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

diff --git a/sections/applications.tex b/sections/applications.tex
@@ -19,7 +19,7 @@ instance: any piece-wise smooth function \(H : I \times (-\epsilon, \epsilon)
 \to M\) determines a variation \(\{ \gamma_t \}_t\) given by \(\gamma_t(s) =
 H(s, t)\). This is representative of the theory that lies ahead, in the sense
 that most of the results we'll discuss in the following are minor refinements
-to the classical theory. Instead, the value of the theory we will develop in
+of the classical theory. Instead, the value of the theory we will develop in
 here lies in its conceptual simplicity: instead of relying in ad-hoc methods we
 can now use the standard tools of calculus to study the critical points of the
 energy functional \(E\).
@@ -29,7 +29,7 @@ a smooth function \(E \in C^\infty(H^1(I, M))\) we can study its classical
 ``critical points'' -- i.e. curves \(\gamma\) with a variation \(\{ \gamma_t
 \}_t\) such that \(\left.\frac\dd\dt\right|_{t = 0} E(\gamma_t) = 0\) -- by
 looking at its derivative. The first variation of energy thus becomes a
-particular case of the formula for \(d E\), and the second variation of energy
+particular case of a formula for \(d E\), and the second variation of energy
 becomes a particular case of a formula for the Hessian of \(E\) at a critical
 point. Without further ado, we prove\dots
 
@@ -136,7 +136,7 @@ We begin with a technical lemma.
                                     X & \mapsto X_0 &
                                     X & \mapsto X_1
   \end{align*}
-  are surjective maps for all \(\gamma \in H^1(\gamma^* TM)\).
+  are surjective maps for all \(\gamma \in H^1(I, M)\).
 \end{proof}
 
 We can now show\dots
@@ -195,9 +195,8 @@ of \(E\!\restriction_{\Omega_{p q} M}\) and \(E\!\restriction_{\Lambda M}\).
   \(\dot\gamma(0) = \dot\gamma(1)\) and \(X_0 = X_1\). This establishes that
   the geodesics are indeed critical points of the restrictions of \(E\).
 
-  Suppose \(\gamma \in \Omega_{p q} M\) is a critical point of
-  \(E\!\restriction_{\Omega_{p q} M}\) and let \(Y, Z \in H^1(\gamma^* TM)\) be
-  such that
+  Suppose \(\gamma \in \Omega_{p q} M\) is a critical point and let \(Y, Z \in
+  H^1(\gamma^* TM)\) be such that
   \begin{align*}
     \frac\nabla\dt Y & = \partial \gamma &
                  Y_0 & = 0 &
@@ -216,10 +215,10 @@ of \(E\!\restriction_{\Omega_{p q} M}\) and \(E\!\restriction_{\Lambda M}\).
   \]
   and
   \[
-    0
-    = d E_\gamma X
+    \langle \partial \gamma, \partial \gamma - Z \rangle_0
     = \left\langle \partial \gamma, \frac\nabla\dt X \right\rangle_0
-    = \langle \partial \gamma, \partial \gamma - Z \rangle_0,
+    = d E_\gamma X
+    = 0,
   \]
   which implies \(\norm{\partial \gamma - Z}_0^2 = 0\). In other words,
   \(\partial \gamma = Z \in H^1(\gamma^* TM)\) and therefore \(\frac\nabla\dt
@@ -315,11 +314,12 @@ technical lemma, whose proof amounts to an uninspiring exercise in analysis --
 see lemma 2.4.6 of \cite{klingenberg}.
 
 \begin{lemma}\label{thm:inclusion-submnfds-is-compact}
-  Let \(\Omega_{p q}^0 M \subset C^0(I, M)\) be the space of curves joining
-  \(p\) to \(q\). Then the inclusion \(\Omega_{p q} M \longhookrightarrow
-  \Omega_{p q}^0 M\) is continuous and compact. Likewise, if \(\Lambda^0 M
-  \subset C^0(I, M)\) is the space of free loops then the inclusion \(\Lambda M
-  \longhookrightarrow \Lambda^0 M\) is continuous and compact.
+  Let \(\Omega_{p q}^0 M \subset C^0(I, M)\) be the space of continuous curves
+  joining \(p\) to \(q\). Then the inclusion \(\Omega_{p q} M
+  \longhookrightarrow \Omega_{p q}^0 M\) is continuous and compact. Likewise,
+  if \(M\) is compact and \(\Lambda^0 M \subset C^0(I, M)\) is the space of
+  continuous free loops then the inclusion \(\Lambda M \longhookrightarrow
+  \Lambda^0 M\) is continuous and compact.
 \end{lemma}
 
 As a first consequence, we prove\dots
@@ -408,9 +408,9 @@ As a first consequence, we prove\dots
       \cdot \norm{X}_0
   \end{equation}
 
