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
Commit
a7b92956d4e6a2942139249cdd7cb0e50f4e11d2 Parent
96e981f59ad0c4ea7de2f1a3eee40a15b0794c26 Author
Pablo <pablo-escobar@riseup.net >
Date
Fri, 5 Aug 2022 14:09:09 +0000
Removed the dependencies on my personal packages
Diffstat
4 files changed, 189 insertions, 114 deletions
diff --git a/preamble.tex b/preamble.tex
@@ -0,0 +1,68 @@
+\documentclass[a4paper]{article}
+\usepackage[total={6in, 9in}]{geometry}
+\usepackage{amsmath, amssymb, amsthm, stmaryrd, mathrsfs, gensymb, dsfont}
+\usepackage{mathtools, adjustbox}
+\usepackage[scr=esstix,cal=boondox]{mathalfa}
+\usepackage{enumitem, xfrac, xcolor, cancel, multicol, tabularx, relsize}
+\usepackage{hyperref}
+\usepackage[backend=biber]{biblatex}
+\usepackage{pgfplots, tikz, tikz-cd}
+\usepackage{graphicx, wrapfig}
+\usepackage[normalem]{ulem}
+
+% Configure graphics
+\pgfplotsset{compat=1.18}
+
+% Use \blacksquare for \qed
+\renewcommand{\qedsymbol}{\ensuremath{\blacksquare}}
+
+% Get propper inequality symbols
+\renewcommand{\leq}{\leqslant}
+\renewcommand{\le}{\leqslant}
+\renewcommand{\geq}{\geqslant}
+\renewcommand{\ge}{\geqslant}
+\renewcommand{\preceq}{\preccurlyeq}
+\renewcommand{\succeq}{\succcurlyeq}
+
+% Add missing arrows
+\newcommand{\longhookrightarrow}{\lhook\joinrel\longrightarrow}
+
+% Fix the subset symbol
+\renewcommand{\subset}{\subseteq}
+
+% Display long arrows instead of short ones
+\renewcommand{\to}{\longrightarrow}
+\renewcommand{\mapsto}{\longmapsto}
+
+% Absolute value and vector norm
+\newcommand{\abs}[1]{\left|\nobreak#1\nobreak\right|}
+\newcommand{\norm}[1]{\left\lVert\nobreak#1\nobreak\right\rVert}
+
+\newcommand{\dt}{\mathrm{d} t}
+\newcommand{\dd}{\mathrm{d}}
+
+% Isomorphism arrow
+\newcommand{\isoto}{\xlongrightarrow{\sim}}
+
+% Configure how link look
+\hypersetup{
+ colorlinks,
+ citecolor=black,
+ filecolor=black,
+ linkcolor=black
+}
+
+% Configure the enumerate environment to use bold roman numerals
+\setenumerate[0]{label={\normalfont\bfseries(\roman*)}}
+
+% Useful theorem definitions
+\newtheorem{theorem}{Theorem}[section]
+\newtheorem{lemma}{Lemma}[section]
+\newtheorem{corollary}{Corollary}[section]
+\newtheorem{proposition}{Proposition}[section]
+\theoremstyle{remark}
+\newtheorem*{note}{Note}
+\theoremstyle{definition}
+\newtheorem{definition}{Definition}[section]
+\newtheorem{example}{Example}[section]
+
diff --git a/sections/applications.tex b/sections/applications.tex
@@ -36,7 +36,7 @@ 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 \\
+ E : H^1(I, M) & \to \mathbb{R} \\
\gamma
& \mapsto \frac{1}{2} \norm{\partial \gamma}_0
= \frac{1}{2} \int_0^1 \norm{\dot\gamma(t)}^2 \; \dt
@@ -144,10 +144,10 @@ 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 \(X_0 = X_1\).
+ 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}
@@ -261,7 +261,7 @@ 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
+ smooth functional \(f : N \to \mathbb{R}\), we call the symmetric tensor
\[
d^2 f(X, Y) = \nabla d f (X, Y) = X Y f - df \nabla_X Y
\]
@@ -333,16 +333,17 @@ As a first consequence, we prove\dots
= \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.
+ 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 \(K_\gamma = - \left( \Id - \frac{\nabla^2}{\dt^2} \right)^{-1}
- \circ (\Id + R_\gamma)\). 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.
