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
git clone: git://git.pablopie.xyz/global-analysis-and-the-banach-manifold-of-class-h1-curvers
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Commit: 98369abb834e133b31178412abf5a58552235b9b
Parent: e5185e2c53b504cb959693bb23b0b13ae519391a
Author: Pablo <pablo-escobar@riseup.net>
Date: Sun, 31 Jul 2022 14:33:45 +0000
Fixed some typos, rephrased some sentences and removed unnecessary whitespace
Diffstat

1 file changed, 46 insertions, 46 deletions
Status	File Name	N° Changes	Insertions	Deletions
Modified	sections/applications.tex	92	46	46
diff --git a/sections/applications.tex b/sections/applications.tex
@@ -16,14 +16,13 @@ Morse index theorem and the Jacobi-Darboux theorem. We start by defining\dots
 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 \(E\).
+\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
+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
@@ -38,7 +37,7 @@ point. Without further ado, we prove\dots
   The energy functional
   \begin{align*}
     E : H^1(I, M) & \to \RR \\
-    \gamma 
+    \gamma
     & \mapsto \frac{1}{2} \norm{\partial \gamma}_0
     = \frac{1}{2} \int_0^1 \norm{\dot\gamma(t)}^2 \; \dt
   \end{align*}
@@ -57,9 +56,9 @@ point. Without further ado, we prove\dots
         \left\langle
         \partial \gamma, (\nabla_X^0 \partial)_\gamma
         \right\rangle_0 \\
-      & = \tilde X \langle \partial \gamma, \partial \gamma \rangle -
+      & = (\tilde X \langle \partial, \partial \rangle_0)(\gamma) -
         \left\langle
-        \partial (\nabla_X^0 \partial)_\gamma, \partial \gamma
+        (\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 \\
@@ -72,8 +71,8 @@ point. Without further ado, we prove\dots
 \end{proof}
 
 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 \(E\).
+\(\nabla\) 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
@@ -258,7 +257,7 @@ 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,
+derivative'' of \(E\) in a meaningful sense by looking at the Hessian form,
 which we define in the following.
 
 \begin{definition}
@@ -274,7 +273,8 @@ 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
+  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
@@ -290,7 +290,7 @@ compute the Hessian of \(E\) at a critical point.
   Given the symmetry of \(d^2 E\), it suffices to take \(X \in T_\gamma
   \Omega_{p q} M\) and show
   \[
-    (d^2 E\!\restriction_{\Omega_{p q} M})(X, X)
+    d^2 E_\gamma(X, X)
     = \norm{\frac\nabla\dt X}_0^2 - \langle R_\gamma X, X \rangle_0
   \]
 
@@ -298,7 +298,7 @@ compute the Hessian of \(E\) at a critical point.
   endpoints and variational field \(X\) and compute
   \[
     \begin{split}
-      (d^2 E\!\restriction_{\Omega_{p q} M})_\gamma(X, X)
+      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
@@ -340,7 +340,7 @@ As a first consequence, we prove\dots
 
 \begin{proof}
   Consider \(K_\gamma = - \left( \Id - \frac{\nabla^2}{\dt^2} \right)^{-1}
-  \circ (\Id + R_\gamma)\). We will show that \(K_\gamma\) is compact and
+  \circ (\Id + R_\gamma)\). We will show that \(K_\gamma\) is compact and that
   \(A_\gamma = \Id + K_\gamma\) for \(\gamma\) in both \(\Omega_{p q} M\) and
   \(\Lambda M\) -- in which case assume \(M\) is compact.
 
@@ -417,13 +417,13 @@ As a first consequence, we prove\dots
   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.
+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 \(A_\gamma = \Id + K_\gamma\). 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 \(E\!\restriction_{\Omega_{p q} M}\),
@@ -459,13 +459,12 @@ stating 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
-\(\gamma\). 
+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: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.
 
@@ -475,8 +474,8 @@ We are now ready to state Morse's index theorem.
   proper conjugate points of \(\gamma\) in the interior of \(I\).
 \end{theorem}
 
-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
+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,
@@ -514,27 +513,28 @@ 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 \(T_\gamma^- \Omega_{p q}
-  M = 0\). 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 \(T_\gamma \Omega_{p q} M =
+  M = 0\). Furtheremore, by noting 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
-  that \(\norm{X}_1\) is sufficiently small.
+  (\ref{eq:energy-taylor-series}) implies that \(E(\eta) > E(\gamma)\),
+  provided \(\norm{X}_1\) 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
+  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 enought \(\delta\). Furtheremore,
-  from (\ref{eq:energy-taylor-series}) and 
+  Clearly \(i\) is an immersion for small enought \(\delta\). Moreover, from
+  (\ref{eq:energy-taylor-series}) and
   \[
-    E(i(v)) 
-    = E(\gamma) - \frac{1}{2} \delta^2 \sum_i \lambda_i \cdot v_i + \cdots
+    E(i(v))
+    = E(\gamma) - \frac{1}{2} \delta^2 \sum_j \lambda_j \cdot v_j + \cdots
   \]
   we find that \(E(i(v)) < E(\gamma)\) for sufficiently small \(\delta\). In
   particular, \(\ell(i(v)) \le E(i(v))^2 < E(\gamma)^2 = \ell(\gamma)\).
@@ -546,7 +546,7 @@ 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 \(C > 0\). 
+some \(C > 0\).
 
 Secondly, inlike the classical formulation we only consider curves in an
 \(H^1\)-neighborhood of \(\gamma\) -- instead of a neighborhood of \(\gamma\)