diff --git a/sections/introduction.tex b/sections/introduction.tex
@@ -1,4 +1,4 @@
-\section{Introduction}
+\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
@@ -12,21 +12,20 @@ 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}.
+``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 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 single theory, which we
-describe in the following.
+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
@@ -39,16 +38,30 @@ section functor, such as smooth sections or Sobolev sections -- notice that by
taking \(E = M \times N\) the manifold \(\Gamma(E)\) is naturally identified
with a space of functions \(M \to N\), getting back at the original case.
-In this notes we hope to provide a very brief introduction to 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 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
+In this 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 complete
+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 energy functional
+\begin{align*}
+ E : H^1(I, M) & \to \RR \\
+ \gamma & \mapsto \frac{1}{2} \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 \(E : H^1(I, M) \to \RR\) and describe several applications to the
-study of the geodesics of \(M\). Before moving to the next section, however, we
-would like to review the basics of the theory of real Banach manifolds.
+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. We'll assume basic knowlage 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
+theory.
+
+Before moving to the next section we would like to review the basics of the
+theory of real Banach manifolds.
\subsection{Banach Manifolds}
@@ -208,5 +221,5 @@ manifolds endowed with a group structure whose group operations are smooth.
Perhaps more interesting to us is the fact that this are both examples of
function spaces. Having reviewed the basics of the theory of Banach manifolds
we can prooced to our in-depth exploration of a particular example, that of the
-space \(H^1(I, M)\) of class \(H^1\) curves \([0, 1] \to M\).
+space \(H^1(I, M)\).