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

Riemannian Geometry course project on the manifold H¹(I, M) of class H¹ curves on a Riemannian manifold M and its applications to the geodesics problem

Diffstats

3 files changed, 415 insertions, 415 deletions

diff --git a/main.tex b/main.tex
@@ -13,7 +13,7 @@
 
 \input{sections/introduction}
 \input{sections/structure}
-\input{sections/aplications}
+\input{sections/applications}
 
 \newpage
 \printbibliography

diff --git a/sections/aplications.tex /dev/null
@@ -1,414 +0,0 @@
-\section{Applications to the Calculus of Variations}\label{sec:aplications}
-
-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
-standard tools of calculus to study the critical points of the energy
-functional \(E\).
-
-\begin{theorem}
-  The energy functional
-  \begin{align*}
-    E : H^1(I, M) & \to     \RR                                  \\
-           \gamma & \mapsto \frac{1}{2} \norm{\partial \gamma}_0
-  \end{align*}
-  is smooth and \(d E_\gamma X = \left\langle \partial \gamma, \frac\nabla\dt X
-  \right\rangle_0\).
-\end{theorem}
-
-\begin{proof}
-  The fact that \(E\) is smooth should be clear from the smoothness of
-  \(\partial\) and \(\norm\cdot_0\). Furthermore, from the definition of
-  \(\frac\nabla\dt\) we have
-  \[
-    \begin{split}
-      \left\langle \partial \gamma, \frac\nabla\dt X \right\rangle_0
-      & =
-        \left\langle
-        \partial \gamma, (\nabla_X^0 \partial)_\gamma
-        \right\rangle_0 \\
-      & = \tilde X \langle \partial \gamma, \partial \gamma \rangle -
-        \left\langle
-        \partial (\nabla_X^0 \partial)_\gamma, \partial \gamma
-        \right\rangle_0 \\
-      & = 2 \tilde X E(\gamma) -
-        \left\langle \frac\nabla\dt X, \partial \gamma \right\rangle_0 \\
-      & = 2 d E_\gamma X -
-        \left\langle \partial \gamma, \frac\nabla\dt X \right\rangle_0
-    \end{split}
-  \]
-  where \(\tilde X \in \mathfrak{X}(H^1(I, M))\) is any vector field with
-  \(\tilde X_\gamma = X\).
-\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}
-
-% 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 \(\{
-    \gamma_t \}_t\) of \(\gamma\) with variational vector field \(X\) we have
-  \[
-    \left.\frac\dd\dt\right|_{t = 0} E(\gamma_t)
-    = \sum_i
-      \left. \langle \dot\gamma(t), X_t \rangle\right|_{t = t_i}^{t_{i + 1}}
-    - \int_0^1 \left\langle \frac\nabla\dt \dot\gamma(t), X_t \right\rangle
-      \; \dt
-  \]
-\end{corollary}
-
-\begin{corollary}
-  The only critical points of \(E\) in \(H^1(I, M)\) are the constant curves.
-\end{corollary}
-
-\begin{proof}
-  Clearly every constant curve is a critical point. On the other hand, if
-  \(\gamma \in H^1(I, M)\) is such that \(\left\langle \partial \gamma,
-  \frac\nabla\dt X \right\rangle_0 = d E_\gamma X = 0\) for all \(X \in
-  H^1(\gamma^* TM)\) then \(\partial \gamma = 0\) and therefore \(\gamma\) is
-  constant.
-\end{proof}
-
-\subsection{The Critical Points of \(E\)}
-
-\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
-  \begin{align*}
-    U \subset H^1(W_\gamma) & \to     T_{\gamma(0)} M &
-    U \subset H^1(W_\gamma) & \to     T_{\gamma(1)} M \\
-                          X & \mapsto X_0 &
-                          X & \mapsto X_1
-  \end{align*}
-  which are indeed smooth functions. This local representation also shows that
-  \begin{align*}
-    d\sigma_\gamma : H^1(\gamma^* TM) & \to     T_{\gamma(0)} M &
-    d\tau_\gamma   : H^1(\gamma^* TM) & \to     T_{\gamma(1)} M \\
-                                    X & \mapsto X_0 &
-                                    X & \mapsto X_1
-  \end{align*}
-  are surjective maps for all \(\gamma \in H^1(\gamma^* TM)\).
-\end{proof}
-
-\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
-  the subspace of \(H^1(\gamma^* TM)\) consisting of class \(H^1\) vector fields
-  \(X\) along \(\gamma\) with \(X_0 = X_1 = 0\). Likewise, the space \(\Lambda
-  M \subset H^1(I, M)\) of free loops is a submanifold whose tangent at
-  \(\gamma\) is given by all \(X \in H^1(\gamma^* TM)\) with \(X_0 = X_1\).
