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
b38ce71bf13d0c6f315b8310a0f60733eb7525cb Parent
a04f80212cc42d3a88db28f0f0abe66181209d17 Author
Pablo <pablo-escobar@riseup.net >
Date
Wed, 27 Jul 2022 22:53:36 +0000
Started to work on the section about the metric of H¹(I, M)
Mostly finished the initial discussion about the canonical isomorphism of vector bundles used in the description of the metric
Diffstat
3 files changed, 185 insertions, 33 deletions
diff --git a/sections/introduction.tex b/sections/introduction.tex
@@ -41,10 +41,10 @@ with a space of functions \(M \to N\), getting back at the original case.
In this notes we hope to provide a very brief introduction to modern theory the
calculus of variations by exploring one of the simplest concrete examples of
the previously described program: we study the differential structure of the
-Banach manifold \(H^1(I, M)\) of class \(H^1\) curves in a complete
-finite-dimensional 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}
+Banach manifold \(H^1(I, M)\) of class \(H^1\) curves in a finite-dimensional
+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 \\
\gamma & \mapsto \frac{1}{2} \int_0^1 \norm{\dot\gamma(t)}^2 \; \dt,
@@ -60,10 +60,18 @@ In section~\ref{sec:structure} we will describe the differential structure of
section~\ref{sec:aplications} we study the critical points of the energy
functional \(E\) and describe how the fundamental results of the classical
theory of the calculus of variations in the context of Riemannian manifolds can
-be reproduced in our new setting. We'll assume basic knowledge of differential
-and Riemannian geometry, as well as some familiarity with the classical theory
-of the calculus of variations -- see \cite[ch.~5]{gorodski} for the classical
-theory.
+be reproduced in our new setting. Other examples of function spaces are
+explored in detail in \cite[sec.~6]{eells}. The 11th chapter of \cite{palais}
+is also a great reference for the general theory of spaces of sections of fiber
+bundles.
+
+% TODO: Finilize this
+We should point out that due to limitations of space we will primarily focus on
+the broad strokes of the theory ahead. Many results are left unprovred, but we
+will include references to other materials containing proofs. We'll assume
+basic knowledge of differential and Riemannian geometry, as well as some
+familiarity with the classical theory of the calculus of variations -- see
+\cite[ch.~5]{gorodski} for the classical theory.
Before moving to the next section we would like to review the basics of the
theory of real Banach manifolds.
@@ -180,7 +188,7 @@ subject can be found in \cite[ch.~1]{klingenberg} and \cite[ch.~2]{lang}.
\end{align*}
\end{definition}
-\begin{lemma}
+\begin{proposition}\label{thm:tanget-space-topology}
Given \(p \in M\) and a chart \(\varphi_i\) with \(p \in U_i\), \(\phi_{p,
i}\) is a linear isomorphism. For any given charts \(\varphi_i, \varphi_j\),
the pullback of the norms of \(V_i\) and \(V_j\) by \(\varphi_i\) and
@@ -190,7 +198,7 @@ subject can be found in \cite[ch.~1]{klingenberg} and \cite[ch.~2]{lang}.
general $T_p M$ is not a normed space, since the norms induced by two
distinct choices of chard need not to coincide. Nevertheless, the topology
induced by this norms is the same.}.
-\end{lemma}
+\end{proposition}
\begin{proof}
The first statement about \(\phi_{p, i}\) being a linear isomorphism should
@@ -225,12 +233,12 @@ modeled after a single Banach space \(V\). It is sometimes convenient, however,
to allow ourselves the more lenient notion of Banach manifold afforded by
definition~\ref{def:banach-manifold}.
