\section{Introduction}\label{sec:introduction}

Known as \emph{global analysis}, or sometimes \emph{non-linear functional
analysis}, the field of study dedicated to the understanding of
infinite-dimensional manifolds has seen remarkable progress in the past several
decades. Among numerous discoveries, perhaps the greatest achievement in global
analysis in the last century was the recognition of the fact that many
interesting function spaces possess natural differentiable structures -- which
are usually infinite-dimensional.

As it turns out, many local problems regarding maps between finite-dimensional
manifolds can be translated to global questions about the geometry of function
spaces -- hence the name ``\emph{global} analysis''. More specifically, a
remarkable number of interesting geometric objects can be characterized as
``critical points'' of functionals in functions spaces. The usual suspects are,
of course, geodesics and minimal submanifolds in general, but there are many
other interesting examples: harmonic functions, Einstein metrics, periodic
solutions to Hamiltonian vector fields, etc. \cite[ch.~11]{palais}.

Such objects are the domain of the so called \emph{calculus of variations},
which is generally concerned with finding functions that minimize or maximize a
given functional, known as the \emph{action functional}, by subjecting such
functions to ``small variations'' -- which is known as \emph{the variational
method}. The meaning of ``small variations'' have historically been a very
dependent on the context of the problem at hand. Only recently, with the
introduction of the tools of global analysis, the numerous ad-hoc methods under
the umbrella of ``variational method'' have been unified into a coherent
theory, which we describe in the following.

By viewing the class of functions we're interested in as a -- most likely
infinite-dimensional -- manifold \(\mathscr{F}\) and the action functional as a
smooth functional \(f : \mathscr{F} \to \mathbb{R}\) we can find minimizing and
maximizing functions by studying the critical points of \(f\). More generally,
modern calculus of variations is concerned with the study of critical points of
smooth functionals \(\Gamma(E) \to \mathbb{R}\), where \(E \to M\) is a smooth
fiber bundle over a finite-dimensional manifold \(M\) and \(\Gamma\) is a given
section functor, such as smooth sections, continuous sections or Sobolev
sections -- notice that by taking \(E = M \times N\) the manifold \(\Gamma(E)\)
is naturally identified with a space of functions \(M \to N\), which gets us
back to the original case.

In these notes we hope to provide a very brief introduction to modern theory
the calculus of variations by exploring one of the simplest concrete examples
of the previously described program. We study the differential structure of the
Banach manifold \(H^1(I, M)\) of class \(H^1\) curves in a 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 \emph{energy functional}
\begin{align*}
  E : H^1(I, M) & \to     \mathbb{R}                                         \\
         \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*}
  L : H^1(I, M) & \to \mathbb{R}                               \\
         \gamma & \mapsto \int_0^1 \norm{\dot\gamma(t)} \; \dt
\end{align*}

In section~\ref{sec:structure} we will describe the differential structure of
\(H^1(I, M)\) and its canonical Riemannian metric. In
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. Other examples of function spaces are
explored in detail in \cite[sec.~6]{eells}. The 11th chapter of \cite{palais}
is also a great reference for the general theory of spaces of sections of fiber
bundles.

We should point out that we will primarily focus on
the broad strokes of the theory ahead and that we will leave many results
unproved. The reasoning behind this is twofold. First, we don't want to bore
the reader with the numerous technical details of some of the constructions
we'll discuss in the following. Secondly, and this is more important, these
notes are meant to be concise. Hence we do not have the necessary space to
discuss neither technicalities nor more involved applications of the theory we
will develop.

In particular, we leave the intricacies of Palais' and Smale's discussion of
condition (C) -- which can be seen as a substitute for the failure of a proper
Hilbert space to be locally compact \cite[ch.~2]{klingenberg} -- and its
applications to the study of closed geodesics out of these notes. As previously
stated, many results are left unproved, but we will include references to other
materials containing proofs. 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
approach.

Before moving to the next section we would like to review the basics of the
theory of real Banach manifolds.

\subsection{Banach Manifolds}

While it is certainly true that Banach spaces can look alien to someone who has
never ventured outside of the realms of Euclidean space, Banach manifolds are
surprisingly similar to their finite-dimensional counterparts. As we'll soon
see, most of the usual tools of differential geometry can be quite easily
translated to the context of Banach manifolds\footnote{The real difficulties
with Banach manifolds only show up while proving certain results, and are
mainly due to complications regarding the fact that not all closed subspaces of
a Banach space have a closed complement.}. The reason behind this is simple: it
turns out that calculus has nothing to do with \(\mathbb{R}^n\).

