diff --git a/sections/introduction.tex b/sections/introduction.tex
@@ -44,10 +44,15 @@ the previously described program: we study the differential structure of the
Banach manifold \(H^1(I, M)\) of class \(H^1\) curves in a complete
finite-dimensional Riemannian manifold \(M\), which encodes the solution to the
\emph{classic} variational problem: that of geodesics. Hence the particular
-action functional we are interested is the infamous energy functional
+action functional we are interested is the infamous \emph{energy functional}
\begin{align*}
- E : H^1(I, M) & \to \RR \\
- \gamma & \mapsto \frac{1}{2} \int_0^1 \norm{\dot\gamma(t)} \; \dt
+ E : H^1(I, M) & \to \RR \\
+ \gamma & \mapsto \frac{1}{2} \int_0^1 \norm{\dot\gamma(t)}^2 \; \dt,
+\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
\end{align*}
In section~\ref{sec:structure} we will describe the differential structure of
@@ -55,7 +60,7 @@ In section~\ref{sec:structure} we will describe the differential structure of
section~\ref{sec:aplications} we study the critical points of the energy
functional \(E\) and describe how the fundamental results of the classical
theory of the calculus of variations in the context of Riemannian manifolds can
-be reproduced in our new setting. We'll assume basic knowlage of differential
+be reproduced in our new setting. We'll assume basic knowledge of differential
and Riemannian geometry, as well as some familiarity with the classical theory
of the calculus of variations -- see \cite[ch.~5]{gorodski} for the classical
theory.
@@ -220,6 +225,6 @@ This last two examples are examples of Banach Lie groups -- i.e. Banach
manifolds endowed with a group structure whose group operations are smooth.
Perhaps more interesting to us is the fact that this are both examples of
function spaces. Having reviewed the basics of the theory of Banach manifolds
-we can prooced to our in-depth exploration of a particular example, that of the
+we can proceed to our in-depth exploration of a particular example, that of the
space \(H^1(I, M)\).
diff --git a/sections/structure.tex b/sections/structure.tex
@@ -1,7 +1,7 @@
\section{The Structure of \(H^1(I, M)\)}\label{sec:structure}
As promised, in this section we will highlight the differential and Riemannian
-strucutures of the space \(H^1(I, M)\) of class \(H^1\) curves in a complete
+structures of the space \(H^1(I, M)\) of class \(H^1\) curves in a complete
finite-dimensional Riemannian manifold \(M\). The first question we should ask
ourselves is an obvious one: what is \(H^1(I, M)\)? Specifically, what is a
class \(H^1\) curve in \(M\)?
@@ -9,9 +9,9 @@ class \(H^1\) curve in \(M\)?
Given an interval \(I\), recall that a continuous curve \(\gamma : I \to
\RR^n\) is called \emph{a class \(H^1\)} curve if \(\gamma\) is absolutely
continuous, \(\dot \gamma(t)\) exists for almost all \(t \in I\) and
-\(\dot\gamma \in L^2(I, \RR^n)\). A classical result by Lebesgue states that
-the so called \emph{Sobolev space \(H^1(I, \RR^n)\)} of all class \(H^1\)
-curves in \(\RR^n\) is a Hilbert space under the inner product given by
+\(\dot\gamma \in L^2(I, \RR^n)\). It is a well known fact that the so called
+\emph{Sobolev space \(H^1(I, \RR^n)\)} of all class \(H^1\) curves in \(\RR^n\)
+is a Hilbert space under the inner product given by
\[
\langle \gamma, \eta \rangle_1
= \int_0^1 \gamma(t) \cdot \eta(t) + \dot\gamma(t) \cdot \dot\eta(t) \; \dt
@@ -34,10 +34,78 @@ Finally, we may define\dots
\end{note}
We should note that every peace-wise smooth curve \(\gamma : I \to M\) is a
-class \(H^1\) curve. This answer raises and aditional question though: why
+class \(H^1\) curve. This answer raises and additional question though: why
class \(H^1\) curves? The classical theory of the calculus of variations -- as
described in \cite[ch.~5]{gorodski} for instance -- is usually exclusively
concerned with the study of peace-wise smooth curves, so the fact that we are
now interested a larger class of curves, highly non-smooth curves in fact,
\emph{should} come as a surprise to the reader.