+ 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
@@ -365,7 +366,7 @@ As a first consequence, we prove\dots
- \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
+ \left(\operatorname{Id} - \frac{\nabla^2}{\dt^2}\right) X, Y
\right\rangle_0
\end{split}
\end{equation}
@@ -380,11 +381,12 @@ As a first consequence, we prove\dots
- \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
+ - \langle (\operatorname{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
+ \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}
@@ -394,17 +396,18 @@ As a first consequence, we prove\dots
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 \(\langle A_\gamma X, Y \rangle_1 = \langle (\Id + K_\gamma) X, Y
- \rangle_1\).
+ 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 \(\norm{K_\gamma X}_1^2 = -
- \langle (\Id + R_\gamma) X, K_\gamma X \rangle_0\), so that
+ \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{\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
+ \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}
@@ -421,9 +424,9 @@ 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.
+\(E\) then \(A_\gamma = \operatorname{Id} + K_\gamma\). 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 \(E\!\restriction_{\Omega_{p q} M}\),
@@ -492,9 +495,9 @@ however, is the following consequence of Morse's theorem.
\[
i : B^k \to \Omega_{p q} M
\]
- of the unit ball \(B^k = \{v \in \RR^k : \norm{v} < 1\}\) with \(i(0) =
- \gamma\), \(E(i(v)) < E(\gamma)\) and \(L(i(v)) < L(\gamma)\) for
- all nonzero \(v \in B^k\).
+ 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}
@@ -542,13 +545,13 @@ 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 \(L : H^1(I, M) \to \RR\) 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
+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
diff --git a/sections/introduction.tex b/sections/introduction.tex
@@ -29,11 +29,11 @@ theory, which we describe in the following.
By viewing the class of functions we're interested in as a -- most likely
infinite-dimensional -- manifold \(\mathscr{F}\) and the action functional as a
-smooth functional \(f : \mathscr{F} \to \RR\) we can find minimizing and
+smooth functional \(f : \mathscr{F} \to \mathbb{R}\) we can find minimizing and
maximizing functions by studying the critical points of \(f\). More generally,
modern calculus of variations is concerned with the study of critical points of
-smooth functionals \(\Gamma(E) \to \RR\), where \(E \to M\) is a smooth fiber
-bundle over a finite-dimensional manifold \(M\) and \(\Gamma\) is a given
+smooth functionals \(\Gamma(E) \to \mathbb{R}\), where \(E \to M\) is a smooth
+fiber bundle over a finite-dimensional manifold \(M\) and \(\Gamma\) is a given
section functor, such as smooth sections or Sobolev sections -- notice that by
taking \(E = M \times N\) the manifold \(\Gamma(E)\) is naturally identified
with a space of functions \(M \to N\), which gets us back to the original case.
@@ -46,12 +46,12 @@ Riemannian manifold \(M\), which encodes the solution to the \emph{classic}
variational problem: that of geodesics. Hence the particular action functional
we are interested is the infamous \emph{energy functional}
\begin{align*}
- E : H^1(I, M) & \to \RR \\
+ E : H^1(I, M) & \to \mathbb{R} \\
\gamma & \mapsto \frac{1}{2} \int_0^1 \norm{\dot\gamma(t)}^2 \; \dt,
\end{align*}
as well as the \emph{length functional}
\begin{align*}
- L : H^1(I, M) & \to \RR \\
+ L : H^1(I, M) & \to \mathbb{R} \\
\gamma & \mapsto \int_0^1 \norm{\dot\gamma(t)} \; \dt
\end{align*}
@@ -96,15 +96,15 @@ translated to the context of Banach manifolds\footnote{The real difficulties
with Banach manifolds only show up while proving certain results, and are
mainly due to complications regarding the fact that not all closed subspaces of
a Banach space have a closed complement.}. The reason behind this is simple: it
-turns out that calculus has nothing to do with \(\RR^n\).
+turns out that calculus has nothing to do with \(\mathbb{R}^n\).
What we mean by this last statement is that none of the fundamental ingredients
of calculus -- the ones necessary to define differentiable functions in
-\(\RR^n\), namely the fact that \(\RR^n\) is a complete normed space -- are
-specific to \(\RR^n\). In fact, this ingredients are precisely the features of
-a Banach space. Thus we may naturally generalize calculus to arbitrary Banach
-spaces, and consequently generalize smooth manifolds to spaces modeled after
-Banach spaces. We begin by the former.
