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}