-  Given a bounded sequence \((X_n)_n \subset \Omega_{p q} M\), it follow from
-  lemma~\ref{thm:inclusion-submnfds-is-compact} that \((X_n)_n\)
-  is relatively compact as a \(C^0\)-sequence. From
+  Given a bounded sequence \((X_n)_n \subset T_\gamma \Omega_{p q} M\), it
+  follow from lemma~\ref{thm:inclusion-submnfds-is-compact} that \((X_n)_n\) is
+  relatively compact as a \(C^0\)-sequence. From
   (\ref{eq:compact-operator-quota}) we then get that \((K_\gamma X_n)_n\) is
   relatively compact as an \(H^1\)-sequence, as desired. The same argument
   holds for \(\Lambda M\) if \(M\) is compact -- so that we can once more apply
@@ -439,7 +439,8 @@ essential for stating Morse's index theorem.
   M = \ker A_\gamma\) and \(T_\gamma^+ \Omega_{p q} M\) is the proper Hilbert
   subspace given by the closure of the subspace spanned by eigenvectors with
   positive eigenvalues. The same holds for critical points \(\gamma\) of
-  \(E\!\restriction_{\Lambda M}\) and \(T_\gamma \Lambda M\).
+  \(E\!\restriction_{\Lambda M}\) and \(T_\gamma \Lambda M\) if \(M\) is
+  compact.
 \end{corollary}
 
 \begin{definition}\label{def:morse-index}
@@ -483,7 +484,7 @@ however, is the following consequence of Morse's theorem.
   \begin{enumerate}
     \item If there are no conjugate points of \(\gamma\) then there exists a
       neighborhood \(U \subset \Omega_{p q} M\) of \(\gamma\) such that
-      \(E(\eta) > E(\gamma)\) for all \(\eta \in U\) with \(\eta \ne 0\).
+      \(E(\eta) > E(\gamma)\) for all \(\eta \in U\) with \(\eta \ne \gamma\).
 
     \item Let \(k > 0\) be the sum of the multiplicities of the conjugate
       points of \(\gamma\) in the interior of \(I\). Then there exists an
@@ -493,15 +494,15 @@ however, is the following consequence of Morse's theorem.
       \]
       of the unit ball \(B^k = \{v \in \RR^k : \norm{v} < 1\}\) with \(i(0) =
       \gamma\), \(E(i(v)) < E(\gamma)\) and \(\ell(i(v)) < \ell(\gamma)\) for
-      all \(v \in B^k \setminus \{ 0 \}\).
+      all nonzero \(v \in B^k\).
   \end{enumerate}
 \end{theorem}
 
 \begin{proof}
   First of all notice that given \(\eta \in U_\gamma\) with \(\eta =
-  \exp_\gamma(X)\), \(X \in H^1(\gamma^* TM)\), the Taylor series for
-  \(E(\eta)\) is given by \(E(\eta) = E(\gamma) + \frac{1}{2} d^2 E_\gamma(X,
-  X) + \cdots\). More precisely,
+  \exp_\gamma(X)\), \(X \in H^1(W_\gamma)\), the Taylor series for \(E(\eta)\)
+  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}
     \to 0
@@ -510,7 +511,7 @@ however, is the following consequence of Morse's theorem.
 
   Let \(\gamma\) be as in \textbf{(i)}. Since \(\gamma\) has no conjugate
   points, it follows from Morses's index theorem that \(T_\gamma^- \Omega_{p q}
-  M = 0\). Furthermore, by noting that any piece-wise smooth \(X \in \ker
+  M = 0\). Furthermore, by noticing that any piece-wise smooth \(X \in \ker
   A_\gamma\) is Jacobi field vanishing at \(p\) and \(q\) one can also show
   \(T_\gamma^0 \Omega_{p q} M = 0\). Hence \(T_\gamma \Omega_{p q} M =
   T_\gamma^+ \Omega_{p q} M\) and therefore \(d^2 E\!\restriction_{\Omega_{p q}
@@ -534,16 +535,16 @@ however, is the following consequence of Morse's theorem.
     = E(\gamma) - \frac{1}{2} \delta^2 \sum_j \lambda_j \cdot v_j + \cdots
   \]
   we find that \(E(i(v)) < E(\gamma)\) for sufficiently small \(\delta\). In
-  particular, \(\ell(i(v)) \le E(i(v))^2 < E(\gamma)^2 = \ell(\gamma)\).
+  particular, \(\ell(i(v))^2 \le E(i(v)) < E(\gamma) = \ell(\gamma)^2\).
 \end{proof}
 
 We should point out that part \textbf{(i)} of theorem~\ref{thm:jacobi-darboux}
 is weaker than the classical formulation of the Jacobi-Darboux theorem -- such
 as in theorem 5.5.3 of \cite{gorodski} for example -- in two aspects. First, we
-do not compare the length of curves in question. This could be amended by
-showing that the length functional \(\ell : H^1(I, M) \to \RR\) is smooth and
-that its Hessian \(d^2 \ell_\gamma\) is given by \(C \cdot d^2 E_\gamma\) for
-some \(C > 0\).
+do not compare the length of curves \(\gamma\) and \(\eta \in U\). This could
+be amended by showing that the length functional \(\ell : H^1(I, M) \to \RR\)
+is smooth and that its Hessian \(d^2 \ell_\gamma\) is given by \(C \cdot d^2
+E_\gamma\) for some \(C > 0\).
 
 Secondly, unlike the classical formulation we only consider curves in an
 \(H^1\)-neighborhood of \(\gamma\) -- instead of a neighborhood of \(\gamma\)