-\end{theorem}
-
-\begin{proof}
-  To see that this are submanifolds, it suffices to note that \(\Omega_{p q}
-  M\) and \(\Lambda M\) are the inverse images of the closed submanifolds
-  \(\{(p, q)\}, \{(p, p) : p \in M\} \subset M \times M\) under the submersion
-  \((\sigma, \tau) : H^1(I, M) \to M \times M\).
-
-  The characterization of their tangent bundles should also be clear: any curve
-  \((-\epsilon, \epsilon) \to H^1(I, M)\) passing through \(\gamma \in
-  \Omega_{p q} M\) whose image is contained in \(\Omega_{p q} M\) is a
-  variation of \(\gamma\) with fixed endpoints, so its variational vector field
-  \(X\) satisfies \(X_0 = X_1 = 0\). Likewise, any variation of a loop \(\gamma
-  \in \Lambda M\) trough loops -- i.e. a curve \((-\epsilon, \epsilon) \to
-  \Lambda M\) passing through \(\gamma\) -- satisfies \(X_0 = X_1\).
-\end{proof}
-
-\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.
-\end{theorem}
-
-\begin{proof}
-  We start by supposing that \(\gamma\) is a geodesic. Since \(\gamma\) is
-  smooth,
-  \[
-    d E_\gamma X
-    = \int_0^1
-      \left\langle \dot\gamma(t), \frac\nabla\dt X \right\rangle \; \dt
-    = \int_0^1
-      \frac\dd\dt \langle \dot\gamma(t), X_t \rangle -
-      \left\langle \frac\nabla\dt \dot\gamma(t), X \right\rangle \; \dt
-    = \langle \dot\gamma(1), X_1 \rangle - \langle \dot\gamma(0), X_0 \rangle
-  \]
-
-  Now if \(\gamma \in \Omega_{p q} M\) and \(X \in T_\gamma \Omega_{p q} M\)
-  then \(d E_\gamma X = \langle \dot\gamma(1), 0 \rangle - \langle
-  \dot\gamma(0), 0 \rangle = 0\). Likewise, if \(\gamma\) is a closed geodesic
-  and \(X \in T_\gamma \Lambda M\) we find \(d E_\gamma X = 0\) since
-  \(\dot\gamma(0) = \dot\gamma(1)\) 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
-  \begin{align*}
-    \frac\nabla\dt Y & = \partial \gamma &
-                 Y_0 & = 0 &
-    \frac\nabla\dt Z & = 0 &
-                 Z_1 & = Y_1
-  \end{align*}
-
-  Let \(X_t = Y_t - t Z_t\). Then \(X_0 = X_1 = 0\) and \(\frac\nabla\dt X =
-  \partial \gamma - Z\). Furthermore,
-  \[
-    \langle Z, \partial \gamma - Z \rangle_0
-    = \left\langle Z, \frac\nabla\dt X \right\rangle_0
-    = \int_0^1 \frac\dd\dt \langle Z_t, X_t \rangle \; \dt
-    = \langle Z_1, X_1 \rangle - \langle Z_0, X_0 \rangle
-    = 0
-  \]
-  and
-  \[
-    0
-    = d E_\gamma X
-    = \left\langle \partial \gamma, \frac\nabla\dt X \right\rangle_0
-    = \langle \partial \gamma, \partial \gamma - Z \rangle_0,
-  \]
-  which implies \(\norm{\partial \gamma - Z}_0^2 = 0\). In other words,
-  \(\partial \gamma = Z \in H^1(\gamma^* TM)\) and therefore \(\frac\nabla\dt
-  \dot\gamma(t) = \frac\nabla\dt Z = 0\) -- i.e. \(\gamma\) is a geodesic.
-
-  Finally, if \(\gamma \in \Lambda M\) with \(\gamma(0) = \gamma(1) = p\) we
-  may apply the argument above to conclude that \(\gamma\) is a geodesic
-  joining \(p\) to \(q = p\). To see that \(\gamma\) is a closed geodesic apply
-  the same argument again for \(\eta(t) = \gamma(1 + \sfrac{1}{2})\) to
-  conclude that \(\dot\gamma(0) = \dot\eta(\sfrac{1}{2}) = \dot\gamma(1)\).
