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

diff --git a/sections/structure.tex b/sections/structure.tex
@@ -10,7 +10,7 @@ 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
 \(\dot\gamma \in H^0(I, \RR^n) = L^2(I, \RR^n)\). It is a well known fact that
-the so called \emph{Sobolev space \(H^1(I, \RR^n)\)} of all class \(H^1\)
+the so called \emph{Sobolev space \(H^1([0, 1], \RR^n)\)} of all class \(H^1\)
 curves in \(\RR^n\) is a Hilbert space under the inner product given by
 \[
   \langle \gamma, \eta \rangle_1
@@ -47,8 +47,8 @@ described in section~\ref{sec:introduction}, we would like \({C'}^\infty(I,
 \RR^n)\) to be a Banach manifold under which both the energy functional and the
 length functional are smooth maps. As most function spaces, \({C'}^\infty(I,
 \RR^n)\) admits several natural topologies. Some of the most obvious candidates
-are the uniform topology and the topology of \(H^0\) norm, which are the
-topologies induces by the norms
+are the uniform topology and the topology of the \(\norm\cdot_0\) norm, which
+are the topologies induces by the norms
 \begin{align*}
   \norm{\gamma}_\infty & = \sup_t \norm{\gamma(t)}            \\
        \norm{\gamma}_0 & = \int_0^1 \norm{\gamma(t)}^2 \; \dt
@@ -91,20 +91,21 @@ approach \(\ell(\gamma) = \sqrt 2\) as \(n\) approaches \(\infty\).
 
 The issue with this particular example is that while \(\gamma_n \to \gamma\)
 uniformly, \(\dot\gamma_n\) does not converge to \(\dot\gamma\) in the uniform
-topology. This hints at the fact that in order to \(E\) and \(\ell\) to be
+topology. This hints at the fact that in order for \(E\) and \(\ell\) to be
 continuous maps we need to control both \(\gamma\) and \(\dot\gamma\). Hence a
 natural candidate for a norm in \({C'}^\infty(I, \RR^n)\) 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 \(H^0(I, \RR^n)\).
+\rangle_1\) -- here \(\norm{\cdot}_0\) denote the norm of \(H^0(I, \RR^n) =
+L^2(I, \RR^n)\).
 
 The other issue we face is one of completeness. Since \(\RR^n\) has a global
 chart, we expect \({C'}^\infty(I, \RR^n)\) to be affine too. In other words, it
 is natural to expect \({C'}^\infty(I, \RR^n)\) to be Banach space. In
 particular, \({C'}^\infty(I, \RR^n)\) must be complete. This is unfortunately
-not the case for \({C'}^\infty(I, \RR^n)\) the norm \(\norm\cdot_1\), but we
+not the case for \({C'}^\infty(I, \RR^n)\) in the \(\norm\cdot_1\) norm, but we
 can consider its completion. Lo and behold, a classical result by Lebesgue
 establishes that this completion just so happens to coincide with \(H^1(I,
 \RR^n)\).
@@ -125,9 +126,9 @@ of sections of vector bundles over the unit interval \(I\). Explicitly, we
 find\dots
 
 \begin{proposition}
-  Given an Euclidean bundle \(E \to I\), the space \(C^0(E)\) of all continuous
-  sections of \(E\) is is the completion of \({C'}^\infty(E)\) under the norm
-  given by
+  Given an Euclidean bundle \(E \to I\) -- i.e. a vector bundle endowed with a
+  Riemannian metric -- the space \(C^0(E)\) of all continuous sections of \(E\)
+  is the completion of \({C'}^\infty(E)\) under the norm given by
   \[
     \norm{\xi}_\infty = \sup_t \norm{\xi_t}
   \]
@@ -135,7 +136,7 @@ find\dots
 
