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

diff --git a/sections/structure.tex b/sections/structure.tex
@@ -1,10 +1,10 @@
 \section{The Structure of \(H^1(I, M)\)}\label{sec:structure}
 
-As promised, in this section we will highlight the differential and Riemannian
-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 \(M\)?
+Throughout this sections let \(M\) be a complete finite-dimensional Riemannian
+manifold. As promised, in this section we will highlight the differential and
+Riemannian structures of the space \(H^1(I, M)\) of class \(H^1\) curves in a
+\(M\). 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 \(M\)?
 
 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
@@ -62,7 +62,7 @@ approximating the curve
 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 \(n\) approachs \(\infty\).
+approach \(\ell(\gamma) = \sqrt 2\) as \(n\) approaches \(\infty\).
 
 \begin{figure}[h]\label{fig:step-curves}
   \centering
@@ -81,8 +81,8 @@ approach \(\ell(\gamma) = \sqrt 2\) as \(n\) approachs \(\infty\).
     \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 \(\sfrac{1}{n}\) in
+  \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.}
 \end{figure}
 
@@ -92,22 +92,163 @@ 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 
 \[
-  \norm{\gamma}_1^2 = \norm{\gamma}_2^2 + \norm{\dot\gamma}_2^2,
+  \norm{\gamma}_1^2 = \norm{\gamma}_0^2 + \norm{\dot\gamma}_0^2,
 \]
 which is, of course, the norm induced by the inner product \(\langle \, ,
-\rangle_1\) -- here \(\norm{\cdot}_2\) denote the norm of \(L^2(I, \RR^n)\).
+\rangle_1\) -- here \(\norm{\cdot}_0\) 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 \(\bar{{C'}^\infty(I, \RR^n)} \cong H^1(I, \RR^n)\).
+unfortunately not the case for \({C'}^\infty(I, \RR^n)\) the norm
+\(\norm\cdot_1\), but we can consider its completion, which, by a classical
+result by Lebesgue, just so happens to coincide with \(H^1(I, M)\). 
 
-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.
+It's also interesting to note that the completion of \({C'}^\infty(I, \RR^n)\)
+with respect to the norm \(\norm\cdot_\infty\) is the space \(C^0(I, \RR^n)\)
+of all continuous curves \(I \to \RR^n\), and that the natural inclusions
+\begin{equation}\label{eq:continuous-inclusions-rn-curves}
+  {C'}^\infty(I, \RR^n) 
+  \longhookrightarrow H^1(I, \RR^n)
+  \longhookrightarrow C^0(I, \RR^n)
+\end{equation}
+are continuous.
+
+This can be seen as a particular case of a more general result regarding spaces
+of sections of vector bundles over the unit interval \(I\). Explicitly, we
+find\dots
+
+\begin{proposition}
+  Given an Euclidean bundle \(E \to I\) -- i.e. a vector bundle endowed with a
+  Riemannian metric -- the space \(H^1(E)\) of all class class \(H^1\) sections
+  of \(E\) is the completion of the space \({C'}^\infty(E)\) of peace-wise
+  smooth sections of \(E\) under the inner product given by
+  \[
+    \langle \xi, \eta \rangle_1 
+    = \int_0^1 \langle \xi_t, \eta_t \rangle +
+    \langle \nabla \xi_t, \nabla \eta_t \rangle \; \dt
+  \]
+\end{proposition}
+
+\begin{proposition}
+  Given an Euclidean bundle \(E \to I\), the space \(C^0(E)\) of all continuous
+  sections of \(E\) is is the completion of \({C'}^\infty(E)\) under the norm
+  given by
+  \[
+    \norm{\xi}_\infty = \sup_t \norm{\xi_t}
+  \]
+\end{proposition}
+
+\begin{proposition}\label{thm:continuous-inclusions-sections}
+  Given an Euclidean bundle \(E \to I\), the inclusions
+  \[
+    {C'}^\infty(E) \longhookrightarrow H^1(E) \longhookrightarrow C^0(E)
+  \]
+  are continuous. More precisely, \(\norm{\xi}_\infty \le \sqrt 2
+  \norm{\xi}_1\).
+\end{proposition}
+
+\begin{proof}
+  Fix \(\xi \in H^1(E)\) and \(t_0, t_1 \in I\) with \(\norm{\xi}_0 \ge
+  \norm{\xi_{t_0}}\) and \(\norm{\xi}_\infty = \norm{\xi_{t_1}}\).
+  Then
+  \[
+    \begin{split}
+      \norm{\xi}_\infty^2
+      & = \norm{\xi_{t_0}}^2 
+        + \int_{t_0}^{t_1} \frac{\dd}{\dd s} \norm{\xi_s}^2 \; \dd s \\
+      \text{(\(\nabla\) is compatible with the metric)}
+      & = \norm{\xi_{t_0}}^2 
+        + 2 \int_{t_0}^{t_1} \langle \xi_s, \nabla \xi_s \rangle \; \dd s \\
+      \text{(Cauchy-Schwarz)}
+      & \le \norm{\xi_{t_0}}^2 
+        + 2 \int_0^1 \norm{\xi_s} \cdot \norm{\nabla \xi_s} \; \dd s \\
+      & \le \norm{\xi}_0^2 + \norm{\xi}_0^2 + \norm{\nabla \xi}_0^2 \\
+      & \le 2 \norm{\xi}_1^2
+    \end{split}
+  \]
+\end{proof}
+
+\begin{note}
+  Apply proposition~\ref{thm:continuous-inclusions-sections} to the bundle \(I
+  \times \RR^n \to I\) to get the continuity of the maps in
+  (\ref{eq:continuous-inclusions-rn-curves}).
+\end{note}
+
+We are particularly interested in the case of the pullback bundle \(E =
+\gamma^* TM \to I\), where \(\gamma : I \to M\) is a peace-wise smooth curve.
+\begin{center}
+  \begin{tikzcd}
+    \gamma^* TM \arrow{r} \arrow[swap]{d}{\pi} & TM \arrow{d}{\pi} \\
+    I \arrow[swap]{r}{\gamma} & M
+  \end{tikzcd}
+\end{center}
+
+We now have all the necessary tools to describe the differential structure of
+\(H^1(I, M)\).
 
