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
c105646c0a75af7854ce20c989cd4a55c16f2400 Parent
261612545184bb3ab38cd0e7df64400ea9b4549d Author
Pablo <pablo-escobar@riseup.net >
Date
Mon, 25 Jul 2022 14:31:52 +0000
Mostly finished the section on Banach manifolds
Diffstat
6 files changed, 243 insertions, 124 deletions
diff --git a/references.bib b/references.bib
@@ -1,126 +1,26 @@
-@book{etingof,
- title = {Introduction to Representation Theory},
- author = {Pavel Etingof},
- publisher = {American Mathematical Society},
- year = {2011},
- series = {Student Mathematical Library},
-}
-
-@article{frobenius1896gruppencharakteren,
- title={{\"U}ber Gruppencharakteren},
- author={Frobenius, F Georg},
- journal={Wiss. Berlin},
- pages={985--1021},
- year={1896}
-}
-
-@article{griess-1982,
- doi = {10.1007/bf01389186},
- year = {1982},
- month = feb,
- publisher = {Springer Science and Business Media {LLC}},
- volume = {69},
- number = {1},
- pages = {1--102},
- author = {Robert L. Griess},
- title = {The friendly giant},
- journal = {Inventiones mathematicae}
-}
-
-@book{pioneers,
- title = {Pioneers of Representation Theory: Frobenius, Burnside, Schur, and Brauer},
- author = {Charles W. Curtis},
- publisher = {American Mathematical Society, London Mathematical Society},
- isbn = {9780821890028},
+@book{klingenberg,
+ title={Riemannian Geometry},
+ author={Wilhelm Klingenberg},
+ isbn={9783110905120},
+ series={De Gruyter Studies in Mathematics},
+ year={2011},
+ publisher={De Gruyter}
+}
+
+@book{lang,
+ title = {Fundamentals of Differential Geometry},
+ author = {Serge Lang},
+ publisher = {Springer},
+ isbn = {9780387985930},
year = {1999},
- series = {History of Mathematics (Volume 15)},
- edition = {New},
-}
-
-@article{borcherds-1992,
- year = {1992},
- month = dec,
- publisher = {Springer Science and Business Media {LLC}},
- volume = {109},
- number = {1},
- pages = {405--444},
- author = {Richard E. Borcherds},
- title = {Monstrous moonshine and monstrous Lie superalgebras},
- journal = {Inventiones Mathematicae},
- doi = {https://doi.org/10.1007/bf01232032},
-}
-
-@book{doutorado-schur,
- title={{\"U}ber eine Klasse von Matrizen, die sich einer gegebenen Matrix zuordnen lassen},
- author={Schur, I.},
- year={1901}
-}
-
-@book{burnside-groups,
- title={Theory of groups of finite order},
- author={Burnside, W.},
- isbn={9781440035456},
- year={1897},
- publisher={The University Press}
+ series = {Graduate Texts in Mathematics №191},
+ edition = {1},
}
-@article{noether-hyperkomplexe,
- title={Hyperkomplexe Gr{\"o}ssen und Darstellungstheorie},
- author={Noether, Emmy},
- journal={Mathematische Zeitschrift},
- volume={30},
- number={1},
- pages={641--692},
- year={1929},
- publisher={Springer}
+@misc{unitary-group-strong-topology,
+ doi = {10.48550/ARXIV.1309.5891},
+ author = {Martin Schottenloher},
+ title = {The Unitary Group In Its Strong Topology},
+ publisher = {arXiv},
+ year = {2013},
}
-
-@book{noether-tribute,
- title={Emmy Noether: A Tribute to Her Life and Work},
- author={Noether, E. and Brewer, J.W. and Smith, R.G. and Smith, M.K.},
- isbn={9780824715502},
- lccn={lc81015203},
- series={Monographs and textbooks in pure and applied mathematics},
- year={1981},
- publisher={M. Dekker}
-}
-
-@book{frobenius1,
- title={{\"U}ber vertauschbare Matrizen},
- author={Frobenius, G.},
- isbn={9783111098005},
- series={Preussische Akademie der Wissenschaften Berlin: Sitzungsberichte der Preu{\ss}ischen Akademie der Wissenschaften zu Berlin},
- year={1896},
- publisher={Reichsdr.}
-}
-
-@book{frobenius2,
- title={{\"U}ber Gruppencharaktere},
- author={Frobenius, G.},
- isbn={9783111097978},
- series={Preussische Akademie der Wissenschaften Berlin: Sitzungsberichte der Preu{\ss}ischen Akademie der Wissenschaften zu Berlin},
- year={1896},
- publisher={Reichsdr.}
-}
-
-@book{frobenius3,
- title={{\"U}ber die Primfactoren der Gruppendeterminante},
- author={Frobenius, G.},
- year={1896},
-}
-
-@book{frobenius4,
- title={{\"U}ber die Darstellung der endlichen Gruppen durch lineare Substitutionen},
- author={Frobenius, G.},
- year={1897},
-}
-
-
-@book{referenciar,
- title={Abhandlungen der K{\"o}niglich Preussischen Akademie der Wissenschaften: 1837},
- author={Deutsche Akademie der Wissenschaften zu Berlin},
- series={Abhandlungen der K{\"o}niglich Preussischen Akademie der Wissenschaften},
- url={https://books.google.com.br/books?id=oto6AQAAMAAJ},
- year={1839},
- publisher={Verlag der K{\"o}niglichen Akademie der Wissenschaften in Commission bei Georg Reimer.}
-} -
\ No newline at end of file
diff --git a/sections/introduction.tex b/sections/introduction.tex
@@ -1 +1,187 @@
\section{Introduction}
+
+With applications in numerous fields of geometry and mathematical physics, such
+as the calculus of variations and the study of harmonic functions, the study of
+infinite-dimensional manifolds has proven itself a remarkbly powerful tool.
