diff --git a/sections/introduction.tex b/sections/introduction.tex
@@ -2,10 +2,10 @@
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 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
+infinite-dimensional manifolds has seen remarkable progress in the past several
+decades. Among its numerous contributions to geometry at large, perhaps the
+greatest achievement in global analysis in the last century was the recognition
+of the fact that many interesting function spaces possesses natural
differentiable structures -- which are usually infinite-dimensional.
As it turns out, many local problems regarding maps between finite-dimensional
@@ -19,52 +19,53 @@ 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
+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 unbrela of
-``variational method'' have been unified into a single theory.
+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.
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
+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 \(\Gamma(E) \to \RR\), 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 \(M \to N\).
+bundle over a finite-dimensional manifold \(M\) and \(\Gamma\) is a given
+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 breif introduction to the calculus of
-variations by exploring one of simplest concrete examples of the previously
+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
-section~\ref{sec:aplications} we sutdy the critical points of the energy
+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 \(M\). Bofore moving to the next section, however, we
+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.
\subsection{Banach Manifolds}
-While it is certainly true that Banach spaces can look alian to someone who has
+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 geommetry can be quite easily
+see, most of the usual tools of differential geometry can be quite easily
translated to the realm of Banach manifolds\footnote{The real difficulties with
-Banach manifolds only show up while proving certain results, and are maily due
+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 \(\RR^n\).
-What we mean by this last statement is that none of the fundamental ingrediants
+What we mean by this last statement is that none of the fundamental ingredients
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
+specific to \(\RR^n\). In fact, this 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.
@@ -160,20 +161,20 @@ subject can be found in \cite[ch.~1]{klingenberg} and \cite[ch.~2]{lang}.
\end{align*}
\end{definition}
-Notice that a single Banach manifold may be ``modeled after'' multible Banach
+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 conviniant, however,
-to allow ourselves the more leniant notion of Banach manifold aforded by
+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 the \(V_i\)'s are
\emph{separable} Banach spaces, 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 afording ourselves a greater
+modeled after separable spaces, in the interest of affording ourselves a greater
number of examples we will \emph{not} assume the \(V_i\)'s to be separable --
unless explicitly stated otherwise. Speaking of examples\dots
@@ -204,4 +205,8 @@ unless explicitly stated otherwise. Speaking of examples\dots
This 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 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 \([0, 1] \to M\).