-We should also note that some authors assume that the \(V_i\)'s are
-\emph{separable} Banach spaces, in which case the assumption that \(M\) is
+We should also note that some authors assume that both the \(V_i\)'s and \(M\)
+itself are \emph{separable}, in which case the assumption that \(M\) is
Hausdorff is redundant. Although we are primarily interested in manifolds
-modeled after separable spaces, in the interest of affording ourselves a greater
-number of examples we will \emph{not} assume the \(V_i\)'s to be separable --
-unless explicitly stated otherwise. Speaking of examples\dots
+modeled after separable spaces, in the interest of affording ourselves a
+greater number of examples we will \emph{not} assume separability -- unless
+explicitly stated otherwise. Speaking of examples\dots
\begin{example}
Any Banach space \(V\) can be seen as a Banach manifold with atlas given by
diff --git a/sections/structure.tex b/sections/structure.tex
@@ -1,10 +1,10 @@
\section{The Structure of \(H^1(I, M)\)}\label{sec:structure}
-Throughout this sections let \(M\) be a complete finite-dimensional Riemannian
-manifold. As promised, in this section we will highlight the differential and
-Riemannian structures of the space \(H^1(I, M)\) of class \(H^1\) curves in a
-\(M\). The 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 \(M\)?
+Throughout this sections let \(M\) be a finite-dimensional Riemannian manifold.
+As promised, in this section we will highlight the differential and Riemannian
+structures of the space \(H^1(I, M)\) of class \(H^1\) curves in a \(M\). The
+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
@@ -95,7 +95,8 @@ natural candidate for a norm in \({C'}^\infty(I, \RR^n)\) 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 \(L^2(I, \RR^n)\).
+\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 to expect \({C'}^\infty(I, \RR^n)\) to be affine too. In other words,
@@ -109,9 +110,9 @@ It's also interesting to note that the completion of \({C'}^\infty(I, \RR^n)\)
with respect to the norm \(\norm\cdot_\infty\) is the space \(C^0(I, \RR^n)\)
of all continuous curves \(I \to \RR^n\), and that the natural inclusions
\begin{equation}\label{eq:continuous-inclusions-rn-curves}
- {C'}^\infty(I, \RR^n)
- \longhookrightarrow H^1(I, \RR^n)
+ H^1(I, \RR^n)
\longhookrightarrow C^0(I, \RR^n)
+ \longhookrightarrow H^0(I, \RR^n)
\end{equation}
are continuous.
@@ -140,13 +141,22 @@ find\dots
\]
\end{proposition}
+\begin{proposition}
+ Given an Euclidean bundle \(E \to I\), the space \(H^0(E)\) of all square
+ integrable sections of \(E\) is is the completion of \({C'}^\infty(E)\) under
+ the inner product given by
+ \[
+ \langle \xi, \eta \rangle_0 = \int_0^1 \langle \xi_t, \eta_t \rangle \; \dt
+ \]
+\end{proposition}
+
\begin{proposition}\label{thm:continuous-inclusions-sections}
Given an Euclidean bundle \(E \to I\), the inclusions
\[
- {C'}^\infty(E) \longhookrightarrow H^1(E) \longhookrightarrow C^0(E)
+ H^1(E) \longhookrightarrow C^0(E) \longhookrightarrow H^0(E)
\]
are continuous. More precisely, \(\norm{\xi}_\infty \le \sqrt 2
- \norm{\xi}_1\).
+ \norm{\xi}_1\) and \(\norm{\xi}_0 \le \norm{\xi}_\infty\).
\end{proposition}
\begin{proof}
@@ -168,6 +178,14 @@ find\dots
& \le 2 \norm{\xi}_1^2
\end{split}
\]
+
+ Finally, given \(\xi \in H^0(E)\) we have
+ \[
+ \norm{\xi}_0^2
+ = \int_0^1 \norm{\xi_t}^2 \; \dt
+ \le \int_0^1 \norm{\xi}_\infty \; \dt
+ = \norm{\xi}_\infty
+ \]
\end{proof}
\begin{note}
@@ -219,9 +237,10 @@ We begin with a technical lemma.
\longhookrightarrow C^0(\gamma^* TM)\) and is therefore open.
\end{proof}
-Let \(W \subset TM\) be an open neighborhood of the zero section in \(TM\)
-such that \(\exp\!\restriction_W : W \to \exp(W)\) is invertible -- whose
-existence is given by the fact that \(M\) is complete.
