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

diff --git a/sections/applications.tex b/sections/applications.tex
@@ -2,18 +2,45 @@
 
 As promised, in this section we will apply our understanding of the structure
 of \(H^1(I, M)\) to the calculus of variations, and in particular to the
-geodesics problem. The first thing we should point out 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 here lies in its conceptual
-simplicity: instead of relying in ad-hoc methods we can now use the
+geodesics problem. We also describe some further applications, such as the
+Morse index theorem and the Jacobi-Darboux theorem. We start by defining\dots
+
+\begin{definition}\label{def:variation}
+  Given \(\gamma \in H^1(I, M)\), a variation \(\{ \gamma_t \}_t\) of
+  \(\gamma\) is a smooth curve \(\gamma_\cdot : (-\epsilon, \epsilon) \to
+  H^1(I, M)\) with \(\gamma_0 = \gamma\). We call the vector
+  \(\left.\frac\dd\dt\right|_{t = 0} \gamma_t \in H^1(\gamma^* TM)\) \emph{the
+  variational vector field of \(\{ \gamma_t \}_t\)}.
+\end{definition}
+
+We should note that the previous definition encompasses the classical definition
+of a variation of a curve, as defined in \cite[ch.~5]{gorodski} for instance:
+any piece-wise smooth function \(H : I \times (-\epsilon, \epsilon) \to M\)
+determines a variation \(\{ \gamma_t \}_t\) as in
+definition~\ref{def:variation} 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 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 \(E\).
 
-\begin{theorem}
+What we mean by this last statement is that by look at the energy functional as
+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
+becomes a particular case of a formula for the Hessian of \(E\) at a critical
+point. Without further ado, we prove\dots
+
+\begin{theorem}\label{thm:energy-is-smooth}
   The energy functional
   \begin{align*}
-    E : H^1(I, M) & \to     \RR                                  \\
-           \gamma & \mapsto \frac{1}{2} \norm{\partial \gamma}_0
+    E : H^1(I, M) & \to \RR \\
+    \gamma 
+    & \mapsto \frac{1}{2} \norm{\partial \gamma}_0
+    = \frac{1}{2} \int_0^1 \norm{\dot\gamma(t)}^2 \; \dt
   \end{align*}
   is smooth and \(d E_\gamma X = \left\langle \partial \gamma, \frac\nabla\dt X
   \right\rangle_0\).
@@ -44,16 +71,10 @@ functional \(E\).
   \(\tilde X_\gamma = X\).
 \end{proof}
 
-\begin{definition}
-  Given \(\gamma \in H^1(I, M)\), a variation \(\{ \gamma_t \}_t\) of
-  \(\gamma\) is a smooth curve \(\gamma_\cdot : (-\epsilon, \epsilon) \to
-  H^1(I, M)\) with \(\gamma_0 = \gamma\). We call the vector
-  \(\left.\frac\dd\dt\right|_{t = 0} \gamma_t \in H^1(\gamma^* TM)\) \emph{the
-  variational vector field of \(\{ \gamma_t \}_t\)}.
-\end{definition}
+As promised, by applying the chain rule and using the compatibility of
+\(\nabla\) with the metric we may thus arrive at the classical formula for the
+first variation of energy \(E\).
 
-% TODO: Add a comment beforehand stating that this follows from the chain rule
-% and the compatibility with the metric
 \begin{corollary}
   Given a piece-wise smooth curve \(\gamma : I \to M\) with
   \(\gamma\!\restriction_{[t_i, t_{i + 1}]}\) smooth and a variation \(\{
@@ -67,6 +88,8 @@ functional \(E\).
   \]
 \end{corollary}
 
+Another interesting consequence of theorem~\ref{thm:energy-is-smooth} is\dots
+
 \begin{corollary}
   The only critical points of \(E\) in \(H^1(I, M)\) are the constant curves.
 \end{corollary}
@@ -79,17 +102,28 @@ functional \(E\).
   constant.
 \end{proof}
 
+Another way to put is to say that the problem of characterizing the critical
+points of \(E\) in \(H^1(I, M)\) is not interesting at all. This shouldn't
+really come as a surprise, as most interesting results from the classical
+theory are concerned with particular classes of variations of a curves, such as
+variations with fixed endpoints or variations through loops. In the next
+section we introduce two submanifolds of \(H^1(I, M)\), corresponding to the
+classes of variations previously described, and classify the critical points of
+the restrictions of \(E\) to such submanifolds.
+
 \subsection{The Critical Points of \(E\)}
 
