\section{The Structure of \(H^1(I, M)\)}\label{sec:structure}

Throughout this section let \(M\) be a 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 \(\gamma : I \to
\mathbb{R}^n\) is called \emph{a class \(H^1\)} curve if \(\gamma\) is
absolutely continuous, \(\dot \gamma(t)\) exists for almost all \(t \in I\) and
\(\dot\gamma \in H^0(I, \mathbb{R}^n) = L^2(I, \mathbb{R}^n)\). It is a well
known fact that the so called \emph{Sobolev space \(H^1([0, 1],
\mathbb{R}^n)\)} of all class \(H^1\) curves in \(\mathbb{R}^n\) is a Hilbert
space under the inner product given by
\[
  \langle \gamma, \eta \rangle_1
  = \int_0^1 \gamma(t) \cdot \eta(t) + \dot\gamma(t) \cdot \dot\eta(t) \; \dt
\]

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 \(\varphi_i \circ \gamma :
  J \to \mathbb{R}^n\) is a class \(H^1\) curve for any chart \(\varphi_i : U_i
  \subset M \to \mathbb{R}^n\) -- 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}
  From now on we fix \(I = [0, 1]\).
\end{note}

Notice in particular that every piece-wise smooth curve \(\gamma : I \to M\) is
a 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 piece-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 \(M = \mathbb{R}^n\).
Denote by \({C'}^\infty(I, \mathbb{R}^n)\) the space of piece-wise curves in
\(\mathbb{R}^n\). As described in section~\ref{sec:introduction}, we would like
\({C'}^\infty(I, \mathbb{R}^n)\) to be a Banach manifold under which both the
energy functional and the length functional are smooth maps. As most function
spaces, \({C'}^\infty(I, \mathbb{R}^n)\) admits several natural topologies.
Some of the most obvious candidates are the uniform topology and the topology
of the \(\norm\cdot_0\) norm, which are the topologies induced by the norms
\begin{align*}
  \norm{\gamma}_\infty & = \sup_t \norm{\gamma(t)}                   \\
       \norm{\gamma}_0 & = \sqrt{\int_0^1 \norm{\gamma(t)}^2 \; \dt}
\end{align*}
respectively.

The problem with the first candidate is that \(L : {C'}^\infty(I, \mathbb{R}^n)
\to \mathbb{R}\) is not a continuous map under the uniform topology. This can
be readily seen by approximating the curve
\begin{align*}
  \gamma : I & \to     \mathbb{R}^2      \\
           t & \mapsto (t, 1 - t)
\end{align*}
with ``staircase curves'' \(\gamma_n : I \to \mathbb{R}^n\) 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 \(L(\gamma_n) = 2\) does
not approach \(L(\gamma) = \sqrt 2\) as \(n\) approaches \(\infty\).

\begin{figure}[h]
  \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 $\sfrac{1}{n}$ in
  width and height.}
  \label{fig:step-curves}
\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 for \(E\) and \(L\) to be
continuous maps we need to control both \(\gamma\) and \(\dot\gamma\). Hence a
natural candidate for a norm in \({C'}^\infty(I, \mathbb{R}^n)\) is
\[
  \norm{\gamma}_1^2 = \norm{\gamma}_0^2 + \norm{\dot\gamma}_0^2,
\]
which is, of course, the norm induced by the inner product \(\langle \, ,
\rangle_1\) -- here \(\norm{\cdot}_0\) denotes the norm of \(H^0(I,
\mathbb{R}^n) = L^2(I, \mathbb{R}^n)\).

The other issue we face is one of completeness. Since \(\mathbb{R}^n\) has a
global chart, we expect \({C'}^\infty(I, \mathbb{R}^n)\) to be affine too. In
other words, it is natural to expect \({C'}^\infty(I, \mathbb{R}^n)\) to be
Banach space. In particular, \({C'}^\infty(I, \mathbb{R}^n)\) must be complete.
This is unfortunately not the case for \({C'}^\infty(I, \mathbb{R}^n)\) in the
\(\norm\cdot_1\) norm, but we can consider its completion. Lo and behold, a
classical result by Lebesgue establishes that this completion just so happens
to coincide with \(H^1(I, \mathbb{R}^n)\).