+Let \(W \subset TM\) be an open neighborhood of the zero section in \(TM\) such
+that \(\exp\!\restriction_W : W \to \exp(W)\) is invertible -- whose existence
+follows from the fact that the injectivity radius depends continuously on \(p
+\in M\).
\begin{definition}
Given \(\gamma \in {C'}^\infty(I, M)\) let \(W_\gamma, W_{\gamma, t} \subset
@@ -246,9 +265,134 @@ Finally, we find\dots
Given \(\gamma \in {C'}^\infty(I, M)\), the map \(\exp_\gamma : H^1(W_\gamma)
\to U_\gamma\) is bijective. The collection \(\{(U_\gamma, \exp_\gamma^{-1} :
U_\gamma \to H^1(\gamma^* TM))\}_{\gamma \in {C'}^\infty(I, M)}\) is an atlas
- for \(H^1(I, M)\) under the final topology of the maps \(\exp_\gamma\). This
- atlas gives \(H^1(\gamma^* TM)\) the structure of a \emph{separable} Hilbert
+ for \(H^1(I, M)\) under the final topology of the maps \(\exp_\gamma\) --
+ i.e. the coarsest topology such that such maps are continuous. This atlas
+ gives \(H^1(\gamma^* TM)\) the structure of a \emph{separable} Banach
manifold.
\end{theorem}
-It should be obvious from the definition
+The fact that \(\exp_\gamma\) is bijective should be clear from the definition
+of \(U_\gamma\) and \(W_\gamma\). That each \(\exp_\gamma^{-1}\) is a
+homeomorphism is also clear from the definition of the topology of \(H^1(I,
+M)\). Moreover, since \({C'}^\infty(I, M)\) is dense, \(\{U_\gamma\}_{\gamma
+\in {C'}^\infty(I, M)}\) is an open cover of \(H^1(I, M)\). The real difficulty
+of this proof is showing that the transition maps
+\[
+ \exp_\eta^{-1} \circ \exp_\gamma :
+ \exp_\gamma^{-1}(U_\gamma \cap U_\eta) \subset H^1(\gamma^* TM)
+ \to H^1(\eta^* TM)
+\]
+are diffeomorphisms, as well as showing that \(H^1(I, M)\) is separable. We
+leave this details we leave as an exercise to the reader -- see theorem 2.3.12
+of \cite{klingenberg} for a full proof.
+
+The charts \(\exp_\gamma^{-1}\) are modeled after separable Hilbert spaces,
+with tipical representatives \(H^1(\gamma^* TM) \cong H^1(I,
+\RR^n)\)\footnote{Any trivialization of $\gamma^* TM$ induces an isomorphism
+$H^1(\gamma^* TM) \isoto H^1(I, \RR^n)$.}.
+
+It's interesting to note that this construction is functorial. More
+precisely\dots
+
+\begin{theorem}
+ Given finite-dimensional Riemannian manifolds \(M\) and \(N\) and a smooth
+ map \(f : M \to N\), the map
+ \begin{align*}
+ H^1(I, f) : H^1(I, M) & \to H^1(I, N) \\
+ \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 \(g\).
+\end{theorem}
+
+We would also like to point out that this is a particular case of a more
+general construction: that of the Banach manifold \(H^1(E)\) of class \(H^1\)
+sections of a smooth fiber bundle \(E \to I\) -- not necessarity a vector
+bundle. Our construction of \(H^1(I, M)\) is equivalent to of of the manifold
+\(H^1(I \times M)\) in the sense that the canonical map
+\[
+ \arraycolsep=1pt
+ \begin{array}{rl}
+ \tilde{\cdot} : H^1(I, M) \to & H^1(I \times M) \\
+ \gamma \mapsto &
+ \begin{array}[t]{rl}
+ \tilde\gamma : I & \to I \times M \\
+ t & \mapsto (t, \gamma(t))
+ \end{array}
+ \end{array}
+\]
+can be easily checked to be a diffeomorphism.
+
+The space \(H^1(E)\) is modeled after the Hilbert spaces \(H^1(F)\) of class
+\(H^1\) sections of open sub-bundles \(F \subset E\) which have the structure
+of a vector bundle -- the so called \emph{vector bundle neighborhoods of
+\(E\)}. This construction is highlighted in great detail and generality in the
+first section of \cite[ch.~11]{palais}, but unfortunately we cannot afford such
+a divergion in this short notes. Having saied that, we are now finally ready to
+discuss the Riemannian structure of \(H^1(I, M)\).
