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

 \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. 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 \(\{ \gamma_t \}_t\) 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
 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 \(E\).
 
 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 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
 
 \begin{theorem}\label{thm:energy-is-smooth}
   The energy functional
   \begin{align*}
     E : H^1(I, M) & \to \mathbb{R} \\
     \gamma
     & \mapsto \frac{1}{2} \norm{\partial \gamma}_0^2
     = \frac{1}{2} \int_0^1 \norm{\dot\gamma(t)}^2 \; \dt
   \end{align*}
   is smooth and \(d E_\gamma X = \left\langle \partial \gamma, \frac\nabla\dt X
   \right\rangle_0\).
 \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
   \[
     \begin{split}
       \left\langle \partial \gamma, \frac\nabla\dt X \right\rangle_0
       & =
         \left\langle
         \partial \gamma, (\nabla_X^0 \partial)_\gamma
         \right\rangle_0 \\
       & = (\tilde X \langle \partial, \partial \rangle_0)(\gamma) -
         \left\langle
         (\nabla_X^0 \partial)_\gamma, \partial \gamma
         \right\rangle_0 \\
       & = 2 \tilde X E(\gamma) -
         \left\langle \frac\nabla\dt X, \partial \gamma \right\rangle_0 \\
       & = 2 d E_\gamma X -
         \left\langle \partial \gamma, \frac\nabla\dt X \right\rangle_0
     \end{split}
   \]
   where \(\tilde X \in \mathfrak{X}(H^1(I, M))\) is any vector field with
   \(\tilde X_\gamma = X\).
 \end{proof}
 
 As promised, by applying the chain rule and using the compatibility of
 \(\nabla\) with the metric we arrive at the classical formula for the first
 variation of energy \(E\).
 
 \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}
 
 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}
 
 \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}
 
 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
   \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 \(\gamma \in H^1(I, M)\).
 \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
   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 these 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}
 
 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 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}
   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 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
   \[
     \langle \partial \gamma, \partial \gamma - Z \rangle_0
     = \left\langle \partial \gamma, \frac\nabla\dt X \right\rangle_0
     = d E_\gamma X
     = 0,
   \]
   which implies \(\norm{\partial \gamma - Z}_0^2 = 0\). In other words,
   \(\partial \gamma = Z \in H^1(\gamma^* TM)\) and therefore \(\frac\nabla\dt
   \dot\gamma(t) = \frac\nabla\dt Z = 0\) -- 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}
 
 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 meaningful sense by looking at the Hessian form,
 which we define in the following.
 
 \begin{definition}
   Given a -- possibly infinite-dimensional -- Riemannian manifold \(N\) and a
   smooth functional \(f : N \to \mathbb{R}\), 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}
 
 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 \(\gamma\) is a critical point of \(E\!\restriction_{\Omega_{p q} M}\)
   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
   \[
     d^2 E_\gamma(X, X)
     = \norm{\frac\nabla\dt X}_0^2 - \langle R_\gamma X, X \rangle_0
   \]
 
   To that end, we fix a variation \(\{ \gamma_t \}_t\) of \(\gamma\) with fixed
   endpoints and variational field \(X\) and compute
   \[
     \begin{split}
       d^2 E_\gamma(X, X)
       & = \left.\frac{\dd^2}{\dt^2}\right|_{t = 0} E(\gamma_t) \\
       \text{(second variation of energy)}
       & = \int_0^1 \norm{\frac\nabla\dt X}^2
         - \langle R(X_t, \dot\gamma(t)) \dot\gamma(t), X_t \rangle \; \dt \\
       & = \norm{\frac\nabla\dt X}_0^2 - \langle R_\gamma X, X \rangle_0
     \end{split}
   \]
 \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 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
 
 \begin{proposition}\label{thm:morse-index-e-is-finite}
   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 \(A_\gamma = \operatorname{Id} + K_\gamma\) 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 \(K_\gamma = - \left( \operatorname{Id} - \frac{\nabla^2}{\dt^2}
   \right)^{-1} \circ (\operatorname{Id} + R_\gamma)\). We will show that
   \(K_\gamma\) is compact and that \(A_\gamma = \operatorname{Id} + K_\gamma\)
   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(\operatorname{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
   \[
     \begin{split}
       \langle A_\gamma X, Y \rangle_1
       & = \left\langle \frac\nabla\dt X, \frac\nabla\dt Y \right\rangle_0
         - \langle R_\gamma X, Y \rangle_0 \\
       & = \langle X, Y \rangle_1 - \langle X, Y \rangle_0
         - \langle R_\gamma X, Y \rangle_0 \\
       & = \langle X, Y \rangle_1
         - \langle (\operatorname{Id} + R_\gamma) X, Y \rangle_0 \\
       & = \langle X, Y \rangle_1
         - \left\langle
           \left( \operatorname{Id} - \frac{\nabla^2}{\dt^2} \right)^{-1}
           \circ (\operatorname{Id} + R_\gamma) X, Y
           \right\rangle_1 \\
       & = \langle X, Y \rangle_1 + \langle K_\gamma X, Y \rangle_1 \\
     \end{split}
   \]
 
   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 \(\langle A_\gamma X, Y \rangle_1 = \langle (\operatorname{Id} +
   K_\gamma) X, Y \rangle_1\).
 