+\(\mathbb{R}^n\), namely the fact that \(\mathbb{R}^n\) is a complete normed
+space -- are specific to \(\mathbb{R}^n\). In fact, this ingredients are
+precisely the features of a Banach space. Thus we may naturally generalize
+calculus to arbitrary Banach spaces, and consequently generalize smooth
+manifolds to spaces modeled after Banach spaces. We begin by the former.
\begin{definition}
Let \(V\) and \(W\) be Banach spaces and \(U \subset V\) be an open subset. A
@@ -252,19 +252,20 @@ explicitly stated otherwise. Speaking of examples\dots
\begin{example}
Any Banach space \(V\) can be seen as a Banach manifold with atlas given by
- \(\{(V, \id : V \to V)\}\) -- sometimes called \emph{an affine Banach
- manifold}. In fact, any open subset \(U \subset V\) of a Banach space \(V\)
- is a Banach manifold under a global chart \(\id : U \to V\).
+ \(\{(V, \operatorname{id} : V \to V)\}\) -- sometimes called \emph{an affine
+ Banach manifold}. In fact, any open subset \(U \subset V\) of a Banach space
+ \(V\) is a Banach manifold under a global chart \(\operatorname{id} : U \to
+ V\).
\end{example}
\begin{example}
The group of units \(A^\times\) of a Banach algebra \(A\) is an open subset,
so that it constitutes a Banach manifold modeled after \(A\)
\cite[sec.~3]{eells}. In particular, given a Banach space \(V\) the group
- \(\GL(V)\) of continuous linear isomorphisms \(V \to V\) is a -- possibly
- non-separable -- Banach manifold modeled after the space \(\mathcal{L}(V) =
- \mathcal{L}(V, V)\) under the operator norm: \(\GL(V) =
- \mathcal{L}(V)^\times\).
+ \(\operatorname{GL}(V)\) of continuous linear isomorphisms \(V \to V\) is a
+ -- possibly non-separable -- Banach manifold modeled after the space
+ \(\mathcal{L}(V) = \mathcal{L}(V, V)\) under the operator norm:
+ \(\operatorname{GL}(V) = \mathcal{L}(V)^\times\).
\end{example}
\begin{example}
diff --git a/sections/structure.tex b/sections/structure.tex
@@ -7,11 +7,12 @@ first question we should ask ourselves is an obvious one: what is \(H^1(I,
M)\)? Specifically, what is a class \(H^1\) curve in \(M\)?
Given an interval \(I\), recall that a continuous curve \(\gamma : I \to
-\RR^n\) is called \emph{a class \(H^1\)} curve if \(\gamma\) is absolutely
-continuous, \(\dot \gamma(t)\) exists for almost all \(t \in I\) and
-\(\dot\gamma \in H^0(I, \RR^n) = L^2(I, \RR^n)\). It is a well known fact that
-the so called \emph{Sobolev space \(H^1([0, 1], \RR^n)\)} of all class \(H^1\)
-curves in \(\RR^n\) is a Hilbert space under the inner product given by
+\mathbb{R}^n\) is called \emph{a class \(H^1\)} curve if \(\gamma\) is
+absolutely continuous, \(\dot \gamma(t)\) exists for almost all \(t \in I\) and
+\(\dot\gamma \in H^0(I, \mathbb{R}^n) = L^2(I, \mathbb{R}^n)\). It is a well
+known fact that the so called \emph{Sobolev space \(H^1([0, 1],
+\mathbb{R}^n)\)} of all class \(H^1\) curves in \(\mathbb{R}^n\) is a Hilbert
+space under the inner product given by
\[
\langle \gamma, \eta \rangle_1
= \int_0^1 \gamma(t) \cdot \eta(t) + \dot\gamma(t) \cdot \dot\eta(t) \; \dt
@@ -22,10 +23,10 @@ Finally, we may define\dots
\begin{definition}
Given an \(n\)-dimensional manifold \(M\), a continuous curve \(\gamma : I
\to M\) is called \emph{a class \(H^1\)} curve if \(\varphi_i \circ \gamma :
- J \to \RR^n\) is a class \(H^1\) curve for any chart \(\varphi_i : U_i
- \subset M \to \RR^n\) -- i.e. if \(\gamma\) can be locally expressed as a
- class \(H^1\) curve in terms of the charts of \(M\). We'll denote by \(H^1(I,
- M)\) the set of all class \(H^1\) curves \(I \to M\).
