamma\) such that
-      \(E(\eta) > E(\gamma)\) and \(\ell(\eta) > \ell(\gamma)\) for all \(\eta
-      \in U\) with \(\eta \ne 0\).
+      \(E(\eta) > E(\gamma)\) for all \(\eta \in U\) with \(\eta \ne 0\).
 
     \item Let \(k > 0\) be the sum of the multiplicities of the conjugate
       points of \(\gamma\) in the interior of \(I\). Then there exists an
@@ -525,8 +524,7 @@ however, is the following consequence of Morse's theorem.
 
   As for part \textbf{(ii)}, fix \(\delta > 0\) and an orthonormal basis
   \(\{X_i : 1 \le i \le k\}\) of \(T_\gamma^- \Omega_{p q} M\) consisting of
-  eigenvectors of \(A_\gamma\) with negative eigenvalues \(- \lambda_i\).
-  Define
+  eigenvectors of \(A_\gamma\) with negative eigenvalues \(- \lambda_i\). Let
   \begin{align*}
     i : B^k & \to \Omega_{p q} M \\
     v & \mapsto \exp_\gamma(\delta (v_1 \cdot X_1 + \cdots + v_k \cdot X_k))
@@ -538,5 +536,6 @@ however, is the following consequence of Morse's theorem.
     E(i(v)) 
     = E(\gamma) - \frac{1}{2} \delta^2 \sum_i \lambda_i \cdot v_i + \cdots
   \]
-  we find that \(E(i(v)) < E(\gamma)\) for sufficiently small \(\delta\).
+  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}