+We begin with a technical lemma.
+
 \begin{lemma}
   The maps \(\sigma, \tau: H^1(I, M) \to M\) with \(\sigma(\gamma) =
   \gamma(0)\) and \(\tau(\gamma) = \gamma(1)\) are submersions.
 \end{lemma}
 
 \begin{proof}
-  To see that \(\sigma\) and \(\tau\) are smooth it suffices to observe that their
-  local representation in \(U_\gamma\) for \(\gamma \in {C'}^\infty(I, M)\) is
-  given by the maps
+  To see that \(\sigma\) and \(\tau\) are smooth it suffices to observe that
+  their local representation in \(U_\gamma\) for \(\gamma \in {C'}^\infty(I,
+  M)\) is given by the maps
   \begin{align*}
     U \subset H^1(W_\gamma) & \to     T_{\gamma(0)} M &
     U \subset H^1(W_\gamma) & \to     T_{\gamma(1)} M \\
@@ -106,6 +140,8 @@ functional \(E\).
   are surjective maps for all \(\gamma \in H^1(\gamma^* TM)\).
 \end{proof}
 
+We can now show\dots
+
 \begin{theorem}
   The subspace \(\Omega_{p q} M \subset H^1(I, M)\) of curves joining \(p, q
   \in M\) is a submanifold whose tangent space \(T_\gamma \Omega_{p q} M\) is
@@ -130,11 +166,14 @@ functional \(E\).
   \Lambda M\) passing through \(\gamma\) -- satisfies \(X_0 = X_1\).
 \end{proof}
 
+Finally, as promised we will provide a characterization of the critical points
+of \(E\!\restriction_{\Omega_{p q} M}\) and \(E\!\restriction_{\Lambda M}\).
+
 \begin{theorem}\label{thm:critical-points-char-in-submanifolds}
   The critical points of \(E\!\restriction_{\Omega_{p q} M}\) are precisely the
-  geodesics joining \(p\) and \(q\). The critical points of
-  \(E\!\restriction_{\Lambda M}\) are the closed geodesics -- including the
-  constant maps.
+  geodesics of \(M\) joining \(p\) and \(q\). The critical points of
+  \(E\!\restriction_{\Lambda M}\) are the closed geodesics of \(M\) --
+  including the constant maps.
 \end{theorem}
 
 \begin{proof}
@@ -194,8 +233,34 @@ functional \(E\).
   conclude that \(\dot\gamma(0) = \dot\eta(\sfrac{1}{2}) = \dot\gamma(1)\).
 \end{proof}
 
+We should point out that the first part of
+theorem~\ref{thm:critical-points-char-in-submanifolds} is a particular case of
+a result regarding critical points of the restriction of \(E\) to the
+submanifold \(H_{N_0, N_1}^1(I, M) \subset H^1(I, M)\) of curves joining
+submanifolds \(N_0, N_1 \subset M\): the critical points of
+\(E\!\restriction_{H_{N_0, N_1}^1(I, M)}\) are the geodesics \(\gamma\) joining
+\(N_0\) to \(N_1\) with \(\dot\gamma(0) \in T_{\gamma(0)} N_0^\perp\) and
+\(\dot\gamma(1) \in T_{\gamma(1)} N_1^\perp\). The proof of this result is
+essentially the same as that of
+theorem~\ref{thm:critical-points-char-in-submanifolds}, given that \(T_\gamma
+H_{N_0, N_1}^1(I, M)\) is subspace of \(H^1\) vector fields \(X\) along
+\(\gamma\) with \(X_0 \in T_{\gamma(0)} N_0\) and \(X_1 \in T_{\gamma(1)}
+N_1\).
+
 \subsection{Second Order Derivatives of \(E\)}
 
+Having establish a clear connection between geodesics and critical points of
+\(E\), the only thing we're missing to complete our goal of providing a modern
+account of the classical theory is a refurnishing of the formula for second
+variation of energy. Intuitively speaking, the second variation of energy
+should be a particular case of a formula for the second derivative of \(E\).
+The issue we face is, of course, that in general there is no such thing as
+``the second derivative'' of a smooth function between manifolds.
+
+Nevertheless, the metric of \(H^1(I, M)\) allow us to discuss ``the second
+derivative'' of \(E\) in a meaning sense via the concept of the Hessian form,
+which we define in the following.
+
 \begin{definition}
   Given a -- possibly infinite-dimensional -- Riemannian manifold \(N\) and a
   smooth functional \(f : N \to \RR\), we call the symmetric tensor
@@ -205,16 +270,8 @@ functional \(E\).
   \emph{the Hessian of \(f\)}.
 \end{definition}
 