+ J \to \mathbb{R}^n\) is a class \(H^1\) curve for any chart \(\varphi_i : U_i
+ \subset M \to \mathbb{R}^n\) -- i.e. if \(\gamma\) can be locally expressed
+ as a class \(H^1\) curve in terms of the charts of \(M\). We'll denote by
+ \(H^1(I, M)\) the set of all class \(H^1\) curves \(I \to M\).
\end{definition}
\begin{note}
@@ -40,31 +41,31 @@ concerned with the study of piece-wise smooth curves, so the fact that we are
now interested a larger class of curves -- highly non-smooth curves, in fact --
\emph{should} come as a surprise to the reader.
-To answer this second question we return to the case of \(M = \RR^n\). Denote
-by \({C'}^\infty(I, \RR^n)\) the space of piece-wise curves in \(\RR^n\). As
-described in section~\ref{sec:introduction}, we would like \({C'}^\infty(I,
-\RR^n)\) to be a Banach manifold under which both the energy functional and the
-length functional are smooth maps. As most function spaces, \({C'}^\infty(I,
-\RR^n)\) admits several natural topologies. Some of the most obvious candidates
-are the uniform topology and the topology of the \(\norm\cdot_0\) norm, which
-are the topologies induces by the norms
+To answer this second question we return to the case of \(M = \mathbb{R}^n\).
+Denote by \({C'}^\infty(I, \mathbb{R}^n)\) the space of piece-wise curves in
+\(\mathbb{R}^n\). As described in section~\ref{sec:introduction}, we would like
+\({C'}^\infty(I, \mathbb{R}^n)\) to be a Banach manifold under which both the
+energy functional and the length functional are smooth maps. As most function
+spaces, \({C'}^\infty(I, \mathbb{R}^n)\) admits several natural topologies.
+Some of the most obvious candidates are the uniform topology and the topology
+of the \(\norm\cdot_0\) norm, which are the topologies induces by the norms
\begin{align*}
\norm{\gamma}_\infty & = \sup_t \norm{\gamma(t)} \\
\norm{\gamma}_0 & = \int_0^1 \norm{\gamma(t)}^2 \; \dt
\end{align*}
respectively.
-The problem with the first candidate is that \(L : {C'}^\infty(I, \RR^n) \to
-\RR\) is not a continuous map under the uniform topology. This can be readily
-seen by approximating the curve
+The problem with the first candidate is that \(L : {C'}^\infty(I, \mathbb{R}^n)
+\to \mathbb{R}\) is not a continuous map under the uniform topology. This can
+be readily seen by approximating the curve
\begin{align*}
- \gamma : I & \to \RR^2 \\
+ \gamma : I & \to \mathbb{R}^2 \\
t & \mapsto (t, 1 - t)
\end{align*}
-with ``staircase curves'' \(\gamma_n : I \to \RR^n\) for larger and larger
-values of \(n\), as shown in figure~\ref{fig:step-curves}: clearly \(\gamma_n
-\to \gamma\) in the uniform topology, but \(L(\gamma_n) = 2\) does not
-approach \(L(\gamma) = \sqrt 2\) as \(n\) approaches \(\infty\).
+with ``staircase curves'' \(\gamma_n : I \to \mathbb{R}^n\) for larger and
+larger values of \(n\), as shown in figure~\ref{fig:step-curves}: clearly
+\(\gamma_n \to \gamma\) in the uniform topology, but \(L(\gamma_n) = 2\) does
+not approach \(L(\gamma) = \sqrt 2\) as \(n\) approaches \(\infty\).
\begin{figure}[h]
\centering
@@ -93,31 +94,31 @@ The issue with this particular example is that while \(\gamma_n \to \gamma\)
uniformly, \(\dot\gamma_n\) does not converge to \(\dot\gamma\) in the uniform
topology. This hints at the fact that in order for \(E\) and \(L\) to be
continuous maps we need to control both \(\gamma\) and \(\dot\gamma\). Hence a
-natural candidate for a norm in \({C'}^\infty(I, \RR^n)\) is
+natural candidate for a norm in \({C'}^\infty(I, \mathbb{R}^n)\) is
\[
\norm{\gamma}_1^2 = \norm{\gamma}_0^2 + \norm{\dot\gamma}_0^2,
\]
which is, of course, the norm induced by the inner product \(\langle \, ,
-\rangle_1\) -- here \(\norm{\cdot}_0\) denote the norm of \(H^0(I, \RR^n) =
-L^2(I, \RR^n)\).