It's also interesting to note that the completion of \({C'}^\infty(I,
\mathbb{R}^n)\) with respect to the norms \(\norm\cdot_\infty\) and
\(\norm\cdot_0\) are \(C^0(I, \mathbb{R}^n)\) and \(H^0(I, \mathbb{R}^n)\),
respectively, and that the natural inclusions
\begin{equation}\label{eq:continuous-inclusions-rn-curves}
  H^1(I, \mathbb{R}^n)
  \longhookrightarrow C^0(I, \mathbb{R}^n)
  \longhookrightarrow H^0(I, \mathbb{R}^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 \(C^0(E)\) of all continuous sections of \(E\)
  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:h0-bundle-is-complete}
  Given an Euclidean bundle \(E \to I\), the space \(H^0(E)\) of all square
  integrable sections of \(E\) is the completion of \({C'}^\infty(E)\) under
  the inner product given by
  \[
    \langle \xi, \eta \rangle_0 = \int_0^1 \langle \xi_t, \eta_t \rangle \; \dt
  \]
\end{proposition}

\begin{proposition}
  Given an Euclidean bundle \(E \to I\), the space \(H^1(E)\) of all class
  \(H^1\) sections of \(E\) is the completion of the space \({C'}^\infty(E)\)
  of piece-wise smooth sections of \(E\) under the inner product given by
  \[
    \langle \xi, \eta \rangle_1
    = \langle \xi, \eta \rangle_0 +
    \left\langle
      \nabla_{\frac\dd\dt} \xi, \nabla_{\frac\dd\dt} \eta
    \right\rangle_0
  \]
\end{proposition}

\begin{proposition}\label{thm:continuous-inclusions-sections}
  Given an Euclidean bundle \(E \to I\), the inclusions
  \[
    H^1(E) \longhookrightarrow C^0(E) \longhookrightarrow H^0(E)
  \]
  are continuous. More precisely, \(\norm{\xi}_\infty \le \sqrt 2
  \norm{\xi}_1\) and \(\norm{\xi}_0 \le \norm{\xi}_\infty\).
\end{proposition}

\begin{proof}
  Given \(\xi \in H^0(E)\) we have
  \[
    \norm{\xi}_0^2
    = \int_0^1 \norm{\xi_t}^2 \; \dt
    \le \int_0^1 \norm{\xi}_\infty^2 \; \dt
    = \norm{\xi}_\infty^2
  \]

  Now given \(\xi \in H^1(E)\) fix \(t_0, t_1 \in I\) with \(\norm{\xi}_\infty
  = \norm{\xi_{t_1}}\) and \(\norm{\xi_{t_0}} \le \norm{\xi}_0\). If \(t_0 <
  t_1\) then
  \[
    \begin{split}
      \norm{\xi}_\infty^2
      & = \norm{\xi_{t_0}}^2
        + \int_{t_0}^{t_1} \frac{\dd}\dt \norm{\xi_t}^2 \; \dt \\
      & \le \norm{\xi}_0^2
        + \int_{t_0}^{t_1} \frac{\dd}\dt \norm{\xi_t}^2 \; \dt \\
      \text{(\(\nabla\) is compatible with the metric)}
      & = \norm{\xi}_0^2 + \int_{t_0}^{t_1}
        2 \left\langle \xi_t, \nabla_{\frac\dd\dt} \xi_t \right\rangle
        \; \dt \\
      \text{(Cauchy-Schwarz)}
      & \le \norm{\xi}_0^2 + \int_0^1
        2 \norm{\xi_t} \cdot \norm{\nabla_{\frac\dd\dt} \xi_t} \; \dt \\
      & \le \norm{\xi}_0^2
        + \int_0^1 \norm{\xi_t}^2 + \norm{\nabla_{\frac\dd\dt} \xi_t}^2
        \; \dt \\
      & = \norm{\xi}_0^2
        + \norm{\xi}_0^2 + \norm{\nabla_{\frac\dd\dt} \xi}_0^2 \\
      & \le 2 \norm{\xi}_1^2
    \end{split}
  \]

