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

diff --git a/main.tex b/main.tex
@@ -2,9 +2,6 @@
 \usepackage{titling}
 \addbibresource{references.bib}
 
-% Configure how footnote numbers are displayed
-\renewcommand{\thefootnote}{[\arabic{footnote}]}
-
 \title{Global Analysis \& the Banach Manifold of Class \(H^1\) Curves}
 \author{Pablo}
 \date{July 2022}

diff --git a/sections/applications.tex b/sections/applications.tex
@@ -470,9 +470,9 @@ We are now ready to state Morse's index theorem.
 \begin{theorem}[Morse]
   Let \(\gamma \in \Omega_{p q} M\) be a critical point of \(E\). Then the
   index of \(\gamma\) is given of the sum of the multiplicities of the proper
-  conjugate points of \(\gamma\)\footnote{By ``conjugate points of $\gamma$''
-  we of course mean points conjugate to $\gamma(0) = p$ along $\gamma$.} in the
-  interior of \(I\).
+  conjugate points\footnote{By ``conjugate points of a geodesic $\gamma$'' we
+  of course mean points conjugate to $\gamma(0) = p$ along $\gamma$.} of
+  \(\gamma\) in the interior of \(I\).
 \end{theorem}
 
 Unfortunately we do not have the space to include the proof of Morse's theorem
@@ -507,7 +507,9 @@ however, is the following consequence of Morse's theorem.
   is given by \(E(\eta) = E(\gamma) + \frac{1}{2} d^2 E_\gamma(X, X) +
   \cdots\). More precisely,
   \begin{equation}\label{eq:energy-taylor-series}
-    \frac{\abs{E(\eta) - E(\gamma) - \frac12 d^2 E_\gamma(X, X)}}{\norm{X}_1^2}
+    \frac
+      {\abs{E(\exp_\gamma(X)) - E(\gamma) - \frac{1}{2} d^2 E_\gamma(X, X)}}
+      {\norm{X}_1^2}
     \to 0
   \end{equation}
   as \(X \to 0\).

diff --git a/sections/introduction.tex b/sections/introduction.tex
@@ -129,13 +129,13 @@ manifolds to spaces modeled after Banach spaces. We begin by the former.
   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 \(C^{n - 1}\) is called
-  \emph{differentiable of class \(C^n\)} if the map
+  \emph{differentiable of class \(C^n\)} if the map\footnote{Here we consider
+  the \emph{projective tensor product} of Banach spaces. See
+  \cite[ch.~1]{klingenberg}.}
   \[
     d^{n - 1} f :
     U \to \mathcal{L}(V, \mathcal{L}(V, \cdots \mathcal{L}(V, W)))
     \cong \mathcal{L}(V^{\otimes n}, W)
-    \footnote{Here we consider the \emph{projective tensor product} of Banach
-    spaces. See \cite[ch.~1]{klingenberg}.}
   \]
   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
@@ -151,7 +151,7 @@ chain rule}.
   : U_2 \to V_3\), the composition map \(g \circ f : U_1 \to V_3\) is smooth
   and its derivative is given by
   \[
-    (d g \circ f)_p = d g_{f(p)} \circ d f_p
+    d (g \circ f)_p = d g_{f(p)} \circ d f_p
   \]
 \end{lemma}
 
@@ -223,13 +223,13 @@ in \cite[ch.~1]{klingenberg} and \cite[ch.~2]{lang}.
       (\phi_{p, i} \circ \phi_{p, j}^{-1}) v
       & = (\varphi_i \circ \varphi_j^{-1} \circ \gamma_v)'(0) \\
       \text{(chain rule)}
-      & = (d \varphi_i \circ \varphi_j^{-1})_{\varphi_j(p)} \dot\gamma_v(0) \\
-      & = (d \varphi_i \circ \varphi_j^{-1})_{\varphi_j(p)} v
+      & = d (\varphi_i \circ \varphi_j^{-1})_{\varphi_j(p)} \dot\gamma_v(0) \\
+      & = d (\varphi_i \circ \varphi_j^{-1})_{\varphi_j(p)} v
     \end{split}
   \]
   where \(v \in V_j\) and \(\gamma_v : (-\epsilon, \epsilon) \to V_j\) is any
   smooth curve with \(\gamma_v(0) = \varphi_j(p)\) and \(\dot\gamma_v(0) = v\):
-  \(\phi_{p, i} \circ \phi_{p, j}^{-1} = (d \varphi_i \circ
+  \(\phi_{p, i} \circ \phi_{p, j}^{-1} = d (\varphi_i \circ
   \varphi_j^{-1})_{\varphi_j(p)}\) is continuous by definition.
 \end{proof}
 