-\begin{lemma}
-  Given a geodesic \(\gamma : I \to M\) with \(\gamma(0) = p\) and \(\gamma(1)
-  = q\) and a variation \(\{ \gamma_t \}_t\) of \(\gamma\) with variational
-  vector field \(X\),
-  \[
-    \left.\frac{\dd^2}{\dt^2}\right|_{t = 0} E(\gamma_t)
-    = \int_0^1 \norm{\frac\nabla\dt X}^2
-    - \langle R(X_t, \dot\gamma(t)) \dot\gamma(t), X_t \rangle \; \dt
-  \]
-\end{lemma}
+We can now apply the classical formula for the second variation of energy to
+compute the Hessian of \(E\) at a critical point.
 
 \begin{theorem}
   If \(\gamma \in \Omega_{p q} M\) is a critical point of \(E\) then
@@ -243,6 +300,7 @@ functional \(E\).
     \begin{split}
       (d^2 E\!\restriction_{\Omega_{p q} M})_\gamma(X, X)
       & = \left.\frac{\dd^2}{\dt^2}\right|_{t = 0} E(\gamma_t) \\
+      \text{(second variation of energy)}
       & = \int_0^1 \norm{\frac\nabla\dt X}^2
         - \langle R(X_t, \dot\gamma(t)) \dot\gamma(t), X_t \rangle \; \dt \\
       & = \norm{\frac\nabla\dt X}_0^2 - \langle R_\gamma X, X \rangle_0
@@ -250,6 +308,12 @@ functional \(E\).
   \]
 \end{proof}
 
+Next we discuss some further applications of the theory we've developed so far.
+In particular, we will work towards Morse's index theorem and and describe how
+one can apply it to establish the Jacobi-Darboux theorem. We begin with a
+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
@@ -258,7 +322,9 @@ functional \(E\).
   \longhookrightarrow \Lambda^0 M\) is continuous and compact.
 \end{lemma}
 
-\begin{proposition}
+As a first consequence, we prove\dots
+
+\begin{proposition}\label{thm:energy-is-morse-function}
   Given a critical point \(\gamma\) of \(E\) in \(\Omega_{p q} M\), the
   self-adjoint operator \(A_\gamma : T_\gamma \Omega_{p q} M \to T_\gamma
   \Omega_{p q} M\) given by
@@ -274,7 +340,7 @@ functional \(E\).
 
 \begin{proof}
   Consider \(K_\gamma = - \left( \Id - \frac{\nabla^2}{\dt^2} \right)^{-1}
-  \circ (R_\gamma + \Id)\). We will show that \(K_\gamma\) is compact and
+  \circ (\Id + R_\gamma)\). We will show that \(K_\gamma\) is compact and
   \(A_\gamma = \Id + K_\gamma\) for \(\gamma\) in both \(\Omega_{p q} M\) and
   \(\Lambda M\) -- in which case assume \(M\) is compact.
 
@@ -334,6 +400,7 @@ functional \(E\).
   As for the compactness of \(K_\gamma\) in the case of \(\Omega_{p q} M\),
   from (\ref{eq:compact-partial-result}) we get \(\norm{K_\gamma X}_1^2 = -
   \langle (\Id + R_\gamma) X, K_\gamma X \rangle_0\), so that
+  proposition~\ref{thm:continuous-inclusions-sections} implies
   \begin{equation}\label{eq:compact-operator-quota}
     \norm{K_\gamma X}_1^2
     \le \norm{\Id + R_\gamma} \cdot \norm{K_\gamma X}_\infty \cdot \norm{X}_0
@@ -348,9 +415,16 @@ functional \(E\).
   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
   lemma~\ref{thm:inclusion-submnfds-is-compact}.
-
 \end{proof}
 
+Once again, the first part of this theory is a particular case of a broader
+result regarding the space of curves joining submanifolds of \(M\): if \(N
+\subset M\) is a totally geodesic manifold of codimension \(1\) and \(\gamma
+\in H_{N, \{q\}}^1(I, M)\) is a critical point of the restriction of \(E\) then
+\(A_\gamma = \Id + K_\gamma\). This results aren't that appealing on their
+own, but they allow us to establish the following result, which is essential for
+stating Morse's index theorem.
+
 \begin{corollary}
   Given a critical point \(\gamma\) of \(E\!\restriction_{\Omega_{p q} M}\),
   \(A_\gamma\) has either finitely many eigenvalues including \(1\) or
@@ -372,7 +446,7 @@ functional \(E\).
   \(T_\gamma \Lambda M\).
 \end{corollary}
 