-
-The other issue we face is one of completeness. Since \(\RR^n\) has a global
-chart, we expect \({C'}^\infty(I, \RR^n)\) to be affine too. In other words, it
-is natural to expect \({C'}^\infty(I, \RR^n)\) to be Banach space. In
-particular, \({C'}^\infty(I, \RR^n)\) must be complete. This is unfortunately
-not the case for \({C'}^\infty(I, \RR^n)\) in the \(\norm\cdot_1\) norm, but we
-can consider its completion. Lo and behold, a classical result by Lebesgue
-establishes that this completion just so happens to coincide with \(H^1(I,
-\RR^n)\).
-
-It's also interesting to note that the completion of \({C'}^\infty(I, \RR^n)\)
-with respect to the norms \(\norm\cdot_\infty\) and \(\norm\cdot_0\) are
-\(C^0(I, \RR^n)\) and \(H^0(I, \RR^n)\), respectively, and that the natural
-inclusions
+\rangle_1\) -- here \(\norm{\cdot}_0\) denote the norm of \(H^0(I,
+\mathbb{R}^n) = L^2(I, \mathbb{R}^n)\).
+
+The other issue we face is one of completeness. Since \(\mathbb{R}^n\) has a
+global chart, we expect \({C'}^\infty(I, \mathbb{R}^n)\) to be affine too. In
+other words, it is natural to expect \({C'}^\infty(I, \mathbb{R}^n)\) to be
+Banach space. In particular, \({C'}^\infty(I, \mathbb{R}^n)\) must be complete.
+This is unfortunately not the case for \({C'}^\infty(I, \mathbb{R}^n)\) in the
+\(\norm\cdot_1\) norm, but we can consider its completion. Lo and behold, a
+classical result by Lebesgue establishes that this completion just so happens
+to coincide with \(H^1(I, \mathbb{R}^n)\).
+
+It's also interesting to note that the completion of \({C'}^\infty(I,
+\mathbb{R}^n)\) with respect to the norms \(\norm\cdot_\infty\) and
+\(\norm\cdot_0\) are \(C^0(I, \mathbb{R}^n)\) and \(H^0(I, \mathbb{R}^n)\),
+respectively, and that the natural inclusions
\begin{equation}\label{eq:continuous-inclusions-rn-curves}
- H^1(I, \RR^n)
- \longhookrightarrow C^0(I, \RR^n)
- \longhookrightarrow H^0(I, \RR^n)
+ H^1(I, \mathbb{R}^n)
+ \longhookrightarrow C^0(I, \mathbb{R}^n)
+ \longhookrightarrow H^0(I, \mathbb{R}^n)
\end{equation}
are continuous.
@@ -197,7 +198,7 @@ find\dots
\begin{note}
Apply proposition~\ref{thm:continuous-inclusions-sections} to the trivial
- bundle \(I \times \RR^n \to I\) to get the continuity of the maps in
+ bundle \(I \times \mathbb{R}^n \to I\) to get the continuity of the maps in
(\ref{eq:continuous-inclusions-rn-curves}).
\end{note}
@@ -276,9 +277,9 @@ Finally, we find\dots
i.e. the coarsest topology such that such maps are continuous. This atlas
gives \(H^1(I, M)\) the structure of a \emph{separable} Banach manifold
modeled after separable Hilbert spaces, with typical representatives
- \(H^1(\gamma^* TM) \cong H^1(I, \RR^n)\)\footnote{Any trivialization of
- $\gamma^* TM$ induces an isomorphism $H^1(\gamma^* TM) \isoto H^1(I,
- \RR^n)$.}.
+ \(H^1(\gamma^* TM) \cong H^1(I, \mathbb{R}^n)\)\footnote{Any trivialization
+ of $\gamma^* TM$ induces an isomorphism $H^1(\gamma^* TM) \isoto H^1(I,
+ \mathbb{R}^n)$.}.