diff --git a/sections/structure.tex b/sections/structure.tex
@@ -48,10 +48,10 @@ Denote by \({C'}^\infty(I, \mathbb{R}^n)\) the space of piece-wise curves in
 energy functional and the length functional are smooth maps. As most function
 spaces, \({C'}^\infty(I, \mathbb{R}^n)\) admits several natural topologies.
 Some of the most obvious candidates are the uniform topology and the topology
-of the \(\norm\cdot_0\) norm, which are the topologies induces by the norms
+of the \(\norm\cdot_0\) norm, which are the topologies induced by the norms
 \begin{align*}
-  \norm{\gamma}_\infty & = \sup_t \norm{\gamma(t)}            \\
-       \norm{\gamma}_0 & = \int_0^1 \norm{\gamma(t)}^2 \; \dt
+  \norm{\gamma}_\infty & = \sup_t \norm{\gamma(t)}                   \\
+       \norm{\gamma}_0 & = \sqrt{\int_0^1 \norm{\gamma(t)}^2 \; \dt}
 \end{align*}
 respectively.
 
@@ -99,7 +99,7 @@ natural candidate for a norm in \({C'}^\infty(I, \mathbb{R}^n)\) is
   \norm{\gamma}_1^2 = \norm{\gamma}_0^2 + \norm{\dot\gamma}_0^2,
 \]
 which is, of course, the norm induced by the inner product \(\langle \, ,
-\rangle_1\) -- here \(\norm{\cdot}_0\) denote the norm of \(H^0(I,
+\rangle_1\) -- here \(\norm{\cdot}_0\) denotes the norm of \(H^0(I,
 \mathbb{R}^n) = L^2(I, \mathbb{R}^n)\).
 
 The other issue we face is one of completeness. Since \(\mathbb{R}^n\) has a
@@ -168,12 +168,12 @@ find\dots
 
 \begin{proof}
   Given \(\xi \in H^0(E)\) we have
-  \[
+  \begin{equation}\label{eq:zero-norm-le-infty-norm}
     \norm{\xi}_0^2
     = \int_0^1 \norm{\xi_t}^2 \; \dt
-    \le \int_0^1 \norm{\xi}_\infty \; \dt
-    = \norm{\xi}_\infty
-  \]
+    \le \int_0^1 \norm{\xi}_\infty^2 \; \dt
+    = \norm{\xi}_\infty^2
+  \end{equation}
 
