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

diff --git a/sections/structure.tex b/sections/structure.tex
@@ -129,7 +129,7 @@ find\dots
   \]
 \end{proposition}
 
-\begin{proposition}
+\begin{proposition}\label{thm:h0-bundle-is-complete}
   Given an Euclidean bundle \(E \to I\), the space \(H^0(E)\) of all square
   integrable sections of \(E\) is is the completion of \({C'}^\infty(E)\) under
   the inner product given by
@@ -345,7 +345,7 @@ M) \isoto H^1(\gamma^* TM)\), as described in
 proposition~\ref{thm:tanget-space-topology}. In fact, this isomorphisms may be
 extended to a canonical isomorphism of vector bundles, as seen in\dots
 
-\begin{lemma}
+\begin{lemma}\label{thm:alpha-fiber-bundles-definition}
   Given \(i = 0, 1\), the collection \(\{(\psi_{i, \gamma}(H^1(U_\gamma) \times
   H^i(\gamma^* TM)), \psi_{i, \gamma}^{-1})\}_{\gamma \in {C'}^\infty(I,
   M)}\) with
@@ -357,12 +357,15 @@ extended to a canonical isomorphism of vector bundles, as seen in\dots
       (X, Y) \mapsto &
       \begin{array}[t]{rl}
         \psi_{i, \gamma}(X) : I & \to \exp_\gamma(X)^* TM \\
-        t & \mapsto (d \exp_{\gamma(t)})_{X_t} Y_t
+        t & \mapsto (d \exp)_{X_t} Y_t
       \end{array}
     \end{array}
   \]
   gives \(\coprod_{\gamma \in {C'}^\infty(I, M)} H^i(\gamma^* TM) \to H^1(I,
-  M)\) the structure of a smooth vector bundle.
+  M)\) the structure of a smooth vector bundle\footnote{Here we use the
+  canonical identification $T_{\gamma(t)} M \cong T_{X_t}^\vee TM$ to apply
+  the vector $Y_t \in T_{\gamma(t)} M$ to the map $(d \exp)_{X_t} : T_{X_t} TM
+  \to T_{\exp_{\gamma(t)}(X_t)} M$.}.
 \end{lemma}
 
 \begin{proposition}
@@ -396,7 +399,116 @@ extended to a canonical isomorphism of vector bundles, as seen in\dots
   {C'}^\infty(I, M)\) is given by \(\phi_\gamma\).
 \end{proof}
 