+Known as \emph{global analysis}, or sometimes \emph{non-linear functional
+analysis}, the field of study dedicated to the understanding of such manifolds
+has seen remarble progress in the past several decades.
+
+Besides the study of Banach Lie groups and algebras, the primary motivation
+behind the study of infinite-dimensional manifolds is the fact that they
+provide a framework to attack local problems regarding maps between manifolds
+by means of global tools -- hence the name ``\emph{global} analysis''.
+Explicitly, by endowing certain function spaces with differential structures we
+may translate local questions about this maps to global questions about such
+manifolds.
+
+At first it may seem like this has nothing to do with infinite-dimensional
+manifolds, but in practice carying out the process of assigning a meaningful
+differential strucutre to a function space usually requires us to drop the
+assumption of finite-dimensionality. In this notes we hope to provide a very
+breif introduction to the field of global analysis by exploring a concrete
+example of the program described in the above: 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 \(M\).
+
+In section~\ref{sec:structure} we will describe the differential structure of
+\(H^1(I, M)\) and its canonical Riemannian metric. In
+section~\ref{sec:aplications} we describe several applications to variational
+calculus and in particular the study of the geodesics of \(M\). Bofore moving
+to the next section, however, we would like to review the basics of the theory
+of Banach manifolds.
+
+\subsection{Banach Manifolds}
+
+While it is certainly true that Banach spaces can look alian to someone who has
+never ventured outside of the realms of Euclidean space, Banach manifolds are
+surprisingly similar to their finite-dimensional counterparts. As we'll soon
+see, most of the usual tools of differential geommetry can be quite easily
+translated to the realm of Banach manifolds\footnote{The real difficulties with
+Banach manifolds only show up while proving certain results, and are maily 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 \(\RR^n\).
+
+What we mean by this last statement is that none of the fundamental ingrediants
+of calculus in \(\RR^n\), namely the fact that it is a complete normed space,
+are specific to \(\RR^n\). In fact, the ingrediants previously described 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 manifolds and \(U \subset V\) be an open
+ subset. A continuous map \(f : U \to W\) is called \emph{differentiable at
+ \(p \in U\)} if there exists a continuous linear operator \(d f_p \in
+ \mathscr{B}(V, W)\) such that
+ \[
+ \frac{\norm{f(p + h) - f(p) - d f_p h}}{\norm{h}} \to 0
+ \]
+ as \(h \to 0\) in \(U\).
+\end{definition}
+
+\begin{definition}
+ Given Banach spaces \(V\) and \(W\) and an open subset \(U \subset V\), a
+ continuous map \(f : U \to W\) is called \emph{differentiable of class
+ \(C^1\)} if \(f\) is differentiable at \(p\) for all \(p \in U\) and the
+ derivative map
+ \begin{align*}
+ df: U & \to \mathscr{B}(V, W) \\
+ p & \mapsto d f_p
+ \end{align*}
+ is continuous. Since \(\mathscr{B}(V, W)\) is a Banach space under the
+ operator norm, we may recursively define functions of class \(C^n\) for \(n >
+ 1\): a function \(f : U \to W\) of class \(C^n\) is called
+ \emph{differentiable of class \(C^{n + 1}\)} if the map
+ \[
+ d^n f :
+ U \to \mathscr{B}(V, \mathscr{B}(V, \cdots \mathscr{B}(V, W)))
+ \cong \mathscr{B}(V^{\otimes n}, W)
+ \footnote{Here we consider the \emph{projective tensor product} of Banach
+ spaces. See \cite[ch.~1]{klingenberg}.}
+ \]
+ is of class \(C^1\). Finally, a map \(f : U \to W\) is called
+ \emph{differentiable of class \(C^\infty\)} or \emph{smooth} if \(f\) is of
+ class \(C^n\) for all \(n > 0\).