-\end{proof}
-
-\subsection{Second Order Derivatives of \(E\)}
-
-\begin{definition}
-  Given a -- possibly infinite-dimensional -- Riemannian manifold \(N\) and a
-  smooth functional \(f : N \to \RR\), we call the symmetric tensor
-  \[
-    d^2 f(X, Y) = \nabla d f (X, Y) =  X Y f - df \nabla_X Y
-  \]
-  \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}
-
-\begin{theorem}
-  If \(\gamma \in \Omega_{p q} M\) is a critical point of \(E\) then
-  \begin{equation}\label{eq:second-variation-general}
-    (d^2 E\!\restriction_{\Omega_{p q} M})_\gamma(X, Y)
-    = \left\langle \frac\nabla\dt X, \frac\nabla\dt Y \right\rangle_0
-    - \langle R_\gamma X, Y \rangle_0,
-  \end{equation}
-  where \(R_\gamma : H^1(\gamma^* TM) \to H^1(\gamma^* TM)\) is given by
-  \((R_\gamma X)_t = R(X_t, \dot\gamma(t)) \dot\gamma(t)\). Formula
-  (\ref{eq:second-variation-general}) also holds for critical points of \(E\)
-  in \(\Lambda M\).
-\end{theorem}
-
-\begin{proof}
-  Given the symmetry of \(d^2 E\), it suffices to take \(X \in T_\gamma
-  \Omega_{p q} M\) and show
-  \[
-    (d^2 E\!\restriction_{\Omega_{p q} M})(X, X)
-    = \norm{\frac\nabla\dt X}_0^2 - \langle R_\gamma X, X \rangle_0
-  \]
-
-  To that end, we fix a variation \(\{ \gamma_t \}_t\) of \(\gamma\) with fixed
-  endpoints and variational field \(X\) and compute
-  \[
-    \begin{split}
-      (d^2 E\!\restriction_{\Omega_{p q} M})_\gamma(X, X)
-      & = \left.\frac{\dd^2}{\dt^2}\right|_{t = 0} E(\gamma_t) \\
-      & = \int_0^1 \norm{\frac\nabla\dt X}^2
-        - \langle R(X_t, \dot\gamma(t)) \dot\gamma(t), X_t \rangle \; \dt \\
-      & = \norm{\frac\nabla\dt X}_0^2 - \langle R_\gamma X, X \rangle_0
-    \end{split}
-  \]
-\end{proof}
-
-\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.
-\end{lemma}
-
-\begin{proposition}
-  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
-  \[
-    \langle A_\gamma X, Y \rangle_1
-    = \langle X, A_\gamma Y \rangle_1
-    = d^2 E_\gamma(X, Y)
-  \]
-  has the form \(A_\gamma = \Id + K_\gamma\) where \(K_\gamma : T_\gamma
-  \Omega_{p q} M \to T_\gamma \Omega_{p q} M\) is a compact operator. The same
-  holds for \(\Lambda M\) if \(M\) is compact.
-\end{proposition}
-
-\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
-  \(A_\gamma = \Id + K_\gamma\) for \(\gamma\) in both \(\Omega_{p q} M\) and
-  \(\Lambda M\) -- in which case assume \(M\) is compact.
-
-  Let \(\gamma \in \Omega_{p q} M\) be a critical point. By
-  theorem~\ref{thm:critical-points-char-in-submanifolds} we know that
-  \(\gamma\) is a geodesic. Let \(X, Y \in \Gamma(\gamma^* TM)\) with \(X_0 =
-  X_1 = Y_0 = Y_1 = 0\). Then
-  \begin{equation}\label{eq:compact-partial-result}
-    \begin{split}
-      \langle X, Y \rangle_1
-      & = \langle X, Y \rangle_0
-        + \left\langle \frac\nabla\dt X, \frac\nabla\dt Y \right\rangle \\
-      & = \langle X, Y \rangle_0
-        + \int_0^1
-          \left\langle \frac\nabla\dt X, \frac\nabla\dt Y \right\rangle
-          \; \dt \\
-      & = \langle X, Y \rangle_0
-        + \int_0^1
-          \frac\dd\dt \langle X_t, Y_t \rangle -
-          \left\langle \frac{\nabla^2}{\dt^2} X, Y \right\rangle \; \dt \\
-      & = \langle X, Y \rangle_0
-        - \left\langle \frac{\nabla^2}{\dt^2} X, Y \right\rangle_0
-        + \left.\langle X_t, Y_t \rangle\right|_{t = 0}^1 \\
-      & = \left\langle
-          \left(\Id - \frac{\nabla^2}{\dt^2}\right) X, Y
-          \right\rangle_0
-    \end{split}
-  \end{equation}
-
-  Since \(\Gamma(\gamma^* TM) \subset H^1(\gamma^* TM)\) is dense,
-  (\ref{eq:compact-partial-result}) extends to all of \(T_\gamma \Omega_{p q}