  Similarly, if \(t_0 > t_1\) then
  \[
    \norm{\xi}_\infty^2
    = \norm{\xi_{t_0}}^2 + \int_{t_0}^{t_1} \frac{\dd}\dt \norm{\xi_t}^2 \; \dt
    = \norm{\xi_{t_0}}^2 + \int_{1 - t_0}^{1 - t_1}
      \frac{\dd}\dt \norm{\xi_{1 - t}}^2 \; \dt
    \le 2 \norm{\xi}_1^2
  \]
\end{proof}

\begin{note}
  Apply proposition~\ref{thm:continuous-inclusions-sections} to the trivial
  bundle \(I \times \mathbb{R}^n \to I\) 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 piece-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
follows from the fact that the injectivity radius depends continuously on \(p
\in M\).

\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 \(\exp_\gamma\) --
  i.e. the coarsest topology such that such maps are continuous. This atlas
  gives \(H^1(I, M)\) the structure of a \emph{separable} Banach manifold
  modeled after separable Hilbert spaces, with typical
  representatives\footnote{Any trivialization of $\gamma^* TM$ induces an
  isomorphism $H^1(\gamma^* TM) \isoto H^1(I \times \mathbb{R}^n) \cong H^1(I,
  \mathbb{R}^n)$.} \(H^1(\gamma^* TM) \cong H^1(I, \mathbb{R}^n)\).
\end{theorem}

The fact that \(\exp_\gamma\) is bijective should be clear from the definition
of \(U_\gamma\) and \(W_\gamma\). That each \(\exp_\gamma^{-1}\) is a
homeomorphism is also clear from the definition of the topology of \(H^1(I,
M)\). Moreover, since \({C'}^\infty(I, M)\) is dense, \(\{U_\gamma\}_{\gamma
\in {C'}^\infty(I, M)}\) is an open cover of \(H^1(I, M)\). The real difficulty
of this proof is showing that the transition maps
\[
  \exp_\eta^{-1} \circ \exp_\gamma :
  \exp_\gamma^{-1}(U_\gamma \cap U_\eta) \subset H^1(\gamma^* TM)
  \to H^1(\eta^* TM)
\]
are diffeomorphisms, as well as showing that \(H^1(I, M)\) is separable. We
leave these details as an exercise to the reader -- see theorem 2.3.12 of
\cite{klingenberg} for a full proof.

It's interesting to note that this construction is functorial. More
precisely\dots

\begin{theorem}
  Given finite-dimensional Riemannian manifolds \(M\) and \(N\) and a smooth
  map \(f : M \to N\), the map
  \begin{align*}
    H^1(I, f) : H^1(I, M) & \to     H^1(I, N) \\
                   \gamma & \mapsto f \circ \gamma
  \end{align*}
  is smooth. In addition, \(H^1(I, f \circ g) = H^1(I, f) \circ H^1(I, g)\) and
  \(H^1(I, \operatorname{id}) = \operatorname{id}\) for any composable smooth
  maps \(f\) and \(g\). We thus have a functor \(H^1(I, -) : \mathbf{Rmnn} \to
  \mathbf{BMnfd}\) from the category \(\mathbf{Rmnn}\) of finite-dimensional
  Riemannian manifolds and smooth maps onto the category \(\mathbf{BMnfd}\) of
  Banach manifolds and smooth maps.
\end{theorem}

We would also like to point out that this is a particular case of a more
general construction: that of the Banach manifold \(H^1(E)\) of class \(H^1\)
sections of a smooth fiber bundle \(E \to I\) -- not necessarily a vector
bundle. Our construction of \(H^1(I, M)\) is equivalent to that of the manifold
\(H^1(I \times M)\), in the sense that the canonical map
\[
  \arraycolsep=1pt
  \begin{array}{rl}
    \tilde{\cdot} : H^1(I, M) \to & H^1(I \times M) \\
    \gamma \mapsto &
    \begin{array}[t]{rl}
      \tilde\gamma : I & \to I \times M \\
      t & \mapsto (t, \gamma(t))
    \end{array}
  \end{array}
\]
can be easily checked to be a diffeomorphism.