   Now given \(\xi \in H^1(E)\) fix \(t_0, t_1 \in I\) with \(\norm{\xi}_\infty
   = \norm{\xi_{t_1}}\). Then
@@ -183,14 +183,21 @@ find\dots
       & = \norm{\xi_{t_0}}^2
         + \int_{t_0}^{t_1} \frac{\dd}{\dd s} \norm{\xi_s}^2 \; \dd s \\
       \text{(\(\nabla\) is compatible with the metric)}
-      & = \norm{\xi_{t_0}}^2 + 2 \int_{t_0}^{t_1}
-        \left\langle \xi_s, \nabla_{\frac\dd{\dd s}} \xi_s \right\rangle
+      & = \norm{\xi_{t_0}}^2 + \int_{t_0}^{t_1}
+        2 \left\langle \xi_s, \nabla_{\frac\dd{\dd s}} \xi_s \right\rangle
         \; \dd s \\
       \text{(Cauchy-Schwarz)}
-      & \le \norm{\xi_{t_0}}^2 + 2 \int_0^1
-        \norm{\xi_s} \cdot \norm{\nabla_{\frac\dd{\dd s}} \xi_s} \; \dd s \\
+      & \le \norm{\xi_{t_0}}^2 + \int_0^1
+        2 \norm{\xi_s} \cdot \norm{\nabla_{\frac\dd{\dd s}} \xi_s} \; \dd s \\
+      & \le \norm{\xi}_\infty^2
+        + \int_0^1 \norm{\xi_s}^2 + \norm{\nabla_{\frac\dd\dt} \xi_s}^2
+        \; \dd s \\
       & \le \norm{\xi}_\infty^2
         + \norm{\xi}_0^2 + \norm{\nabla_{\frac\dd\dt} \xi}_0^2 \\
+      % TODO: This is actually wrong. Fix this.
+      \text{(because of equation (\ref{eq:zero-norm-le-infty-norm}))}
+      & \le \norm{\xi}_0^2
+        + \norm{\xi}_0^2 + \norm{\nabla_{\frac\dd\dt} \xi}_0^2 \\
       & \le 2 \norm{\xi}_1^2
     \end{split}
   \]
@@ -276,10 +283,10 @@ Finally, we find\dots
   for \(H^1(I, M)\) under the final topology of the maps \(\exp_\gamma\) --
   i.e. the coarsest topology such that such maps are continuous. This atlas
   gives \(H^1(I, M)\) the structure of a \emph{separable} Banach manifold
-  modeled after separable Hilbert spaces, with typical representatives
-  \(H^1(\gamma^* TM) \cong H^1(I, \mathbb{R}^n)\)\footnote{Any trivialization
-  of $\gamma^* TM$ induces an isomorphism $H^1(\gamma^* TM) \isoto H^1(I,
-  \mathbb{R}^n)$.}.
+  modeled after separable Hilbert spaces, with typical
+  representatives\footnote{Any trivialization of $\gamma^* TM$ induces an
+  isomorphism $H^1(\gamma^* TM) \isoto H^1(I, \mathbb{R}^n)$.} \(H^1(\gamma^*
+  TM) \cong H^1(I, \mathbb{R}^n)\).
 \end{theorem}
 
 The fact that \(\exp_\gamma\) is bijective should be clear from the definition
@@ -371,9 +378,9 @@ extended to a canonical isomorphism of vector bundles, as seen in\dots
   \]
   gives \(\coprod_{\gamma \in {C'}^\infty(I, M)} H^i(\gamma^* TM) \to H^1(I,
   M)\) the structure of a smooth vector bundle\footnote{Here we use the
-  canonical identification $T_{\gamma(t)} M \cong T_{X_t}^\vee TM$ to apply
-  the vector $Y_t \in T_{\gamma(t)} M$ to the map $(d \exp)_{X_t} : T_{X_t} TM
-  \to T_{\exp_{\gamma(t)}(X_t)} M$.}.
+  canonical identification $T_{\gamma(t)} M \cong T_{X_t} TM$ to apply the
+  vector $Y_t \in T_{\gamma(t)} M$ to the map $(d \exp)_{X_t} : T_{X_t} TM \to
+  T_{\exp_{\gamma(t)}(X_t)} M$.}.
 \end{lemma}
 