What we mean by this last statement is that none of the fundamental ingredients
of calculus -- the ones necessary to define differentiable functions in
\(\mathbb{R}^n\), namely the fact that \(\mathbb{R}^n\) is a complete normed
space -- are specific to \(\mathbb{R}^n\). In fact, these ingredients are
precisely the features of a Banach space. Thus we may naturally generalize
calculus to arbitrary Banach spaces, and consequently generalize smooth
manifolds to spaces modeled after Banach spaces. We begin by the former.

\begin{definition}
  Let \(V\) and \(W\) be Banach spaces and \(U \subset V\) be an open subset. A
  continuous map \(f : U \to W\) is called \emph{differentiable at \(p \in U\)}
  if there exists a continuous linear operator \(d f_p \in \mathcal{L}(V, W)\)
  such that
  \[
    \frac{\norm{f(p + h) - f(p) - d f_p h}}{\norm{h}} \to 0
  \]
  as \(h \to 0\) in \(V\).
\end{definition}

\begin{definition}
  Given Banach spaces \(V\) and \(W\) and an open subset \(U \subset V\), a
  continuous map \(f : U \to W\) is called \emph{differentiable of class
  \(C^1\)} if \(f\) is differentiable at \(p\) for all \(p \in U\) and the
  \emph{derivative map}
  \begin{align*}
    df: U & \to     \mathcal{L}(V, W) \\
        p & \mapsto d f_p
  \end{align*}
  is continuous. Since \(\mathcal{L}(V, W)\) is a Banach space under the
  operator norm, we may recursively define functions of class \(C^n\) for \(n >
  1\): a function \(f : U \to W\) of class \(C^{n - 1}\) is called
  \emph{differentiable of class \(C^n\)} if the map\footnote{Here we consider
  the \emph{projective tensor product} of Banach spaces. See
  \cite[ch.~1]{klingenberg}.}
  \[
    d^{n - 1} f :
    U \to \mathcal{L}(V, \mathcal{L}(V, \cdots \mathcal{L}(V, W)))
    \cong \mathcal{L}(V^{\otimes n}, W)
  \]
  is of class \(C^1\). Finally, a map \(f : U \to W\) is called
  \emph{differentiable of class \(C^\infty\)} or \emph{smooth} if \(f\) is of
  class \(C^n\) for all \(n > 0\).
\end{definition}

The following lemma is also of huge importance, and it is known as \emph{the
chain rule}.

\begin{lemma}\label{thm:chain-rule}
  Given Banach spaces \(V_1\), \(V_2\) and \(V_3\), open subsets \(U_1 \subset
  V_1\) and \(U_2 \subset V_2\) and two smooth maps \(f : U_1 \to U_2\) and \(g
  : U_2 \to V_3\), the composition map \(g \circ f : U_1 \to V_3\) is smooth
  and its derivative is given by
  \[
    d (g \circ f)_p = d g_{f(p)} \circ d f_p
  \]
\end{lemma}

As promised, these simple definitions allows us to expand the usual tools of
differential geometry to the infinite-dimensional setting. In fact, in most
cases it suffices to simply copy the definition of the finite-dimensional case.
For instance, as in the finite-dimensional case we may call a map between
Banach manifolds \(M\) and \(N\) \emph{smooth} if it can be locally expressed
as a smooth function between open subsets of the model spaces. As such, we will
only provide the most important definitions: those of a Banach manifold and its
tangent space at a given point. Complete accounts of the subject can be found
in \cite[ch.~1]{klingenberg} and \cite[ch.~2]{lang}.

\begin{definition}\label{def:banach-manifold}
  A Banach manifold \(M\) is a Hausdorff topological space endowed with a
  maximal atlas \(\{(U_i, \varphi_i)\}_i\), i.e. an open cover \(\{U_i\}_i\) of
  \(M\) and homeomorphisms \(\varphi_i : U_i \to \varphi_i(U_i) \subset V_i\)
  -- known as \emph{charts} -- where
  \begin{enumerate}
    \item Each \(V_i\) is a Banach space
    \item For each \(i\) and \(j\), \(\varphi_i \circ \varphi_j^{-1} :
      \varphi_j(U_i \cap U_j) \subset V_j \to \varphi_i(U_i \cap U_j) \subset
      V_i\) is a smooth map
    \item \(\{(U_i, \varphi_i)\}_i\) is maximal with respect to the items above
  \end{enumerate}
\end{definition}

\begin{definition}
  Given a Banach manifold \(M\) with maximal atlas \(\{(U_i, \varphi_i)\}_i\)
  and \(p \in M\), the tangent space \(T_p M\) of \(M\) at \(p\) is the
  quotient of the space \(\{ \gamma : (- \epsilon, \epsilon) \to M \mid \gamma
  \ \text{is smooth}, \gamma(0) = p \}\) by the equivalence relation that
  identifies two curves \(\gamma\) and \(\eta\) such that
  \((\varphi_i \circ \gamma)'(0) = (\varphi_i \circ \eta)'(0)\) for all \(i\)
  with \(p \in U_i\).
\end{definition}

\begin{definition}
  Given \(p \in M\) and a chart \(\varphi_i : U_i \to V_i\) with \(p \in U_i\),
  let
  \begin{align*}
    \phi_{p, i} :    T_p M & \to     V_i                          \\
                  [\gamma] & \mapsto (\varphi_i \circ \gamma)'(0)
  \end{align*}
\end{definition}