+
+\subsection{The Metric of \(H^1(I, M)\)}
+
+We begin our discussion of the Riemannian structure of \(H^1(I, M)\) by looking
+at its tangent bundle. Notice that for each \(\gamma \in {C'}^\infty(I, M)\)
+the chart \(\exp_\gamma^{-1} : U_\gamma \to H^1(\gamma^* TM)\) induces a
+canonical isomorphism \(\phi_\gamma : T_\gamma H^1(I, M) \isoto H^1(\gamma^*
+TM)\), as described in propostion~\ref{thm:tanget-space-topology}. In fact,
+this isomorphisms may be extended to a cononical isomorphism of vector bundles,
+as seen in\dots
+
+\begin{lemma}
+ Given \(i = 0, 1\), the collection \(\{(\psi_{i, \gamma}(H^1(U_\gamma) \times
+ H^i(\gamma^* TM)), \psi_{i, \gamma}^{-1})\}_{\gamma \in {C'}^\infty(I,
+ M)}\) with
+ \[
+ \arraycolsep=1pt
+ \begin{array}{rl}
+ \psi_{i, \gamma} : H^1(U_\gamma) \times H^i(\gamma^* TM) \to
+ & \prod_{\eta \in H^1(I, M)} H^i(\eta^* TM) \\
+ (X, Y) \mapsto &
+ \begin{array}[t]{rl}
+ \psi_{i, \gamma}(X) : I & \to \exp_\gamma(X)^* TM \\
+ t & \mapsto (d \exp_{\gamma(t)})_{X_t} Y_t
+ \end{array}
+ \end{array}
+ \]
+ gives \(\coprod_{\gamma \in {C'}^\infty(I, M)} H^i(\gamma^* TM) \to H^1(I,
+ M)\) the structure of a smooth vector bundle.
+\end{lemma}
+
+\begin{proposition}
+ There is a canonical isomorphism of vector bundles
+ \[
+ T H^1(I, M) \isoto \coprod_{\gamma \in H^1(I, M)} H^1(\gamma^* TM)
+ \]
+ whose restriction \(T_\gamma H^1(I, M) \isoto H^1(\gamma^* TM)\) is given by
+ \(\phi_\gamma\) for all \(\gamma \in {C'}^\infty(I, M)\).
+\end{proposition}
+
+\begin{proof}
+ Note that the sets \(H^1(U_\gamma) \times T_\gamma H^1(I, M)\) are precisely
+ the images of the canonical charts
+ \[
+ \varphi_\gamma :
+ \varphi_\gamma^{-1}(H^1(U_\gamma) \times T_\gamma H^1(I, M))
+ \subset T H^1(I, M)
+ \to H^1(U_\gamma) \times T_\gamma H^1(I, M)
+ \]
+ of \(T H^1(I, M)\).
+
+ By composing charts we get local vector bundle isomorphism \(\psi_{1, \gamma}
+ \circ (\id, \varphi_\gamma) \circ \varphi_\gamma :
+ \varphi_\gamma^{-1}(H^1(U_\gamma) \times T_\gamma H^1(I, M)) \isoto \psi_{1,
+ \gamma}(H^1(U_\gamma) \times H^1(\gamma^* TM))\). Because of the fact that
+ \(\varphi_\gamma\) and \(\psi_{1, \gamma}^{-1}\) are charts, this
+ isomorphisms agree in the intersections, so they may be glued together into a
+ global vector bundle isomorphism. Furthermore, by construction the
+ restriction of this isomorphism to \(T_\gamma H^1(I, M)\) with \(\gamma \in
+ {C'}^\infty(I, M)\) is given by \(\phi_\gamma\).
+\end{proof}
+
+This las result will be the basis for our analysis of the Riemannian structure
+of \(H^1(I, M)\). We may now describe the metric of \(H^1(I, M)\) in terms of
+the fibers \(T_\gamma H^1(I, M) \cong H^1(\gamma^* TM)\).
+