-  M\). Hence given \(X, Y \in T_\gamma \Omega_{p q} M\) we have
-  \[
-    \begin{split}
-      \langle A_\gamma X, Y \rangle_1
-      & = \left\langle \frac\nabla\dt X, \frac\nabla\dt Y \right\rangle_0
-        - \langle R_\gamma X, Y \rangle_0 \\
-      & = \langle X, Y \rangle_1 - \langle X, Y \rangle_0
-        - \langle R_\gamma X, Y \rangle_0 \\
-      & = \langle X, Y \rangle_1 - \langle (\Id + R_\gamma) X, Y \rangle_0 \\
-      & = \langle X, Y \rangle_1
-        - \left\langle
-          \left( \Id - \frac{\nabla^2}{\dt^2} \right)^{-1}
-          \circ (\Id + R_\gamma) X, Y
-          \right\rangle_1 \\
-      & = \langle X, Y \rangle_1 + \langle K_\gamma X, Y \rangle_1 \\
-    \end{split}
-  \]
-
-  Now consider a critical point \(\gamma \in \Lambda M\) -- i.e. a closed
-  geodesic. Equation (\ref{eq:compact-partial-result}) also holds for \(X, Y
-  \in \Gamma(\gamma^* TM)\) with \(X_0 = X_1\) and \(Y_0 = Y_1\), so it holds
-  for all \(X, Y \in T_\gamma \Lambda M\). Hence by applying the same argument
-  we get \(\langle A_\gamma X, Y \rangle_1 = \langle (\Id + K_\gamma) X, Y
-  \rangle_1\).
-
-  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
-  \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
-    \le \sqrt{2} \norm{\Id + R_\gamma} \cdot \norm{K_\gamma X}_1
-      \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
-  (\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
-  lemma~\ref{thm:inclusion-submnfds-is-compact}.
-
-\end{proof}
-
-\begin{corollary}
-  Given a critical point \(\gamma\) of \(E\!\restriction_{\Omega_{p q} M}\),
-  \(A_\gamma\) has either finitely many eigenvalues including \(1\) or
-  infinitely many eigenvalues not equal to \(1\), in which case the only
-  accumulation point of the set of eigenvalues of \(A_\gamma\) is \(1\) and
-  \(1\) is a spectral value but not an eigenvalue. In particular, there is an
-  orthogonal decomposition
-  \[
-    T_\gamma \Omega_{p q} M
-    = T_\gamma^- \Omega_{p q} M
-    \oplus T_\gamma^0 \Omega_{p q} M
-    \oplus T_\gamma^+ \Omega_{p q} M,
-  \]
-  where \(T_\gamma^- \Omega_{p q} M\) is the finite-dimensional subspace
-  spanned by eigenvectors with negative eigenvalues, \(T_\gamma^0 \Omega_{p q}
-  M = \ker A_\gamma\) and \(T_\gamma^+ \Omega_{p q} M\) is the proper Hilbert
-  subspace spanned by eigenvectors with positive eigenvalues. The same holds
-  for critical points \(\gamma\) of \(E\!\restriction_{\Lambda M}\) and
-  \(T_\gamma \Lambda M\).
-\end{corollary}
-
-\begin{definition}
-  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
-  critical point \(\gamma\) of \(E\!\restriction_{\Lambda M}\) \emph{the
-  \(\Lambda\)-index of \(\gamma\)}. Whenever the submanifold \(\gamma\) lies in
-  is clear from context we refer to the \(\Omega\)-index or the
-  \(\Lambda\)-index of \(\gamma\) simply by \emph{the index of \(\gamma\)}.
-\end{definition}
-
-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 \(\gamma\) in the interior of \(I\).
-\end{theorem}
-
-Consequence:
-
-% TODO: Sketch a proof
-\begin{theorem}[Jacobi-Darboux]
-  Let \(\gamma \in \Omega_{p q} M\) be a critical point of \(E\).
-  \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
-      \(E(\eta) > E(\gamma)\) and \(\ell(\eta) > \ell(\gamma)\) for all \(\eta
-      \in U\) with \(\eta \ne 0\).
-
-    \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
-      immersion
-      \[
-        i : B^k \to \Omega_{p q} M
-      \]
-      of the unit ball \(B^k = \{v \in \RR^k : \norm{v} < 1\}\) with \(F(0) =
-      \gamma\), \(E(i(v)) < E(\gamma)\) and \(\ell(i(v)) < \ell(\gamma)\) for
-      all \(v \in B^k \setminus \{ 0 \}\).
