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 \(\coprod_\gamma H^1(\gamma^* TM) \isoto T H^1(I, M)\).