 \begin{proposition}
@@ -394,17 +401,43 @@ extended to a canonical isomorphism of vector bundles, as seen in\dots
     \subset T H^1(I, M)
     \to H^1(W_\gamma) \times T_\gamma H^1(I, M)
   \]
-  of \(T H^1(I, M)\).
-
-  By composing charts we get local vector bundle isomorphism \(\psi_{1, \gamma}
-  \circ (\operatorname{id}, \varphi_\gamma) \circ \varphi_\gamma :
-  \varphi_\gamma^{-1}(H^1(W_\gamma) \times T_\gamma H^1(I, M)) \isoto \psi_{1,
-  \gamma}(H^1(W_\gamma) \times H^1(\gamma^* TM))\). Because of the fact that
-  \(\varphi_\gamma\) and \(\psi_{1, \gamma}^{-1}\) are charts, this
-  isomorphisms agree in the intersections, so they may be glued together into a
-  global vector bundle isomorphism. Furthermore, by construction the
-  restriction of this isomorphism to \(T_\gamma H^1(I, M)\) with \(\gamma \in
-  {C'}^\infty(I, M)\) is given by \(\phi_\gamma\).
+  of \(T H^1(I, M)\) -- i.e. the charts given by\footnote{Once more, we use the
+  canonical identification $T_X H^1(W_\gamma) \cong H^1(\gamma^* TM)$ to apply
+  the vector $\phi_\gamma(Y) \in H^1(\gamma^* TM)$ to $(d \exp_\gamma)_X : T_X
+  H^1(W_\gamma) \to T_{\exp_\gamma(X)} H^1(I, M)$.}
+  \begin{align*}
+    \varphi_\gamma^{-1} : H^1(W_\gamma) \times T_\gamma H^1(I, M)
+    & \to T H^1(I, M) \\
+    (X, Y) & \mapsto (d \exp_\gamma)_X \phi_\gamma(Y)
+  \end{align*}
+
+  By composing charts we get fiber-preserving, fiber-wise linear diffeomorphism
+  \[
+    \varphi_\gamma^{-1}(H^1(W_\gamma) \times T_\gamma H^1(I, M))
+    \subset T H^1(I, M)
+    \isoto
+    \psi_{1, \gamma}(H^1(W_\gamma) \times H^1(\gamma^* TM)),
+  \]
+  which takes \(\varphi^{-1}(X, Y) \in T_{exp_\gamma(X)} H^1(I, M)\) to
+  \(\psi_{1, \gamma}(X, \phi_\gamma(Y)) \in H^1(\exp_\gamma(X)^* TM)\). With
+  enought patience, one can deduce from the fact that \(\varphi_\gamma\) and
+  \(\psi_{1, \gamma}^{-1}\) are charts that this maps agree in the intersection
+  of the open subsets \(\varphi_\gamma^{-1}(H^1(W_\gamma) \times T_\gamma
+  H^1(I, M))\), so that they may be glued together into a global smooth map
+  \(\Phi : T H^1(I, M) \to \coprod_{\eta \in H^1(I, M)} H^1(\eta^* TM)\).
+
+  Since this map is a fiber-preserving, fiber-wise linear local diffeomorphism,
+  this is an isomorphism of vector bundles.
+  Furthermore, by construction
+  \[
+    \Phi(X)_t
+    = \psi_{1, \gamma}(0, \phi_\gamma(X))_t
+    = (d \exp)_{0_{\gamma(t)}} \phi_\gamma(X)_t
+    = \phi_\gamma(X)_t
+  \]
+  for each \(\gamma \in {C'}^\infty(I, M)\) and \(X \in T_\gamma H^1(I, M)\).
+  In other words, \(\Phi\!\restriction_{T_\gamma H^1(I, M)} = \phi_\gamma\) as
+  required.
 \end{proof}
 
 At this point it may be tempting to think that we could now define the metric
@@ -516,11 +549,14 @@ At this point it should be obvious that definition~\ref{def:h1-metric} does
 indeed endow \(H^1(I, M)\) with the structure of a Riemannian manifold: the
 inner products \(\langle \, , \rangle_1 : H^1(\gamma^* TM) \times H^1(\gamma^*
 TM) \to \mathbb{R}\) may be glued together into a single positive-definite
-section \(\langle \, , \rangle_1 \in \Gamma( \operatorname{Sym}^2
-\coprod_\gamma H^1(\gamma^* TM))\) -- whose smoothness follows from
+section \(\langle \, , \rangle_1 \in \Gamma\left(\operatorname{Sym}^2
+\coprod_\gamma H^1(\gamma^* TM)\right)\) -- whose smoothness follows from
 theorem~\ref{thm:h0-has-metric-extension},
 proposition~\ref{thm:partial-is-smooth-sec} and
 proposition~\ref{thm:covariant-derivative-h0} -- which is then mapped to a
 positive-definite section of \(\operatorname{Sym}^2 T H^1(I, M)\) by the
-induced isomorphism \(\operatorname{Sym}^2 \coprod_\gamma H^1(\gamma^* TM)
-\isoto \operatorname{Sym}^2 T H^1(I, M)\).
+induced isomorphism
+\[
+  \Gamma\left(\operatorname{Sym}^2 \coprod_\gamma H^1(\gamma^* TM)\right)
+  \isoto \Gamma(\operatorname{Sym}^2 T H^1(I, M))
+\]