diff --git a/sections/structure.tex b/sections/structure.tex
@@ -22,11 +22,10 @@ Finally, we may define\dots
\begin{definition}
Given an \(n\)-dimensional manifold \(M\), a continuous curve \(\gamma : I
\to M\) is called \emph{a class \(H^1\)} curve if \(\varphi_i \circ \gamma :
- J \to \RR^n\) for any chart \(\varphi_i : U_i \subset M \to \RR^n\) and
- \(J \subset I\) a maximal subinterval where \(\varphi_i \circ \gamma\) is
- defined -- i.e. if \(\gamma\) can be locally expressed as a class \(H^1\)
- curve in terms of the charts of \(M\). We'll denote by \(H^1(I, M)\) of all
- class \(H^1\) curves \(I \to M\).
+ J \to \RR^n\) is a class \(H^1\) curve for any chart \(\varphi_i : U_i
+ \subset M \to \RR^n\) -- i.e. if \(\gamma\) can be locally expressed as a
+ class \(H^1\) curve in terms of the charts of \(M\). We'll denote by \(H^1(I,
+ M)\) the set of all class \(H^1\) curves \(I \to M\).
\end{definition}
\begin{note}
@@ -67,7 +66,7 @@ values of \(n\), as shown in figure~\ref{fig:step-curves}: clearly \(\gamma_n
\to \gamma\) 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}
+\begin{figure}[h]
\centering
\begin{tikzpicture}
\draw (4, 1) -- (1, 4);
@@ -87,6 +86,7 @@ approach \(L(\gamma) = \sqrt 2\) as \(n\) approaches \(\infty\).
\caption{A diagonal line representing the curve $\gamma$ overlaps a
staircase-like curve $\gamma_n$, whose steps measure $\sfrac{1}{n}$ in
width and height.}
+ \label{fig:step-curves}
\end{figure}
The issue with this particular example is that while \(\gamma_n \to \gamma\)