global-analysis-and-the-banach-manifold-of-class-h1-curvers

diff --git a/references.bib b/references.bib
@@ -44,3 +44,12 @@
   year={1966}
 }
 
+@book{gorodski,
+  title   = {An introduction to Riemannian geometry},
+  author  = {Claudio Gorodski},
+  edition = {Preliminary version 3},
+  year    = {2022},
+  month   = jun,
+  url     = {https://www.ime.usp.br/~gorodski/teaching/mat5771-2022/master07-05-2022.pdf},
+}
+

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 \(H^1(I, M)\).
 

diff --git a/sections/structure.tex b/sections/structure.tex
@@ -1 +1,43 @@
 \section{The Structure of \(H^1(I, M)\)}\label{sec:structure}
+
+As promised, in this section we will highlight the differential and Riemannian
+strucutures of the space \(H^1(I, M)\) of class \(H^1\) curves in a complete
+finite-dimensional Riemannian manifold \(M\). 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 \(\gamma : I \to
+\RR^n\) is called \emph{a class \(H^1\)} curve if \(\gamma\) is absolutely
+continuous, \(\dot \gamma(t)\) exists for almost all \(t \in I\) and
+\(\dot\gamma \in L^2(I, \RR^n)\). A classical result by Lebesgue states that
+the so called \emph{Sobolev space \(H^1(I, \RR^n)\)} of all class \(H^1\)
+curves in \(\RR^n\) 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 \(\varphi_i \circ \gamma :
+  J \to \RR^n\) for any chart \(\varphi_i : U_i \subset M \to \RR^n\) and
+  \(J \subset I\) a maximal subinterval where \(\varphi_i \circ \gamma\) is
+  defined -- 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 \(H^1(I, M)\) of all
+  class \(H^1\) curves \(I \to M\).
+\end{definition}
+
+\begin{note}
+  From now on we fix \(I = [0, 1]\).
+\end{note}
+
+We should note that every peace-wise smooth curve \(\gamma : I \to M\) is a
+class \(H^1\) curve. This answer raises and aditional 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 peace-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.
+