-  \end{enumerate}
-\end{theorem}

diff --git /dev/null b/sections/applications.tex
@@ -0,0 +1,414 @@
+\section{Applications to the Calculus of Variations}\label{sec:aplications}
+
+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
+standard tools of calculus to study the critical points of the energy
+functional \(E\).
+
+\begin{theorem}
+  The energy functional
+  \begin{align*}
+    E : H^1(I, M) & \to     \RR                                  \\
+           \gamma & \mapsto \frac{1}{2} \norm{\partial \gamma}_0
+  \end{align*}
+  is smooth and \(d E_\gamma X = \left\langle \partial \gamma, \frac\nabla\dt X
+  \right\rangle_0\).
+\end{theorem}
+
+\begin{proof}
+  The fact that \(E\) is smooth should be clear from the smoothness of
+  \(\partial\) and \(\norm\cdot_0\). Furthermore, from the definition of
+  \(\frac\nabla\dt\) we have
+  \[
+    \begin{split}
+      \left\langle \partial \gamma, \frac\nabla\dt X \right\rangle_0
+      & =
+        \left\langle
+        \partial \gamma, (\nabla_X^0 \partial)_\gamma
+        \right\rangle_0 \\
+      & = \tilde X \langle \partial \gamma, \partial \gamma \rangle -
+        \left\langle
+        \partial (\nabla_X^0 \partial)_\gamma, \partial \gamma
+        \right\rangle_0 \\
+      & = 2 \tilde X E(\gamma) -
+        \left\langle \frac\nabla\dt X, \partial \gamma \right\rangle_0 \\
+      & = 2 d E_\gamma X -
+        \left\langle \partial \gamma, \frac\nabla\dt X \right\rangle_0
+    \end{split}
+  \]
+  where \(\tilde X \in \mathfrak{X}(H^1(I, M))\) is any vector field with
+  \(\tilde X_\gamma = X\).
+\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}
+
+% 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 \(\{
+    \gamma_t \}_t\) of \(\gamma\) with variational vector field \(X\) we have
+  \[
+    \left.\frac\dd\dt\right|_{t = 0} E(\gamma_t)
+    = \sum_i
+      \left. \langle \dot\gamma(t), X_t \rangle\right|_{t = t_i}^{t_{i + 1}}
+    - \int_0^1 \left\langle \frac\nabla\dt \dot\gamma(t), X_t \right\rangle
+      \; \dt
+  \]
+\end{corollary}
+
+\begin{corollary}
+  The only critical points of \(E\) in \(H^1(I, M)\) are the constant curves.
+\end{corollary}
+
+\begin{proof}
+  Clearly every constant curve is a critical point. On the other hand, if
+  \(\gamma \in H^1(I, M)\) is such that \(\left\langle \partial \gamma,
+  \frac\nabla\dt X \right\rangle_0 = d E_\gamma X = 0\) for all \(X \in
+  H^1(\gamma^* TM)\) then \(\partial \gamma = 0\) and therefore \(\gamma\) is
+  constant.
+\end{proof}
+
+\subsection{The Critical Points of \(E\)}
+
+\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
+  \begin{align*}
+    U \subset H^1(W_\gamma) & \to     T_{\gamma(0)} M &
+    U \subset H^1(W_\gamma) & \to     T_{\gamma(1)} M \\
+                          X & \mapsto X_0 &
+                          X & \mapsto X_1
+  \end{align*}
+  which are indeed smooth functions. This local representation also shows that
+  \begin{align*}
+    d\sigma_\gamma : H^1(\gamma^* TM) & \to     T_{\gamma(0)} M &
+    d\tau_\gamma   : H^1(\gamma^* TM) & \to     T_{\gamma(1)} M \\
+                                    X & \mapsto X_0 &
+                                    X & \mapsto X_1
+  \end{align*}
+  are surjective maps for all \(\gamma \in H^1(\gamma^* TM)\).
+\end{proof}
+
+\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
+  the subspace of \(H^1(\gamma^* TM)\) consisting of class \(H^1\) vector fields
+  \(X\) along \(\gamma\) with \(X_0 = X_1 = 0\). Likewise, the space \(\Lambda
+  M \subset H^1(I, M)\) of free loops is a submanifold whose tangent at
+  \(\gamma\) is given by all \(X \in H^1(\gamma^* TM)\) with \(X_0 = X_1\).