\end{theorem}
The fact that \(\exp_\gamma\) is bijective should be clear from the definition
@@ -307,11 +308,11 @@ precisely\dots
\gamma & \mapsto f \circ \gamma
\end{align*}
is smooth. In addition, \(H^1(I, f \circ g) = H^1(I, f) \circ H^1(I, g)\) and
- \(H^1(I, \id) = \id\) for any composable smooth maps \(f\) and \(g\). We thus
- have a functor \(H^1(I, -) : \categoryname{Rnn} \to \categoryname{BMnd}\)
- from the category \(\categoryname{Rnn}\) of finite-dimensional Riemannian
- manifolds and smooth maps onto the category \(\categoryname{BMnd}\) of Banach
- manifolds and smooth maps.
+ \(H^1(I, \operatorname{id}) = \operatorname{id}\) for any composable smooth
+ maps \(f\) and \(g\). We thus have a functor \(H^1(I, -) : \mathbf{Rnn} \to
+ \mathbf{BMnd}\) from the category \(\mathbf{Rnn}\) of finite-dimensional
+ Riemannian manifolds and smooth maps onto the category \(\mathbf{BMnd}\) of
+ Banach manifolds and smooth maps.
\end{theorem}
We would also like to point out that this is a particular case of a more
@@ -396,7 +397,7 @@ extended to a canonical isomorphism of vector bundles, as seen in\dots
of \(T H^1(I, M)\).
By composing charts we get local vector bundle isomorphism \(\psi_{1, \gamma}
- \circ (\id, \varphi_\gamma) \circ \varphi_\gamma :
+ \circ (\operatorname{id}, \varphi_\gamma) \circ \varphi_\gamma :
\varphi_\gamma^{-1}(H^1(W_\gamma) \times T_\gamma H^1(I, M)) \isoto \psi_{1,
\gamma}(H^1(W_\gamma) \times H^1(\gamma^* TM))\). Because of the fact that
\(\varphi_\gamma\) and \(\psi_{1, \gamma}^{-1}\) are charts, this
@@ -435,7 +436,7 @@ words, we'll show\dots
\begin{proof}
Given \(\gamma \in {C'}^\infty(I, M)\) and \(X \in H^1(\gamma^* TM)\), let
\begin{align*}
- g_X^\gamma : H^0(\gamma^* TM) \times H^0(\gamma^* TM) & \to \RR \\
+ g_X^\gamma : H^0(\gamma^* TM) \times H^0(\gamma^* TM) & \to \mathbb{R} \\
(Y, Z) &
\mapsto \int_0^1
\langle (d\exp)_{X_t} Y_t, (d\exp)_{X_t} Z_t \rangle \; \dt
@@ -514,10 +515,12 @@ We may now finally describe the canonical Riemannian metric of \(H^1(I, M)\).
At this point it should be obvious that definition~\ref{def:h1-metric} does
indeed endow \(H^1(I, M)\) with the structure of a Riemannian manifold: the
inner products \(\langle \, , \rangle_1 : H^1(\gamma^* TM) \times H^1(\gamma^*
-TM) \to \RR\) may be glued together into a single positive-definite section
-\(\langle \, , \rangle_1 \in \Gamma( \Sym^2 \coprod_\gamma H^1(\gamma^* TM))\)
--- whose smoothness follows from theorem~\ref{thm:h0-has-metric-extension},
+TM) \to \mathbb{R}\) may be glued together into a single positive-definite
+section \(\langle \, , \rangle_1 \in \Gamma( \operatorname{Sym}^2
+\coprod_\gamma H^1(\gamma^* TM))\) -- whose smoothness follows from
+theorem~\ref{thm:h0-has-metric-extension},
proposition~\ref{thm:partial-is-smooth-sec} and
proposition~\ref{thm:covariant-derivative-h0} -- which is then mapped to a
-positive-definite section of \(\Sym^2 T H^1(I, M)\) by the induced isomorphism
-\(\Sym^2 \coprod_\gamma H^1(\gamma^* TM) \isoto \Sym^2 T H^1(I, M)\).
+positive-definite section of \(\operatorname{Sym}^2 T H^1(I, M)\) by the
+induced isomorphism \(\operatorname{Sym}^2 \coprod_\gamma H^1(\gamma^* TM)
+\isoto \operatorname{Sym}^2 T H^1(I, M)\).