\begin{proposition}\label{thm:tanget-space-topology}
  Given \(p \in M\) and a chart \(\varphi_i\) with \(p \in U_i\), \(\phi_{p,
  i}\) is a linear isomorphism. For any given charts \(\varphi_i, \varphi_j\),
  the pullback of the norms of \(V_i\) and \(V_j\) by \(\varphi_i\) and
  \(\varphi_j\) respectively define equivalent norms in \(T_p M\). In
  particular, any choice chart gives \(T_p M\) the structure of a topological
  vector space, and this topology is independent of this choice\footnote{In
  general $T_p M$ is not a normed space, since the norms induced by two
  distinct choices of chard need not to coincide. Nevertheless, the topology
  induced by these norms is the same.}.
\end{proposition}

\begin{proof}
  The first statement about \(\phi_{p, i}\) being a linear isomorphism should
  be clear from the definition of \(T_p M\). The second statement about the
  equivalence of the norms is equivalent to checking that \(\phi_{p, i} \circ
  \phi_{p, j}^{-1} : V_j \to V_i\) is continuous for each \(i\) and \(j\) with
  \(p \in U_i\) and \(p \in U_j\).

  But this follows immediately from the identity
  \[
    \begin{split}
      (\phi_{p, i} \circ \phi_{p, j}^{-1}) v
      & = (\varphi_i \circ \varphi_j^{-1} \circ \gamma_v)'(0) \\
      \text{(chain rule)}
      & = d (\varphi_i \circ \varphi_j^{-1})_{\varphi_j(p)} \dot\gamma_v(0) \\
      & = d (\varphi_i \circ \varphi_j^{-1})_{\varphi_j(p)} v
    \end{split}
  \]
  where \(v \in V_j\) and \(\gamma_v : (-\epsilon, \epsilon) \to V_j\) is any
  smooth curve with \(\gamma_v(0) = \varphi_j(p)\) and \(\dot\gamma_v(0) = v\):
  \(\phi_{p, i} \circ \phi_{p, j}^{-1} = d (\varphi_i \circ
  \varphi_j^{-1})_{\varphi_j(p)}\) is continuous by definition.
\end{proof}

Notice that a single Banach manifold may be ``modeled after'' multiple Banach
spaces, in the sense that the \(V_i\)'s of definition~\ref{def:banach-manifold}
may vary with \(i\). Lemma~\ref{thm:chain-rule} implies that for each \(i\) and
\(j\) with \(p \in U_i \cap U_j\), \((d \varphi_i \circ
\varphi_j^{-1})_{\varphi_j(p)} : V_j \to V_i\) is a continuous linear
isomorphism, so that we may assume that each connected component of \(M\) is
modeled after a single Banach space \(V\). It is sometimes convenient, however,
to allow ourselves the more lenient notion of Banach manifold afforded by
definition~\ref{def:banach-manifold}.

We should also note that some authors assume that both the \(V_i\)'s and \(M\)
itself are \emph{separable}, in which case the assumption that \(M\) is
Hausdorff is redundant. Although we are primarily interested in manifolds
modeled after separable spaces, in the interest of affording ourselves a
greater number of examples we will \emph{not} assume separability -- unless
explicitly stated otherwise. Speaking of examples\dots

\begin{example}
  Any Banach space \(V\) can be seen as a Banach manifold with atlas given by
  \(\{(V, \operatorname{id} : V \to V)\}\) -- sometimes called \emph{an affine
  Banach manifold}. In fact, any open subset \(U \subset V\) of a Banach space
  \(V\) is a Banach manifold under a global chart \(\operatorname{id} : U \to
  V\).
\end{example}

\begin{example}
  The group of units \(A^\times\) of a Banach algebra \(A\) is an open subset,
  so that it constitutes a Banach manifold modeled after \(A\)
  \cite[sec.~3]{eells}. In particular, given a Banach space \(V\) the group
  \(\operatorname{GL}(V)\) of continuous linear isomorphisms \(V \to V\) is a
  -- possibly non-separable -- Banach manifold modeled after the space
  \(\mathcal{L}(V) = \mathcal{L}(V, V)\) under the operator norm:
  \(\operatorname{GL}(V) = \mathcal{L}(V)^\times\).
\end{example}

\begin{example}
  Given a complex Hilbert space \(H\), the space \(\operatorname{U}(H)\) of
  unitary operators \(H \to H\) -- endowed with the topology of the operator
  norm -- is a Banach manifold modeled after the closed subspace
  \(\mathfrak{u}(H) \subset \mathcal{L}(H)\) of continuous skew-symmetric
  operators \(H \to H\) \cite[p.~4]{unitary-group-strong-topology}.
\end{example}

These 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 these are both examples of
function spaces. Having reviewed the basics of the theory of Banach manifolds
we can proceed to our in-depth exploration of a particular example, that of the
space \(H^1(I, M)\).