-\begin{definition}
+\begin{definition}\label{def:morse-index}
   Given a critical point \(\gamma\) of \(E\!\restriction_{\Omega_{p q} M}\) we
   call the number \(\dim T_\gamma^- \Omega_{p q} M\) \emph{the \(\Omega\)-index
   of \(\gamma\)}. Likewise, we call \(\dim T_\gamma^- \Lambda M\) for a
@@ -382,7 +456,18 @@ functional \(E\).
   \(\Lambda\)-index of \(\gamma\) simply by \emph{the index of \(\gamma\)}.
 \end{definition}
 
-Morse's index theorem:
+This definition highlights one of the greatest strengths of our approach: while
+the index of a geodesic \(\gamma\) can be defined without the aid of the tools
+developed in here, by using of the Hessian form \(d^2 E_\gamma\) we can place
+definition~\ref{def:morse-index} in the broader context of Morse theory and
+differential topology at large. In fact, the geodesics problems and the energy
+functional where among Morse's original proposed applications.
+Proposition~\ref{thm:energy-is-morse-function} amounts to a proof that \(E\) is
+a Morse function, while definition~\ref{def:morse-index} amounts to the
+definition of the Morse index of the function \(E\) at a critical point
+\(\gamma\). 
+
+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
@@ -390,9 +475,13 @@ Morse's index theorem:
   proper conjugate points of \(\gamma\) in the interior of \(I\).
 \end{theorem}
 
-Consequence:
+Unfortunately we do not have the space to include the proof of this result in
+here, but see theorem 2.5.9 of \cite{klingenberg}. This theorem can be
+generalized for \(H_{N, \{q\}}^1(I, M)\) by replacing the notion of conjugate
+point with the notion of focal points of \(N\) -- see theorem 7.5.4 of
+\cite{gorodski} for the classical approach. What we are really interested in,
+however, is the following consequence of Morse's theorem.
 
-% TODO: Sketch a proof
 \begin{theorem}[Jacobi-Darboux]
   Let \(\gamma \in \Omega_{p q} M\) be a critical point of \(E\).
   \begin{enumerate}
@@ -407,8 +496,47 @@ Consequence:
       \[
         i : B^k \to \Omega_{p q} M
       \]
-      of the unit ball \(B^k = \{v \in \RR^k : \norm{v} < 1\}\) with \(F(0) =
+      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 \(v \in B^k \setminus \{ 0 \}\).
   \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,
+  \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
+  \end{equation}
+  as \(X \to 0\).
+
+  Let \(\gamma\) be as in \textbf{(i)}. Since \(\gamma\) has no conjugate
+  points, it follows from Morses's index theorem that \(T_\gamma^- \Omega_{p q}
+  M = 0\). It is also not hard to check that \(T_\gamma^0 \Omega_{p q} M = 0\)
+  too by noting that any smooth \(X \in \ker A_\gamma\) is Jacobi field
+  vanishing at the endpoints of \(\gamma\). Hence \(T_\gamma \Omega_{p q} M =
+  T_\gamma^+ \Omega_{p q} M\) and therefore \(d^2 E\!\restriction_{\Omega_{p q}
+  M}\) is positive-definite. Now given \(\eta = \exp_\gamma(X)\) as before,
+  (\ref{eq:energy-taylor-series}) implies that \(E(\eta) > E(\gamma)\) provided
+  that \(\norm{X}_1\) is sufficiently small.
+
+  As for part \textbf{(ii)}, fix \(\delta > 0\) and an orthonormal basis
+  \(\{X_i : 1 \le i \le k\}\) of \(T_\gamma^- \Omega_{p q} M\) consisting of
+  eigenvectors of \(A_\gamma\) with negative eigenvalues \(- \lambda_i\).
+  Define
+  \begin{align*}
+    i : B^k & \to \Omega_{p q} M \\
+    v & \mapsto \exp_\gamma(\delta (v_1 \cdot X_1 + \cdots + v_k \cdot X_k))
+  \end{align*}
+
+  Clearly \(i\) is an immersion for small enought \(\delta\). Furtheremore,
+  from (\ref{eq:energy-taylor-series}) and 
+  \[
+    E(i(v)) 
+    = E(\gamma) - \frac{1}{2} \delta^2 \sum_i \lambda_i \cdot v_i + \cdots
+  \]
+  we find that \(E(i(v)) < E(\gamma)\) for sufficiently small \(\delta\).
+\end{proof}