The space \(H^1(E)\) is modeled after the Hilbert spaces \(H^1(F)\) of class
\(H^1\) sections of open sub-bundles \(F \subset E\) which have the structure
of a vector bundle -- the so called \emph{vector bundle neighborhoods of
\(E\)}. This construction is highlighted in great detail and generality in the
first section of \cite[ch.~11]{palais}, but unfortunately we cannot afford such
a diversion in these short notes. Having said that, we are now finally ready to
layout the Riemannian structure of \(H^1(I, M)\).

\subsection{The Metric of \(H^1(I, M)\)}

We begin our discussion of the Riemannian structure of \(H^1(I, M)\) by looking
at its tangent bundle. Notice that for each \(\gamma \in {C'}^\infty(I, M)\)
the chart \(\exp_\gamma^{-1} : U_\gamma \to H^1(\gamma^* TM)\) induces a
canonical isomorphism \(\phi_\gamma = \phi_{\gamma, \gamma} : T_\gamma H^1(I,
M) \isoto H^1(\gamma^* TM)\), as described in
proposition~\ref{thm:tanget-space-topology}. In fact, these isomorphisms may be
extended to a canonical isomorphism of vector bundles, as seen in\dots

\begin{lemma}\label{thm:alpha-fiber-bundles-definition}
  Given \(i = 0, 1\), the collection \(\{(\psi_{i, \gamma}(H^1(W_\gamma) \times
  H^i(\gamma^* TM)), \psi_{i, \gamma}^{-1})\}_{\gamma \in {C'}^\infty(I,
  M)}\) with
  \[
    \arraycolsep=1pt
    \begin{array}{rl}
      \psi_{i, \gamma} : H^1(W_\gamma) \times H^i(\gamma^* TM) \to
      & \coprod_{\eta \in H^1(I, M)} H^i(\eta^* TM) \\
      (X, Y) \mapsto &
      \begin{array}[t]{rl}
        \psi_{i, \gamma}(X) : I & \to \exp_\gamma(X)^* TM \\
        t & \mapsto (d \exp)_{X_t} Y_t
      \end{array}
    \end{array}
  \]
  gives \(\coprod_{\gamma \in {C'}^\infty(I, M)} H^i(\gamma^* TM) \to H^1(I,
  M)\) the structure of a smooth vector bundle\footnote{Here we use the
  canonical identification $T_{\gamma(t)} M \cong T_{X_t} TM$ 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}
  There is a canonical isomorphism of vector bundles
  \[
    T H^1(I, M) \isoto \coprod_{\gamma \in H^1(I, M)} H^1(\gamma^* TM)
  \]
  whose restriction \(T_\gamma H^1(I, M) \isoto H^1(\gamma^* TM)\) is given by
  \(\phi_\gamma\) for all \(\gamma \in {C'}^\infty(I, M)\).
\end{proposition}

\begin{proof}
  Note that the sets \(H^1(W_\gamma) \times T_\gamma H^1(I, M)\) are precisely
  the images of the charts
  \[
    \varphi_\gamma^{-1} :
    \varphi_\gamma(H^1(W_\gamma) \times T_\gamma H^1(I, M))
    \subset T H^1(I, M)
    \to H^1(W_\gamma) \times T_\gamma H^1(I, M)
  \]
  of \(T H^1(I, M)\) given by\footnote{Once more, we use the
  canonical identification $T_X H^1(W_\gamma) \cong H^1(\gamma^* TM)$ to apply
  the vector $\phi_\gamma(Y) \in H^1(\gamma^* TM)$ to $(d \exp_\gamma)_X : T_X
  H^1(W_\gamma) \to T_{\exp_\gamma(X)} H^1(I, M)$.}
  \begin{align*}
    \varphi_\gamma : H^1(W_\gamma) \times T_\gamma H^1(I, M)
    & \to T H^1(I, M) \\
    (X, Y) & \mapsto (d \exp_\gamma)_X \phi_\gamma(Y)
  \end{align*}