 \begin{proposition}\label{thm:h0-bundle-is-complete}
   Given an Euclidean bundle \(E \to I\), the space \(H^0(E)\) of all square
-  integrable sections of \(E\) is is the completion of \({C'}^\infty(E)\) under
+  integrable sections of \(E\) is the completion of \({C'}^\infty(E)\) under
   the inner product given by
   \[
     \langle \xi, \eta \rangle_0 = \int_0^1 \langle \xi_t, \eta_t \rangle \; \dt
@@ -143,10 +144,9 @@ find\dots
 \end{proposition}
 
 \begin{proposition}
-  Given an Euclidean bundle \(E \to I\) -- i.e. a vector bundle endowed with a
-  Riemannian metric -- the space \(H^1(E)\) of all class class \(H^1\) sections
-  of \(E\) is the completion of the space \({C'}^\infty(E)\) of piece-wise
-  smooth sections of \(E\) under the inner product given by
+  Given an Euclidean bundle \(E \to I\), the space \(H^1(E)\) of all class
+  \(H^1\) sections of \(E\) is the completion of the space \({C'}^\infty(E)\)
+  of piece-wise smooth sections of \(E\) under the inner product given by
   \[
     \langle \xi, \eta \rangle_1
     = \langle \xi, \eta \rangle_0 +
@@ -166,9 +166,16 @@ find\dots
 \end{proposition}
 
 \begin{proof}
-  Fix \(\xi \in H^1(E)\) and \(t_0, t_1 \in I\) with \(\norm{\xi}_0 \ge
-  \norm{\xi_{t_0}}\) and \(\norm{\xi}_\infty = \norm{\xi_{t_1}}\).
-  Then
+  Given \(\xi \in H^0(E)\) we have
+  \[
+    \norm{\xi}_0^2
+    = \int_0^1 \norm{\xi_t}^2 \; \dt
+    \le \int_0^1 \norm{\xi}_\infty \; \dt
+    = \norm{\xi}_\infty
+  \]
+
+  Now given \(\xi \in H^1(E)\) fix \(t_0, t_1 \in I\) with \(\norm{\xi}_\infty
+  = \norm{\xi_{t_1}}\). Then
   \[
     \begin{split}
       \norm{\xi}_\infty^2
@@ -181,19 +188,11 @@ find\dots
       \text{(Cauchy-Schwarz)}
       & \le \norm{\xi_{t_0}}^2 + 2 \int_0^1
         \norm{\xi_s} \cdot \norm{\nabla_{\frac\dd{\dd s}} \xi_s} \; \dd s \\
-      & \le \norm{\xi}_0^2
+      & \le \norm{\xi}_\infty^2
         + \norm{\xi}_0^2 + \norm{\nabla_{\frac\dd\dt} \xi}_0^2 \\
       & \le 2 \norm{\xi}_1^2
     \end{split}
   \]
-
-  Finally, given \(\xi \in H^0(E)\) we have
-  \[
-    \norm{\xi}_0^2
-    = \int_0^1 \norm{\xi_t}^2 \; \dt
-    \le \int_0^1 \norm{\xi}_\infty \; \dt
-    = \norm{\xi}_\infty
-  \]
 \end{proof}
 
 \begin{note}
@@ -318,8 +317,8 @@ precisely\dots
 We would also like to point out that this is a particular case of a more
 general construction: that of the Banach manifold \(H^1(E)\) of class \(H^1\)
 sections of a smooth fiber bundle \(E \to I\) -- not necessarily a vector
-bundle. Our construction of \(H^1(I, M)\) is equivalent to of of the manifold
-\(H^1(I \times M)\) in the sense that the canonical map
+bundle. Our construction of \(H^1(I, M)\) is equivalent to that of the manifold
+\(H^1(I \times M)\), in the sense that the canonical map
 \[
   \arraycolsep=1pt
   \begin{array}{rl}
@@ -339,7 +338,9 @@ of a vector bundle -- the so called \emph{vector bundle neighborhoods of
 \(E\)}. This construction is highlighted in great detail and generality in the
 first section of \cite[ch.~11]{palais}, but unfortunately we cannot afford such
 a diversion in this short notes. Having said that, we are now finally ready to
-discuss the Riemannian structure of \(H^1(I, M)\).
+layout the Riemannian structure of \(H^1(I, M)\).
+
+We are finally ready to discuss some applications.
 
