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

diff --git a/main.tex b/main.tex
@@ -1,6 +1,7 @@
 \input{preamble}
+\addbibresource{references.bib}
 
-\title{Global Analysis \& the Hilbert Manifold of \(H^1\) Curves}
+\title{Variational Calculus \& the Hilbert Manifold of \(H^1\) Curves}
 \author{Thiago Brevidelli Garcia -- 4638749}
 \date{July 2022}
 
@@ -10,6 +11,8 @@
 \tableofcontents
 
 \input{sections/introduction}
+\input{sections/diff-structure}
+\input{sections/aplications}
 
 \printbibliography
 

diff --git a/plan.md b/plan.md
@@ -1,5 +1,34 @@
 # Trabalho Gorodski
 
+## Plano
+
+1. Introdução
+  * Espaços de função e a ideia da construção geral do Palais
+    * Muitas coisas são pontos críticos de funcionais
+    * Nosso amado H¹(I, M) e suas aplicações ao cálculo variacional
+    * Falar sobre outras aplicações de espaços de funções
+  * Variedades Banach: o que são e onde habitam?
+    * Falar brevemente sobre cálculo em espaços de Banach e definir as
+      variedades
+    * Exemplos de variedades Banach
+    * Falar do teorema de Henderson?
+2. Por que queremos curvas H¹ e não só curvas suaves por partes?
+3. As cartas do H¹(I, M)
+  * Falar do espaço tangente Tγ H¹(I, M) ≅ H¹(γ\* TM)
+  * Explicitar o isomorfismo entre a construção do Palais e a do Klingenberg
+  * Comentar dos exemplos mais aprofundados na seção 6 do Eells
+  * A métrica de H¹(I, M)
+4. Aplicações ao cálculo variacional
+  * Energia e comprimento são suaves
+  * Pontos críticos do funcional energia
+  * Fórmulas pra primeira 
+  * Hessiana em variedades Riemannianas e a segunda variação da energia
+    * Definição da Hessiana e motivação mínima
+    * Fórmula geral da segunda variação
+  * Compacidade do operador simétrico associado à segunda variação e paranaues
+    sobre índice de Morse
+  * Jacobi-Darboux
+
 ## O que muda de dim finita pra dim infinita?
 
 * Nada! (do ponto de vista de definições)

diff --git a/references.bib b/references.bib
@@ -1,126 +1,26 @@
-@book{etingof,
-   title =     {Introduction to Representation Theory},
-   author =    {Pavel Etingof},
-   publisher = {American Mathematical Society},
-   year =      {2011},
-   series =    {Student Mathematical Library},
-}
-
-@article{frobenius1896gruppencharakteren,
-  title={{\"U}ber Gruppencharakteren},
-  author={Frobenius, F Georg},
-  journal={Wiss. Berlin},
-  pages={985--1021},
-  year={1896}
-}
-
-@article{griess-1982,
-  doi = {10.1007/bf01389186},
-  year = {1982},
-  month = feb,
-  publisher = {Springer Science and Business Media {LLC}},
-  volume = {69},
-  number = {1},
-  pages = {1--102},
-  author = {Robert L. Griess},
-  title = {The friendly giant},
-  journal = {Inventiones mathematicae}
-}
-
-@book{pioneers,
-   title =     {Pioneers of Representation Theory: Frobenius, Burnside, Schur, and Brauer},
-   author =    {Charles W. Curtis},
-   publisher = {American Mathematical Society, London Mathematical Society},
-   isbn =      {9780821890028},
+@book{klingenberg,
+  title={Riemannian Geometry},
+  author={Wilhelm Klingenberg},
+  isbn={9783110905120},
+  series={De Gruyter Studies in Mathematics},
+  year={2011},
+  publisher={De Gruyter}
+}
+
+@book{lang,
+   title =     {Fundamentals of Differential Geometry},
+   author =    {Serge Lang},
+   publisher = {Springer},
+   isbn =      {9780387985930},
    year =      {1999},
-   series =    {History of Mathematics (Volume 15)},
-   edition =   {New},
-}
-
-@article{borcherds-1992,
-  year =      {1992},
-  month =     dec,
-  publisher = {Springer Science and Business Media {LLC}},
-  volume =    {109},
-  number =    {1},
-  pages =     {405--444},
-  author =    {Richard E. Borcherds},
-  title =     {Monstrous moonshine and monstrous Lie superalgebras},
-  journal =   {Inventiones Mathematicae},
-  doi =       {https://doi.org/10.1007/bf01232032},
-}
-
-@book{doutorado-schur,
-  title={{\"U}ber eine Klasse von Matrizen, die sich einer gegebenen Matrix zuordnen lassen},
-  author={Schur, I.},
-  year={1901}
-}
-
-@book{burnside-groups,
-  title={Theory of groups of finite order},
-  author={Burnside, W.},
-  isbn={9781440035456},
-  year={1897},
-  publisher={The University Press}
+   series =    {Graduate Texts in Mathematics    №191},
+   edition =   {1},
 }
 