 \subsection{The Charts of \(H^1(I, M)\)}
+
+We begin with a technical lemma.
+
+\begin{lemma}\label{thm:section-in-open-is-open}
+  Let \(W \subset TM\) be an open neighborhood of the zero section in \(TM\).
+  Given \(\gamma \in {C'}^\infty(I, M)\), denote by \(W_{\gamma, t}\) the set \(W
+  \cap T_\gamma(t) M\) and let \(W_\gamma = \bigcup_t W_{\gamma, t}\). Then
+  \(H^1(W_\gamma) = \{ X \in H^1(\gamma^* TM) : X_t \in W_{\gamma, t} \;
+  \forall t \}\) is an open subset of \(H^1(\gamma^* TM)\).
+\end{lemma}
+
+\begin{proof}
+  Let \(C^0(W_\gamma) = \{ X \in C^0(\gamma^* TM) : X_t \in W_{\gamma, t} \;
+  \forall t \}\). We claim \(C^0(W_\gamma)\) is open in \(C^0(\gamma^* TM)\).
+  Indeed, given \(X \in C^0(W_\gamma)\) there exists \(\delta > 0\) such
+  that 
+  \[
+    \begin{split}
+      \norm{X - Y}_\infty < \delta
+      & \implies \norm{X_t - Y_t} < \delta \; \forall t \\
+      & \implies Y_t \in W_{\gamma, t} \; \forall t \\
+      & \implies Y \in C^0(W_\gamma)
+    \end{split}
+  \]
+
+  Finally, notice that \(H^1(W_\gamma)\) is the inverse image of
+  \(C^0(W_\gamma)\) under the continuous inclusion \(H^1(\gamma^* TM)
+  \longhookrightarrow C^0(\gamma^* TM)\) and is therefore open.
+\end{proof}
+
+Let \(W \subset TM\) be an open neighborhood of the zero section in \(TM\)
+such that \(\exp\!\restriction_W : W \to \exp(W)\) is invertible -- whose
+existence is given by the fact that \(M\) is complete. 
+
+\begin{definition}
+  Given \(\gamma \in {C'}^\infty(I, M)\) let \(W_\gamma, W_{\gamma, t} \subset
+  \gamma^* TM\) be as in lemma~\ref{thm:section-in-open-is-open}, define
+  \[
+    \arraycolsep=1pt
+    \begin{array}{rl}
+      \exp_\gamma : H^1(W_\gamma) & \to H^1(I, M) \\
+      X & 
+      \begin{array}[t]{rl}
+        \mapsto \exp \circ X : I & \to M \\
+        t & \mapsto \exp_{\gamma(t)}(X_t)
+      \end{array}
+    \end{array}
+  \]
+  and let \(U_\gamma = \exp_\gamma(H^1(W_\gamma))\).
+\end{definition}
+
+Finally, we find\dots
+
+\begin{theorem}
+  Given \(\gamma \in {C'}^\infty(I, M)\), the map \(\exp_\gamma : H^1(W_\gamma)
+  \to U_\gamma\) is bijective. The collection \(\{(U_\gamma, \exp_\gamma^{-1} :
+  U_\gamma \to H^1(\gamma^* TM))\}_{\gamma \in {C'}^\infty(I, M)}\) is an atlas
+  for \(H^1(I, M)\) under the final topology of the maps \(\exp_\gamma\). This
+  atlas gives \(H^1(\gamma^* TM)\) the structure of a \emph{separable} Hilbert
+  manifold.
+\end{theorem}
+
+It should be obvious from the definition