   As for the compactness of \(K_\gamma\) in the case of \(\Omega_{p q} M\),
   from (\ref{eq:compact-partial-result}) we get \(\norm{K_\gamma X}_1^2 = -
   \langle (\operatorname{Id} + R_\gamma) X, K_\gamma X \rangle_0\), so that
   proposition~\ref{thm:continuous-inclusions-sections} implies
   \begin{equation}\label{eq:compact-operator-quota}
     \norm{K_\gamma X}_1^2
     \le \norm{\operatorname{Id} + R_\gamma}
       \cdot \norm{K_\gamma X}_\infty \cdot \norm{X}_0
     \le \sqrt{2} \norm{\operatorname{Id} + R_\gamma} \cdot \norm{K_\gamma X}_1
       \cdot \norm{X}_0
   \end{equation}
 
   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
   lemma~\ref{thm:inclusion-submnfds-is-compact}.
 \end{proof}
 
 Once again, the first part of this proposition 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 \(A_\gamma = \operatorname{Id} + K_\gamma\). These 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 \(E\!\restriction_{\Omega_{p q} M}\),
   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 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 \(T_\gamma \Lambda M\) if \(M\) is
   compact.
 \end{corollary}
 
 \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
   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}
 
 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. In
 fact, the geodesics problems and the energy functional where among Morse's
 original proposed applications. Proposition~\ref{thm:morse-index-e-is-finite}
 and definition~\ref{def:morse-index} amount to a proof that the Morse index of
 \(E\) at a critical point \(\gamma\) is finite.
 
 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\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
 in here, but see theorem 2.5.9 of \cite{klingenberg}. The index 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.
 
 \begin{theorem}[Jacobi-Darboux]\label{thm: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
       \(E(\eta) > E(\gamma)\) 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
       immersion
       \[
         i : B^k \to \Omega_{p q} M
       \]
       of the unit ball \(B^k = \{v \in \mathbb{R}^k : \norm{v} < 1\}\) with
       \(i(0) = \gamma\), \(E(i(v)) < E(\gamma)\) and \(L(i(v)) < L(\gamma)\)
       for all nonzero \(v \in B^k\).
   \end{enumerate}
 \end{theorem}
 
 \begin{proof}
   First of all notice that given \(\eta \in U_\gamma\) with \(\eta =
   \exp_\gamma(X)\), \(X \in H^1(W_\gamma)\), 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(\exp_\gamma(X)) - E(\gamma) - \frac{1}{2} d^2 E_\gamma(X, X)}}
       {\norm{X}_1^2}
     \to 0
   \end{equation}
   as \(X \to 0\).
 
   Let \(\gamma\) be as in \textbf{(i)}. Since \(\gamma\) has no conjugate
   points, it follows from Morse's index theorem that \(T_\gamma^- \Omega_{p q}
   M = 0\). 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 \(d^2 E\!\restriction_{\Omega_{p q}
   M}\) is positive-definite. Now given \(\eta = \exp_\gamma(X)\) as before,
   (\ref{eq:energy-taylor-series}) implies that \(E(\eta) > E(\gamma)\),
   provided \(\norm{X}_1\) is sufficiently small.
 
   As for part \textbf{(ii)}, fix an orthonormal basis
   \(\{X_j : 1 \le j \le k\}\) of \(T_\gamma^- \Omega_{p q} M\) consisting of
   eigenvectors of \(A_\gamma\) with negative eigenvalues \(- \lambda_i\). Let
   \(\delta > 0\) and 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 enough \(\delta\). Moreover, from
   (\ref{eq:energy-taylor-series}) and
   \[
     E(i(v))
     = E(\gamma) - \frac{1}{2} \delta^2 \sum_j \lambda_j \cdot v_j + \cdots
   \]
   we find that \(E(i(v)) < E(\gamma)\) for sufficiently small \(\delta\). In
   particular, \(L(i(v))^2 \le E(i(v)) < E(\gamma) = L(\gamma)^2\).
 \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 \(\gamma\) and \(\eta \in U\). This could
 be amended by showing that the length functional \(L : H^1(I, M) \to
 \mathbb{R}\) is smooth and that its Hessian \(d^2 L_\gamma\) 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 \(\gamma\) in \(\Omega_{p q} M\) in the uniform
 topology. On the other hand, part \textbf{(ii)} is definitively an improvement
 of the classical formulation: we can find curves \(\eta = i(v)\) shorter than
 \(\gamma\) already in an \(H^1\)-neighborhood of \(\gamma\).
 
 This concludes our discussion of the applications of our theory to the
 geodesics problem. We hope that these short notes could provide the reader with
 a glimpse of the rich theory of the calculus of variations and global analysis
 at large. We once again refer the reader to \cite[ch.~2]{klingenberg},
 \cite[ch.~11]{palais} and \cite[sec.~6]{eells} for further insight on modern
 variational methods.