 \subsection{The Metric of \(H^1(I, M)\)}
 
@@ -352,13 +353,13 @@ proposition~\ref{thm:tanget-space-topology}. In fact, this isomorphisms may be
 extended to a canonical isomorphism of vector bundles, as seen in\dots
 
 \begin{lemma}\label{thm:alpha-fiber-bundles-definition}
-  Given \(i = 0, 1\), the collection \(\{(\psi_{i, \gamma}(H^1(U_\gamma) \times
+  Given \(i = 0, 1\), the collection \(\{(\psi_{i, \gamma}(H^1(W_\gamma) \times
   H^i(\gamma^* TM)), \psi_{i, \gamma}^{-1})\}_{\gamma \in {C'}^\infty(I,
   M)}\) with
   \[
     \arraycolsep=1pt
     \begin{array}{rl}
-      \psi_{i, \gamma} : H^1(U_\gamma) \times H^i(\gamma^* TM) \to
+      \psi_{i, \gamma} : H^1(W_\gamma) \times H^i(\gamma^* TM) \to
       & \coprod_{\eta \in H^1(I, M)} H^i(\eta^* TM) \\
       (X, Y) \mapsto &
       \begin{array}[t]{rl}
@@ -384,20 +385,20 @@ extended to a canonical isomorphism of vector bundles, as seen in\dots
 \end{proposition}
 
 \begin{proof}
-  Note that the sets \(H^1(U_\gamma) \times T_\gamma H^1(I, M)\) are precisely
+  Note that the sets \(H^1(W_\gamma) \times T_\gamma H^1(I, M)\) are precisely
   the images of the canonical charts
   \[
     \varphi_\gamma :
-    \varphi_\gamma^{-1}(H^1(U_\gamma) \times T_\gamma H^1(I, M))
+    \varphi_\gamma^{-1}(H^1(W_\gamma) \times T_\gamma H^1(I, M))
     \subset T H^1(I, M)
-    \to H^1(U_\gamma) \times T_\gamma H^1(I, M)
+    \to H^1(W_\gamma) \times T_\gamma H^1(I, M)
   \]
   of \(T H^1(I, M)\).
 
   By composing charts we get local vector bundle isomorphism \(\psi_{1, \gamma}
   \circ (\id, \varphi_\gamma) \circ \varphi_\gamma :
-  \varphi_\gamma^{-1}(H^1(U_\gamma) \times T_\gamma H^1(I, M)) \isoto \psi_{1,
-  \gamma}(H^1(U_\gamma) \times H^1(\gamma^* TM))\). Because of the fact that
+  \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
@@ -467,7 +468,7 @@ words, we'll show\dots
 \begin{proposition}\label{thm:partial-is-smooth-sec}
   The map
   \begin{align*}
-    \partial : H^1(I, M) & \to     \coprod_{\gamma} H^0\gamma^* TM \\
+    \partial : H^1(I, M) & \to     \coprod_{\gamma} H^0(\gamma^* TM) \\
                   \gamma & \mapsto \dot\gamma
   \end{align*}
   is a smooth section of \(\coprod_{\gamma \in H^1(I, M)} H^0(\gamma^* TM) \to
@@ -482,8 +483,8 @@ words, we'll show\dots
   \begin{align*}
     \mathfrak{X}(H^1(I, M))
     & \to \Gamma\left(\coprod_\gamma H^0(\gamma^* TM)\right) \\
-    \xi
-    & \mapsto \nabla_\xi^0 \partial
+    \tilde X
+    & \mapsto \nabla_{\tilde X}^0 \partial
   \end{align*}
   is such that
   \[