+\end{theorem}
+
+\begin{proof}
+  To see that this are submanifolds, it suffices to note that \(\Omega_{p q}
+  M\) and \(\Lambda M\) are the inverse images of the closed submanifolds
+  \(\{(p, q)\}, \{(p, p) : p \in M\} \subset M \times M\) under the submersion
+  \((\sigma, \tau) : H^1(I, M) \to M \times M\).
+
+  The characterization of their tangent bundles should also be clear: any curve
+  \((-\epsilon, \epsilon) \to H^1(I, M)\) passing through \(\gamma \in
+  \Omega_{p q} M\) whose image is contained in \(\Omega_{p q} M\) is a
+  variation of \(\gamma\) with fixed endpoints, so its variational vector field
+  \(X\) satisfies \(X_0 = X_1 = 0\). Likewise, any variation of a loop \(\gamma
+  \in \Lambda M\) trough loops -- i.e. a curve \((-\epsilon, \epsilon) \to
+  \Lambda M\) passing through \(\gamma\) -- satisfies \(X_0 = X_1\).
+\end{proof}
+
+\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.
+\end{theorem}
+
+\begin{proof}
+  We start by supposing that \(\gamma\) is a geodesic. Since \(\gamma\) is
+  smooth,
+  \[
+    d E_\gamma X
+    = \int_0^1
+      \left\langle \dot\gamma(t), \frac\nabla\dt X \right\rangle \; \dt
+    = \int_0^1
+      \frac\dd\dt \langle \dot\gamma(t), X_t \rangle -
+      \left\langle \frac\nabla\dt \dot\gamma(t), X \right\rangle \; \dt
+    = \langle \dot\gamma(1), X_1 \rangle - \langle \dot\gamma(0), X_0 \rangle
+  \]
+
+  Now if \(\gamma \in \Omega_{p q} M\) and \(X \in T_\gamma \Omega_{p q} M\)
+  then \(d E_\gamma X = \langle \dot\gamma(1), 0 \rangle - \langle
+  \dot\gamma(0), 0 \rangle = 0\). Likewise, if \(\gamma\) is a closed geodesic
+  and \(X \in T_\gamma \Lambda M\) we find \(d E_\gamma X = 0\) since
+  \(\dot\gamma(0) = \dot\gamma(1)\) 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
+  \begin{align*}
+    \frac\nabla\dt Y & = \partial \gamma &
+                 Y_0 & = 0 &
+    \frac\nabla\dt Z & = 0 &
+                 Z_1 & = Y_1
+  \end{align*}
+
+  Let \(X_t = Y_t - t Z_t\). Then \(X_0 = X_1 = 0\) and \(\frac\nabla\dt X =
+  \partial \gamma - Z\). Furthermore,
+  \[
+    \langle Z, \partial \gamma - Z \rangle_0
+    = \left\langle Z, \frac\nabla\dt X \right\rangle_0
+    = \int_0^1 \frac\dd\dt \langle Z_t, X_t \rangle \; \dt
+    = \langle Z_1, X_1 \rangle - \langle Z_0, X_0 \rangle
+    = 0
+  \]
+  and
+  \[
+    0
+    = d E_\gamma X
+    = \left\langle \partial \gamma, \frac\nabla\dt X \right\rangle_0
+    = \langle \partial \gamma, \partial \gamma - Z \rangle_0,
+  \]
+  which implies \(\norm{\partial \gamma - Z}_0^2 = 0\). In other words,
+  \(\partial \gamma = Z \in H^1(\gamma^* TM)\) and therefore \(\frac\nabla\dt
+  \dot\gamma(t) = \frac\nabla\dt Z = 0\) -- i.e. \(\gamma\) is a geodesic.
+
+  Finally, if \(\gamma \in \Lambda M\) with \(\gamma(0) = \gamma(1) = p\) we
+  may apply the argument above to conclude that \(\gamma\) is a geodesic
+  joining \(p\) to \(q = p\). To see that \(\gamma\) is a closed geodesic apply
+  the same argument again for \(\eta(t) = \gamma(1 + \sfrac{1}{2})\) to
+  conclude that \(\dot\gamma(0) = \dot\eta(\sfrac{1}{2}) = \dot\gamma(1)\).