-This last result will be the basis for our analysis of the Riemannian structure
-of \(H^1(I, M)\): we may now describe the metric of \(H^1(I, M)\) in terms of
-the fibers \(T_\gamma H^1(I, M) \cong H^1(\gamma^* TM)\).
+At this point it may be tempting to think that we could now define the metric
+of \(H^1(I, M)\) in a fiber-wise basis via the identification \(T_\gamma
+H^1(\gamma^* TM) \cong H^1(\gamma^* TM)\). In a very real sense this is what we
+are about to do, but unfortunately there are still technicalities in our way.
+The issue we face is that proposition~\ref{thm:h0-bundle-is-complete} only
+applies for \emph{smooth} vector bundles \(E \to I\), which may not be the case
+for \(E = \gamma^* TM\) if \(\gamma \in H^1(I, M)\) lies outside of
+\({C'}^\infty(I, M)\). In fact, neither \(\langle X , Y \rangle_0\) nor
+\(\langle \, , \rangle_1\) are defined \emph{a priori} for \(X, Y \in
+H^0(\gamma^* TM)\) with \(\gamma \notin {C'}^\infty(I, M)\).
+
+Nevertheless, we can get around this limitation by extending the metric
+\(\langle \, , \rangle_0\) and the covariant derivative \(\frac\nabla\dt =
+\nabla_{\frac\dd\dt}\) to
+those \(H^0(\gamma^* TM)\) with \(\gamma \notin {C'}^\infty(I, M)\). In other
+words, we'll show\dots
+
+\begin{theorem}\label{thm:h0-has-metric-extension}
+  The vector bundle \(\coprod_{\gamma \in H^1(I, M)} H^0(\gamma^* TM) \to
+  H^1(I, M)\) admits a canonical Riemannian metric whose restriction the the
+  fibers \(H^0(\gamma^* TM) = \left.\coprod_{\eta \in H^1(I, M)} H^0(\eta^*
+  TM)\right|_\gamma\) for \(\gamma \in {C'}^\infty(I, M)\) is given by
+  \(\langle \, , \rangle_0\) as defined in
+  proposition~\ref{thm:h0-bundle-is-complete}.
+\end{theorem}
+
+\begin{proof}
+  Given \(\gamma \in {C'}^\infty(I, M)\) and \(X \in H^1(\gamma^*)\), let
+  \begin{align*}
+    g_X^\gamma : H^0(\gamma^* TM) \times H^0(\gamma^* TM) & \to \RR \\
+    (Y, Z) & 
+    \mapsto \int_0^1 
+    \langle (d\exp)_{X_t} Y_t, (d\exp)_{X_t} Z_t \rangle \; \dt
+  \end{align*}
+
+  This is clearly a Riemannian metric in the bundle \(H^1(W_\gamma) \times
+  H^0(\gamma^* TM) \to H^1(W_\gamma)\). Now by composing with the chart
+  \(\psi_{0, \gamma}^{-1}\) as in
+  lemma~\ref{thm:alpha-fiber-bundles-definition} we get a Riemannian metric
+  \(g_\gamma\) in the bundle \(\coprod_{\eta \in U_\gamma} H^0(\eta^* TM) =
+  \left. \coprod_{\eta \in H^1(I, M)} H^0(\eta^* TM) \right|_{U_\gamma} \to
+  U_\gamma\). One can then quickly verify that the \(g^\gamma\)'s agree in the
+  intersection of the \(U_\gamma\)'s, so that they define a global Riemannian
+  metric \(g\) in \(\coprod_{\eta \in H^1(I, M)} H^0(\eta^* TM)\).
+
+  Furthermore, given \(\gamma \in {C'}^\infty(I, M)\) and \(X, Y \in
+  H^0(\gamma^* TM)\) by construction we have
+  \[
+    g_\gamma(X, Y) 
+    = g_0^\gamma(X, Y) 
+    = \int_0^1 \langle (d \exp)_0 X_t, (d \exp)_0 Y_t \rangle \; \dt 
+    = \int_0^1 \langle X_t, Y_t \rangle \; \dt 
+    = \langle X, Y \rangle_0
+  \]
+\end{proof}
+
+\begin{proposition}\label{thm:partial-is-smooth-sec}
+  The map 
+  \begin{align*}
+    \partial : H^1(I, M) & \to     \coprod_{\gamma} H^0\gamma^* TM \\
+                  \gamma & \mapsto \dot\gamma
+  \end{align*}
+  is a smooth section of \(\coprod_{\gamma \in H^1(I, M)} H^0(\gamma^* TM) \to
+  H^1(I, M)\).
+\end{proposition}
+
+\begin{proposition}\label{thm:covariant-derivative-h0}
+  Denote by \(\nabla^0 : \mathfrak{X}(H^1(I, M)) \times
+  \Gamma\left(\coprod_\gamma H^0(\gamma^* TM)\right) \to
+  \Gamma\left(\coprod_{\gamma} H^0(\gamma^* TM)\right)\) the Levi-Civita
+  connection of \(\coprod_\gamma H^0(\gamma^* TM)\). The map
+  \begin{align*}
+    \mathfrak{X}(H^1(I, M)) 
+    & \to \Gamma\left(\coprod_\gamma H^0(\gamma^* TM)\right) \\
+    \xi 
+    & \mapsto \nabla_\xi^0 \partial
+  \end{align*}
+  is such that
+  \[
+    (\nabla_X^0 \partial)_\gamma 
+    = \nabla_{\frac\dd\dt} X
+    = \frac\nabla\dt X
+  \]
+  for all \(\gamma \in {C'}^\infty(I, M)\) and \(X \in H^1(\gamma^* TM) \cong
+  T_\gamma H^1(I, M)\). Given some arbitrary \(\gamma \in H^1(I, M)\) and \(X
+  \in H^1(\gamma^* TM)\) we therefore denote \((\nabla_X^0 \partial)_\gamma\)
+  simply by \(\frac\nabla\dt X\).
+\end{proposition}
+
+The proof of this last two propositions was deemed too technical to be included
+in here, but see proposition 2.3.16 and 2.3.18 of \cite{klingenberg}. We may
+now finally describe the canonical Riemannian metric of \(H^1(I, M)\).
+
+\begin{definition}\label{def:h1-metric}
+  Given \(\gamma \in H^1(I, M)\) and \(X, Y \in H^1(\gamma^* TM)\), let
+  \[
+    \langle X, Y \rangle_1 
+    = \langle X, Y \rangle_0 
+    + \left\langle \frac\nabla\dt X, \frac\nabla\dt Y \right\rangle_0
+  \]
+\end{definition}
 
+At this point it should be obvious that definition~\ref{def:h1-metric} does
+indeed endow \(H^1(I, M)\) with the structure of a Riemannian manifold: the
+inner products \(\langle \, , \rangle_1 : H^1(\gamma^* TM) \times H^1(\gamma^*
+TM) \to \RR\) may be glued together into a single positive-definite 
+section \(\langle \, , \rangle_1 \in \Gamma( \Sym^2 \coprod_\gamma H^1(\gamma^*
+TM))\) -- whose smoothness follows from
+theorem~\ref{thm:h0-has-metric-extension},
+proposition~\ref{thm:partial-is-smooth-sec} and
+proposition~\ref{thm:covariant-derivative-h0} -- which is then mapped to a
+positive-definite section of \(\Sym^2 T H^1(I, M)\) by the canonical
+isomorphism \(\coprod_\gamma H^1(\gamma^* TM) \isoto T H^1(I, M)\).