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

diff --git a/sections/applications.tex b/sections/applications.tex
@@ -493,7 +493,7 @@ however, is the following consequence of Morse's theorem.
         i : B^k \to \Omega_{p q} M
       \]
       of the unit ball \(B^k = \{v \in \RR^k : \norm{v} < 1\}\) with \(i(0) =
-      \gamma\), \(E(i(v)) < E(\gamma)\) and \(\ell(i(v)) < \ell(\gamma)\) for
+      \gamma\), \(E(i(v)) < E(\gamma)\) and \(L(i(v)) < L(\gamma)\) for
       all nonzero \(v \in B^k\).
   \end{enumerate}
 \end{theorem}
@@ -535,15 +535,15 @@ however, is the following consequence of Morse's theorem.
     = E(\gamma) - \frac{1}{2} \delta^2 \sum_j \lambda_j \cdot v_j + \cdots
   \]
   we find that \(E(i(v)) < E(\gamma)\) for sufficiently small \(\delta\). In
-  particular, \(\ell(i(v))^2 \le E(i(v)) < E(\gamma) = \ell(\gamma)^2\).
+  particular, \(L(i(v))^2 \le E(i(v)) < E(\gamma) = L(\gamma)^2\).
 \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 \(\gamma\) and \(\eta \in U\). 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
+be amended by showing that the length functional \(L : H^1(I, M) \to \RR\)
+is smooth and that its Hessian \(d^2 L_\gamma\) is given by \(C \cdot d^2
 E_\gamma\) for some \(C > 0\).
 
 Secondly, unlike the classical formulation we only consider curves in an

diff --git a/sections/introduction.tex b/sections/introduction.tex
@@ -51,8 +51,8 @@ we are interested is the infamous \emph{energy functional}
 \end{align*}
 as well as the \emph{length functional}
 \begin{align*}
-  \ell : H^1(I, M) & \to \RR                                      \\
-            \gamma & \mapsto \int_0^1 \norm{\dot\gamma(t)} \; \dt
+  L : H^1(I, M) & \to \RR                                      \\
+         \gamma & \mapsto \int_0^1 \norm{\dot\gamma(t)} \; \dt
 \end{align*}
 
 In section~\ref{sec:structure} we will describe the differential structure of

diff --git a/sections/structure.tex b/sections/structure.tex
@@ -55,7 +55,7 @@ are the topologies induces by the norms
 \end{align*}
 respectively.
 
-The problem with the first candidate is that \(\ell : {C'}^\infty(I, \RR^n) \to
+The problem with the first candidate is that \(L : {C'}^\infty(I, \RR^n) \to
 \RR\) is not a continuous map under the uniform topology. This can be readily
 seen by approximating the curve
 \begin{align*}
@@ -64,8 +64,8 @@ seen by approximating the curve
 \end{align*}
 with ``staircase curves'' \(\gamma_n : I \to \RR^n\) for larger and larger
 values of \(n\), as shown in figure~\ref{fig:step-curves}: clearly \(\gamma_n
-\to \gamma\) in the uniform topology, but \(\ell(\gamma_n) = 2\) does not
-approach \(\ell(\gamma) = \sqrt 2\) as \(n\) approaches \(\infty\).
+\to \gamma\) in the uniform topology, but \(L(\gamma_n) = 2\) does not
+approach \(L(\gamma) = \sqrt 2\) as \(n\) approaches \(\infty\).
 
 \begin{figure}[h]\label{fig:step-curves}
   \centering
@@ -91,7 +91,7 @@ approach \(\ell(\gamma) = \sqrt 2\) as \(n\) approaches \(\infty\).
 
 The issue with this particular example is that while \(\gamma_n \to \gamma\)
 uniformly, \(\dot\gamma_n\) does not converge to \(\dot\gamma\) in the uniform
-topology. This hints at the fact that in order for \(E\) and \(\ell\) to be
+topology. This hints at the fact that in order for \(E\) and \(L\) to be
 continuous maps we need to control both \(\gamma\) and \(\dot\gamma\). Hence a
 natural candidate for a norm in \({C'}^\infty(I, \RR^n)\) is
 \[