+\end{proof}
+
+\subsection{Second Order Derivatives of \(E\)}
+
+\begin{definition}
+  Given a -- possibly infinite-dimensional -- Riemannian manifold \(N\) and a
+  smooth functional \(f : N \to \RR\), we call the symmetric tensor
+  \[
+    d^2 f(X, Y) = \nabla d f (X, Y) =  X Y f - df \nabla_X Y
+  \]
+  \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}
+
+\begin{theorem}
+  If \(\gamma \in \Omega_{p q} M\) is a critical point of \(E\) then
+  \begin{equation}\label{eq:second-variation-general}
+    (d^2 E\!\restriction_{\Omega_{p q} M})_\gamma(X, Y)
+    = \left\langle \frac\nabla\dt X, \frac\nabla\dt Y \right\rangle_0
+    - \langle R_\gamma X, Y \rangle_0,
+  \end{equation}
+  where \(R_\gamma : H^1(\gamma^* TM) \to H^1(\gamma^* TM)\) is given by
+  \((R_\gamma X)_t = R(X_t, \dot\gamma(t)) \dot\gamma(t)\). Formula
+  (\ref{eq:second-variation-general}) also holds for critical points of \(E\)
+  in \(\Lambda M\).
+\end{theorem}
+
+\begin{proof}
+  Given the symmetry of \(d^2 E\), it suffices to take \(X \in T_\gamma
+  \Omega_{p q} M\) and show
+  \[
+    (d^2 E\!\restriction_{\Omega_{p q} M})(X, X)
+    = \norm{\frac\nabla\dt X}_0^2 - \langle R_\gamma X, X \rangle_0
+  \]
+
+  To that end, we fix a variation \(\{ \gamma_t \}_t\) of \(\gamma\) with fixed
+  endpoints and variational field \(X\) and compute
+  \[
+    \begin{split}
+      (d^2 E\!\restriction_{\Omega_{p q} M})_\gamma(X, X)
+      & = \left.\frac{\dd^2}{\dt^2}\right|_{t = 0} E(\gamma_t) \\
+      & = \int_0^1 \norm{\frac\nabla\dt X}^2
+        - \langle R(X_t, \dot\gamma(t)) \dot\gamma(t), X_t \rangle \; \dt \\
+      & = \norm{\frac\nabla\dt X}_0^2 - \langle R_\gamma X, X \rangle_0
+    \end{split}
+  \]
+\end{proof}
+
+\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.
+\end{lemma}
+
+\begin{proposition}
+  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
+  \[
+    \langle A_\gamma X, Y \rangle_1
+    = \langle X, A_\gamma Y \rangle_1
+    = d^2 E_\gamma(X, Y)
+  \]
+  has the form \(A_\gamma = \Id + K_\gamma\) where \(K_\gamma : T_\gamma
+  \Omega_{p q} M \to T_\gamma \Omega_{p q} M\) is a compact operator. The same
+  holds for \(\Lambda M\) if \(M\) is compact.
+\end{proposition}
+
+\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
+  \(A_\gamma = \Id + K_\gamma\) for \(\gamma\) in both \(\Omega_{p q} M\) and
+  \(\Lambda M\) -- in which case assume \(M\) is compact.
+
+  Let \(\gamma \in \Omega_{p q} M\) be a critical point. By
+  theorem~\ref{thm:critical-points-char-in-submanifolds} we know that
+  \(\gamma\) is a geodesic. Let \(X, Y \in \Gamma(\gamma^* TM)\) with \(X_0 =
+  X_1 = Y_0 = Y_1 = 0\). Then
+  \begin{equation}\label{eq:compact-partial-result}
+    \begin{split}
+      \langle X, Y \rangle_1
+      & = \langle X, Y \rangle_0
+        + \left\langle \frac\nabla\dt X, \frac\nabla\dt Y \right\rangle \\
+      & = \langle X, Y \rangle_0
+        + \int_0^1
+          \left\langle \frac\nabla\dt X, \frac\nabla\dt Y \right\rangle
+          \; \dt \\
+      & = \langle X, Y \rangle_0
+        + \int_0^1
+          \frac\dd\dt \langle X_t, Y_t \rangle -
+          \left\langle \frac{\nabla^2}{\dt^2} X, Y \right\rangle \; \dt \\
+      & = \langle X, Y \rangle_0
+        - \left\langle \frac{\nabla^2}{\dt^2} X, Y \right\rangle_0
+        + \left.\langle X_t, Y_t \rangle\right|_{t = 0}^1 \\
+      & = \left\langle
+          \left(\Id - \frac{\nabla^2}{\dt^2}\right) X, Y
+          \right\rangle_0
+    \end{split}
+  \end{equation}
+
+  Since \(\Gamma(\gamma^* TM) \subset H^1(\gamma^* TM)\) is dense,
+  (\ref{eq:compact-partial-result}) extends to all of \(T_\gamma \Omega_{p q}
