diff --git a/sections/introduction.tex b/sections/introduction.tex
@@ -1,35 +1,53 @@
\section{Introduction}
-With applications in numerous fields of geometry and mathematical physics, such
-as the calculus of variations and the study of harmonic functions, the study of
-infinite-dimensional manifolds has proven itself a remarkbly powerful tool.
Known as \emph{global analysis}, or sometimes \emph{non-linear functional
-analysis}, the field of study dedicated to the understanding of such manifolds
-has seen remarble progress in the past several decades.
-
-Besides the study of Banach Lie groups and algebras, the primary motivation
-behind the study of infinite-dimensional manifolds is the fact that they
-provide a framework to attack local problems regarding maps between manifolds
-by means of global tools -- hence the name ``\emph{global} analysis''.
-Explicitly, by endowing certain function spaces with differential structures we
-may translate local questions about this maps to global questions about such
-manifolds.
-
-At first it may seem like this has nothing to do with infinite-dimensional
-manifolds, but in practice carying out the process of assigning a meaningful
-differential strucutre to a function space usually requires us to drop the
-assumption of finite-dimensionality. In this notes we hope to provide a very
-breif introduction to the field of global analysis by exploring a concrete
-example of the program described in the above: we study the differential
-structure of the Banach manifold \(H^1(I, M)\) of class \(H^1\) curves in a
-complete finite-dimensional Riemannian manifold \(M\).
-
-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 describe several applications to variational
-calculus and in particular the study of the geodesics of \(M\). Bofore moving
-to the next section, however, we would like to review the basics of the theory
-of Banach manifolds.
+analysis}, the field of study dedicated to the understanding of
+infinite-dimensional manifolds has seen remarble progress in the past several
+decades. Among its numerous contributions to the field of geometry at large,
+perhaps the greatest achievement of global analysis in the last century was the
+recognition of the fact that many interesting function spaces posseces 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 linear functionals in functions spaces. The usual
+suspects are, of course, geodesics -- critical points of the energy functional
+\(E\) -- 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 generaly concerned with finding functions that minimize or maximize a
+given functional, known as the 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 unbrela of
+``variational method'' have been unified into a single theory.
+
+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 \(f : \mathscr{F} \to \RR\) we can find minimizing and
+maximizing functions by studying the critical points of \(f\). More generaly,
+modern calculus of variations is interested in studying critical points of
+smooth functionals \(\Gamma(E) \to \RR\), where \(E \to M\) is a smooth fiber
+bundle over a finite-dimensional manifold \(M\) and \(\Gamma(E)\) is a manifold
+of a certain class of sections of \(E\), such as smooth sections or Sobolev
+sections -- notice that by taking \(E = M \times N\) the manifold \(\Gamma(E)\)
+is naturaly identified with a space of functions \(M \to N\).
+
+In this notes we hope to provide a very breif introduction to the calculus of
+variations by exploring one of 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 complete finite-dimensional
+Riemannian manifold \(M\). 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 sutdy the critical points of the energy
+functional \(E : H^1(I, M) \to \RR\) and describe several applications to the
+study of the geodesics of \(M\). Bofore moving to the next section, however, we
+would like to review the basics of the theory of real Banach manifolds.
\subsection{Banach Manifolds}
@@ -44,17 +62,18 @@ space have a closed complement}. The reason behind this is simple: it turns out
that calculus has nothing to do with \(\RR^n\).
What we mean by this last statement is that none of the fundamental ingrediants
-of calculus in \(\RR^n\), namely the fact that it is a complete normed space,
-are specific to \(\RR^n\). In fact, the ingrediants previously described 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.
+of calculus -- the ones necessary to define differentiable functions in
+\(\RR^n\), namely the fact that \(\RR^n\) is a complete normed space -- are
+specific to \(\RR^n\). In fact, this ingrediants 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 manifolds 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 \(d f_p \in
- \mathscr{B}(V, W)\) such that
+ \mathcal{L}(V, W)\) such that
\[
\frac{\norm{f(p + h) - f(p) - d f_p h}}{\norm{h}} \to 0
\]
@@ -67,17 +86,17 @@ manifolds to spaces modeled after Banach spaces. We begin by the former.
\(C^1\)} if \(f\) is differentiable at \(p\) for all \(p \in U\) and the
derivative map
\begin{align*}
- df: U & \to \mathscr{B}(V, W) \\
+ df: U & \to \mathcal{L}(V, W) \\
p & \mapsto d f_p
\end{align*}
- is continuous. Since \(\mathscr{B}(V, W)\) is a Banach space under the
+ 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 \(C^n\) is called
\emph{differentiable of class \(C^{n + 1}\)} if the map
\[
d^n f :
- U \to \mathscr{B}(V, \mathscr{B}(V, \cdots \mathscr{B}(V, W)))
- \cong \mathscr{B}(V^{\otimes n}, W)
+ U \to \mathcal{L}(V, \mathcal{L}(V, \cdots \mathcal{L}(V, W)))
+ \cong \mathcal{L}(V^{\otimes n}, W)
\footnote{Here we consider the \emph{projective tensor product} of Banach
spaces. See \cite[ch.~1]{klingenberg}.}
\]
@@ -104,7 +123,7 @@ N\) 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} \cite[ch.~2]{lang}.
+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
@@ -170,15 +189,15 @@ unless explicitly stated otherwise. Speaking of examples\dots
so that it constitutes a Banach manifold modeled after \(A\). In particular,
given a Banach space \(V\) the group \(\GL(V)\) of continuous linear
isomorphisms \(V \to V\) is a -- possibly non-separable -- Banach manifold
- modeled after the space \(\mathscr{B}(V) = \mathscr{B}(V, V)\) under the
- operator norm: \(\GL(V) = \mathscr{B}(V)^\times\).
+ modeled after the space \(\mathcal{L}(V) = \mathcal{L}(V, V)\) under the
+ operator norm: \(\GL(V) = \mathcal{L}(V)^\times\).
\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
- \(\mathfrak{u}(H) \subset \mathscr{B}(H)\) of continuous skew-symmetric
+ \(\mathfrak{u}(H) \subset \mathcal{L}(H)\) of continuous skew-symmetric
operators \(H \to H\) \cite[p.~4]{unitary-group-strong-topology}.
\end{example}