  By composing charts we get a fiber-preserving, fiber-wise linear
  diffeomorphism
  \[
    \varphi_\gamma(H^1(W_\gamma) \times T_\gamma H^1(I, M))
    \subset T H^1(I, M)
    \isoto
    \psi_{1, \gamma}(H^1(W_\gamma) \times H^1(\gamma^* TM)),
  \]
  which takes \(\varphi_\gamma(X, Y) \in T_{\exp_\gamma(X)} H^1(I, M)\) to
  \(\psi_{1, \gamma}(X, \phi_\gamma(Y)) \in H^1(\exp_\gamma(X)^* TM)\). With
  enough patience, one can deduce from the fact that \(\varphi_\gamma^{-1}\)
  and \(\psi_{1, \gamma}^{-1}\) are charts that these maps agree in the
  intersections of the open subsets \(\varphi_\gamma(H^1(W_\gamma) \times
  T_\gamma H^1(I, M))\), so that they may be glued together into a global
  smooth map \(\Phi : T H^1(I, M) \to \coprod_{\eta \in H^1(I, M)} H^1(\eta^*
  TM)\).

  Since this map is a fiber-preserving, fiber-wise linear local diffeomorphism,
  this is an isomorphism of vector bundles.
  Furthermore, by construction
  \[
    \Phi(X)_t
    = \psi_{1, \gamma}(0, \phi_\gamma(X))_t
    = (d \exp)_{0_{\gamma(t)}} \phi_\gamma(X)_t
    = \phi_\gamma(X)_t
  \]
  for each \(\gamma \in {C'}^\infty(I, M)\) and \(X \in T_\gamma H^1(I, M)\).
  In other words, \(\Phi\!\restriction_{T_\gamma H^1(I, M)} = \phi_\gamma\) as
  required.
\end{proof}

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 \(T_\gamma H^1(I,
M) \cong H^1(\gamma^* TM)\). 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 to 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 \(X \in H^1(\gamma^* TM)\), let
  \begin{align*}
    g_X^\gamma : H^0(\gamma^* TM) \times H^0(\gamma^* TM) & \to \mathbb{R} \\
    (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
  \[
    g_\gamma(X, Y)
    = g_0^\gamma(X, Y)
    = \int_0^1
      \langle
      (d \exp)_{0_{\gamma(t)}} X_t, (d \exp)_{0_{\gamma(t)}} Y_t
      \rangle \; \dt
    = \int_0^1 \langle X_t, Y_t \rangle \; \dt
    = \langle X, Y \rangle_0
  \]
\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) \\
    \tilde X
    & \mapsto \nabla_{\tilde X}^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 proofs of these last two propositions were 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 \(\langle \, , \rangle_1 : H^1(\gamma^* TM) \times H^1(\gamma^*
TM) \to \mathbb{R}\) may be glued together into a single positive-definite
section \(\langle \, , \rangle_1 \in \Gamma\left(\operatorname{Sym}^2
\coprod_\gamma H^1(\gamma^* TM)\right)\) -- 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 \(\operatorname{Sym}^2 T H^1(I, M)\) by the
induced isomorphism
\[
  \Gamma\left(\operatorname{Sym}^2 \coprod_\gamma H^1(\gamma^* TM)\right)
  \isoto \Gamma(\operatorname{Sym}^2 T H^1(I, M))
\]

We are finally ready to discuss some applications.

positive-definite section of \(\operatorname{Sym}^2 T H^1(I, M)\) by the
induced isomorphism
\[
  \Gamma\left(\operatorname{Sym}^2 \coprod_\gamma H^1(\gamma^* TM)\right)
  \isoto \Gamma(\operatorname{Sym}^2 T H^1(I, M))
\]

We are finally ready to discuss some applications.