+To answer this second question we return to the case of \(M = \RR^n\). Denote
+by \({C'}^\infty(I, \RR^n)\) the space of peace-wise curves in \(\RR^n\). As
+described in section~\ref{sec:introduction}, we would like \({C'}^\infty(I,
+\RR^n)\) to be a Banach manifold under which both the energy functional and the
+length functional are smooth maps. As most function spaces, \({C'}^\infty(I,
+\RR^n)\) admits several natural topologies. Perhaps the most obvious candidate
+is the \(L^2\) topology, which is to say, the topology induced by the norm
+\[
+ \norm{\gamma}_\infty = \sup_t \norm{\gamma(t)}
+\]
+
+The problem with this choice is that \(\ell : {C'}^\infty(I, \RR^n) \to \RR\)
+is not a continuous map under the uniform topology. This can be readily seen by
+approximating the curve
+\begin{align*}
+ \gamma : I & \to \RR^2 \\
+ t & \mapsto (t, 1 - t)
+\end{align*}
+with ``step curves'' \(\gamma_n : I \to \RR^n\) for larger and larger values of
+\(n\), as shown in figure~\ref{fig:step-curves}.
+
+% TODO: Add a figure and a caption explaining why length is discontinuous
+\begin{figure}\label{fig:step-curves}
+ \centering
+ \begin{tikzpicture}
+ \draw (4, 1) -- (1, 4);
+ \draw (4, 1) -- (4, 2)
+ -- (3, 2)
+ -- (3, 3) node[right]{$\gamma_n$}
+ -- (2, 3) node[left]{$\gamma$}
+ -- (2, 4)
+ -- (1, 4);
+ \draw[dotted] (4.5, .5) -- (4, 1);
+ \draw[dotted] (.5, 4.5) -- (1, 4);
+ \draw (1, 4.3) -- (2, 4.3);
+ \draw (1, 4.2) -- (1, 4.4);
+ \draw (2, 4.2) -- (2, 4.4);
+ \node[above] at (1.5, 4.3) {$\sfrac{1}{n}$};
+ \end{tikzpicture}
+ \caption{A diagonal line representing the curve \(\gamma\) overlaps a
+ staircase-like curve \(\gamma_n\), whose steps measure \(\sfrac{1}{n}\).}
+\end{figure}
+
+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 to \(E\) and \(\ell\) 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
+\[
+ \norm{\gamma}_1 = \norm{\gamma}_2 + \norm{\dot\gamma}_2,
+\]
+which is, of course, the norm induced by the inner product \(\langle \, ,
+\rangle_1\) -- here \(\norm{\cdot}_2\) denote the norm of \(L^2(I, \RR^n)\).
+
+The other issue we face is one of completeness. Since \(\RR^n\) has a global
+chart, we to expect \({C'}^\infty(I, \RR^n)\) to be affine too. In other words,
+it is natural to expect \({C'}^\infty(I, \RR^n)\) to be Banach space. In
+particular, \({C'}^\infty(I, \RR^n)\) must be a normed Banach space. This is
+unfortunately not the case for the norm \(\norm{ \cdot }_1\), but we can
+consider its completion \(\bar{{C'}^\infty(I, \RR^n)}\). A classical result by
+Lebesgue establish that \(\bar{{C'}^\infty(I, \RR^n)} \cong H^1(I, \RR^n)\).
+
+This hopefully motivates our choice to consider class \(H^1\) maps in \(M\),
+but the is still a number of questions we need tackle. In particular, what
+should \(H^1(I, M)\) be modeled after? What are the charts of \(H^1(I, M)\)?
+This will be the focus of our next subsection.
+
+\subsection{The Charts of \(H^1(I, M)\)}