-@article{noether-hyperkomplexe,
-  title={Hyperkomplexe Gr{\"o}ssen und Darstellungstheorie},
-  author={Noether, Emmy},
-  journal={Mathematische Zeitschrift},
-  volume={30},
-  number={1},
-  pages={641--692},
-  year={1929},
-  publisher={Springer}
+@misc{unitary-group-strong-topology,
+  doi = {10.48550/ARXIV.1309.5891},
+  author = {Martin Schottenloher},
+  title = {The Unitary Group In Its Strong Topology},
+  publisher = {arXiv},
+  year = {2013},
 }
-
-@book{noether-tribute,
-  title={Emmy Noether: A Tribute to Her Life and Work},
-  author={Noether, E. and Brewer, J.W. and Smith, R.G. and Smith, M.K.},
-  isbn={9780824715502},
-  lccn={lc81015203},
-  series={Monographs and textbooks in pure and applied mathematics},
-  year={1981},
-  publisher={M. Dekker}
-}
-
-@book{frobenius1,
-  title={{\"U}ber vertauschbare Matrizen},
-  author={Frobenius, G.},
-  isbn={9783111098005},
-  series={Preussische Akademie der Wissenschaften Berlin: Sitzungsberichte der Preu{\ss}ischen Akademie der Wissenschaften zu Berlin},
-  year={1896},
-  publisher={Reichsdr.}
-}
-
-@book{frobenius2,
-  title={{\"U}ber Gruppencharaktere},
-  author={Frobenius, G.},
-  isbn={9783111097978},
-  series={Preussische Akademie der Wissenschaften Berlin: Sitzungsberichte der Preu{\ss}ischen Akademie der Wissenschaften zu Berlin},
-  year={1896},
-  publisher={Reichsdr.}
-}
-
-@book{frobenius3,
-  title={{\"U}ber die Primfactoren der Gruppendeterminante},
-  author={Frobenius, G.},
-  year={1896},
-}
-
-@book{frobenius4,
-    title={{\"U}ber die Darstellung der endlichen Gruppen durch lineare Substitutionen},
-    author={Frobenius, G.},
-    year={1897},
-}
-    
-
-@book{referenciar,
-  title={Abhandlungen der K{\"o}niglich Preussischen Akademie der Wissenschaften: 1837},
-  author={Deutsche Akademie der Wissenschaften zu Berlin},
-  series={Abhandlungen der K{\"o}niglich Preussischen Akademie der Wissenschaften},
-  url={https://books.google.com.br/books?id=oto6AQAAMAAJ},
-  year={1839},
-  publisher={Verlag der K{\"o}niglichen Akademie der Wissenschaften in Commission bei Georg Reimer.}
-}-
\ No newline at end of file

diff --git a/sections/aplications.tex b/sections/aplications.tex
@@ -0,0 +1 @@
+\section{Aplications to Variational Calculus}\label{sec:aplications}

diff --git a/sections/diff-structure.tex b/sections/diff-structure.tex
@@ -0,0 +1 @@
+\section{The Structure of \(H^1(I, M)\)}\label{sec:structure}

diff --git a/sections/introduction.tex b/sections/introduction.tex
@@ -1 +1,187 @@
 \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. 
+
+\subsection{Banach Manifolds}
+
+While it is certainly true that Banach spaces can look alian 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
+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
+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
+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.
+
+\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
+  \[
+    \frac{\norm{f(p + h) - f(p) - d f_p h}}{\norm{h}} \to 0
+  \]
+  as \(h \to 0\) in \(U\).
+\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
+  derivative map
+  \begin{align*}
+    df: U & \to     \mathscr{B}(V, W) \\
+        p & \mapsto d f_p
+  \end{align*}
+  is continuous. Since \(\mathscr{B}(V, W)\) 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
+  \[
+    d^n f : 
+    U \to \mathscr{B}(V, \mathscr{B}(V, \cdots \mathscr{B}(V, W)))
+    \cong \mathscr{B}(V^{\otimes n}, W)
+    \footnote{Here we consider the \emph{projective tensor product} of Banach
+    spaces. See \cite[ch.~1]{klingenberg}.}
+  \]
+  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}
+
+\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, this 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 straight up copy the definition of the finite-dimensional
+case. For instance, as in the finite-dimensional case we may call a map \(M \to
+N\) 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}.
+
+\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 \(p \in U_i\).
+  The space \(T_p M\) is a topological vector space isomorphic to \(V_i\) for
+  each \(i\) with \(p \in U_i\): each chart \(\varphi_i : U_i \to M\) induces a
+  norm in \(T_p M\)\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.
+  However, since all $V_i$'s are isomorphic as Banach spaces this norms are
+  all mutually equivalent, so that they define a (unique) topology in $T_p M$.}
+  given by the pullback of the norm of \(V_i\) through the linear isomorphism
+  \begin{align*}
+    \phi_i :    T_p M & \isoto  V_i                          \\
+             [\gamma] & \mapsto (\varphi_i \circ \gamma)'(0)
+  \end{align*}
+\end{definition}
+
+Notice that a single Banach manifold may be ``modeled after'' multible 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
+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
+number of examples we will \emph{not} assume the \(V_i\)'s to be separable --
+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
+  \(\{(V, \id : V \to V)\}\) -- 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 \(\id : U \to V\).
+\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 \(A\). 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: \(\GL(V) = \mathscr{B}(V)^\times\).
+\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
+  operators \(H \to H\) \cite[p.~4]{unitary-group-strong-topology}.
+\end{example}
+
+This last two examples are examples of Banach Lie groups -- i.e. Banach
+manifolds endowed with a group structure whose group operations are smooth.
+