+  M\). Hence given \(X, Y \in T_\gamma \Omega_{p q} M\) we have
+  \[
+    \begin{split}
+      \langle A_\gamma X, Y \rangle_1
+      & = \left\langle \frac\nabla\dt X, \frac\nabla\dt Y \right\rangle_0
+        - \langle R_\gamma X, Y \rangle_0 \\
+      & = \langle X, Y \rangle_1 - \langle X, Y \rangle_0
+        - \langle R_\gamma X, Y \rangle_0 \\
+      & = \langle X, Y \rangle_1 - \langle (\Id + R_\gamma) X, Y \rangle_0 \\
+      & = \langle X, Y \rangle_1
+        - \left\langle
+          \left( \Id - \frac{\nabla^2}{\dt^2} \right)^{-1}
+          \circ (\Id + R_\gamma) X, Y
+          \right\rangle_1 \\
+      & = \langle X, Y \rangle_1 + \langle K_\gamma X, Y \rangle_1 \\
+    \end{split}
+  \]
+
+  Now consider a critical point \(\gamma \in \Lambda M\) -- i.e. a closed
+  geodesic. Equation (\ref{eq:compact-partial-result}) also holds for \(X, Y
+  \in \Gamma(\gamma^* TM)\) with \(X_0 = X_1\) and \(Y_0 = Y_1\), so it holds
+  for all \(X, Y \in T_\gamma \Lambda M\). Hence by applying the same argument
+  we get \(\langle A_\gamma X, Y \rangle_1 = \langle (\Id + K_\gamma) X, Y
+  \rangle_1\).
+
+  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
+  \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
+    \le \sqrt{2} \norm{\Id + R_\gamma} \cdot \norm{K_\gamma X}_1
+      \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
+  (\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
+  lemma~\ref{thm:inclusion-submnfds-is-compact}.
+
+\end{proof}
+
+\begin{corollary}
+  Given a critical point \(\gamma\) of \(E\!\restriction_{\Omega_{p q} M}\),
+  \(A_\gamma\) has either finitely many eigenvalues including \(1\) or
+  infinitely many eigenvalues not equal to \(1\), in which case the only
+  accumulation point of the set of eigenvalues of \(A_\gamma\) is \(1\) and
+  \(1\) is a spectral value but not an eigenvalue. In particular, there is an
+  orthogonal decomposition
+  \[
+    T_\gamma \Omega_{p q} M
+    = T_\gamma^- \Omega_{p q} M
+    \oplus T_\gamma^0 \Omega_{p q} M
+    \oplus T_\gamma^+ \Omega_{p q} M,
+  \]
+  where \(T_\gamma^- \Omega_{p q} M\) is the finite-dimensional subspace
+  spanned by eigenvectors with negative eigenvalues, \(T_\gamma^0 \Omega_{p q}
+  M = \ker A_\gamma\) and \(T_\gamma^+ \Omega_{p q} M\) is the proper Hilbert
+  subspace spanned by eigenvectors with positive eigenvalues. The same holds
+  for critical points \(\gamma\) of \(E\!\restriction_{\Lambda M}\) and
+  \(T_\gamma \Lambda M\).
+\end{corollary}
+
+\begin{definition}
+  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
+  critical point \(\gamma\) of \(E\!\restriction_{\Lambda M}\) \emph{the
+  \(\Lambda\)-index of \(\gamma\)}. Whenever the submanifold \(\gamma\) lies in
+  is clear from context we refer to the \(\Omega\)-index or the
+  \(\Lambda\)-index of \(\gamma\) simply by \emph{the index of \(\gamma\)}.
+\end{definition}
+
+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 \(\gamma\) in the interior of \(I\).
+\end{theorem}
+
+Consequence:
+
+% TODO: Sketch a proof
+\begin{theorem}[Jacobi-Darboux]
+  Let \(\gamma \in \Omega_{p q} M\) be a critical point of \(E\).
+  \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
+      \(E(\eta) > E(\gamma)\) and \(\ell(\eta) > \ell(\gamma)\) for all \(\eta
+      \in U\) with \(\eta \ne 0\).
+
+    \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
+      immersion
+      \[
+        i : B^k \to \Omega_{p q} M
+      \]
+      of the unit ball \(B^k = \{v \in \RR^k : \norm{v} < 1\}\) with \(F(0) =
+      \gamma\), \(E(i(v)) < E(\gamma)\) and \(\ell(i(v)) < \ell(\gamma)\) for
+      all \(v \in B^k \setminus \{ 0 \}\).
+  \end{enumerate}
+\end{theorem}