+\end{definition}
+
+\begin{lemma}\label{thm:chain-rule}
+ Given Banach spaces \(V_1\), \(V_2\) and \(V_3\), open subsets \(U_1 \subset
+ V_1\) and \(U_2 \subset V_2\) and two smooth maps \(f : U_1 \to U_2\) and \(g
+ : U_2 \to V_3\), the composition map \(g \circ f : U_1 \to V_3\) is smooth
+ and its derivative is given by
+ \[
+ (d g \circ f)_p = d g_{f(p)} \circ d f_p
+ \]
+\end{lemma}
+
+As promised, this simple definitions allows us to expand the usual tools of
+differential geometry to the infinite-dimensional setting. In fact, in most
+cases it suffices to straight up copy the definition of the finite-dimensional
+case. For instance, as in the finite-dimensional case we may call a map \(M \to
+N\) between Banach manifolds \(M\) and \(N\) \emph{smooth} if it can be locally
+expressed as a smooth function between open subsets of the model spaces. As
+such, we will only provide the most important definitions: those of a Banach
+manifold and its tangent space at a given point. Complete accounts of the
+subject can be found in \cite[ch.~1]{klingenberg} \cite[ch.~2]{lang}.
+
+\begin{definition}\label{def:banach-manifold}
+ A Banach manifold \(M\) is a Hausdorff topological space endowed with a
+ maximal atlas \(\{(U_i, \varphi_i)\}_i\), i.e. an open cover \(\{U_i\}_i\) of
+ \(M\) and homeomorphisms \(\varphi_i : U_i \to \varphi_i(U_i) \subset V_i\)
+ -- known as \emph{charts} -- where
+ \begin{enumerate}
+ \item Each \(V_i\) is a Banach space
+ \item For each \(i\) and \(j\), \(\varphi_i \circ \varphi_j^{-1} :
+ \varphi_j(U_i \cap U_j) \subset V_j \to \varphi_i(U_i \cap U_j) \subset
+ V_i\) is a smooth map
+ \item \(\{(U_i, \varphi_i)\}_i\) is maximal with respect to the items above
+ \end{enumerate}
+\end{definition}
+
+\begin{definition}
+ Given a Banach manifold \(M\) with maximal atlas \(\{(U_i, \varphi_i)\}_i\)
+ and \(p \in M\), the tangent space \(T_p M\) of \(M\) at \(p\) is the
+ quotient of the space \(\{ \gamma : (- \epsilon, \epsilon) \to M \mid \gamma
+ \ \text{is smooth}, \gamma(0) = p \}\) by the equivalence relation that
+ identifies two curves \(\gamma\) and \(\eta\) such that
+ \((\varphi_i \circ \gamma)'(0) = (\varphi_i \circ \eta)'(0)\) for all \(i\)
+ with \(p \in U_i\).
+ The space \(T_p M\) is a topological vector space isomorphic to \(V_i\) for
+ each \(i\) with \(p \in U_i\): each chart \(\varphi_i : U_i \to M\) induces a
+ norm in \(T_p M\)\footnote{In general $T_p M$ is not a normed space, since
+ the norms induced by two distinct choices of chard need not to coincide.
+ However, since all $V_i$'s are isomorphic as Banach spaces this norms are
+ all mutually equivalent, so that they define a (unique) topology in $T_p M$.}
+ given by the pullback of the norm of \(V_i\) through the linear isomorphism
+ \begin{align*}
+ \phi_i : T_p M & \isoto V_i \\
+ [\gamma] & \mapsto (\varphi_i \circ \gamma)'(0)
+ \end{align*}
+\end{definition}
+
+Notice that a single Banach manifold may be ``modeled after'' multible Banach
+spaces, in the sense that the \(V_i\)'s of definition~\ref{def:banach-manifold}
+may vary with \(i\). Lemma~\ref{thm:chain-rule} implies that for each \(i\) and
+\(j\) with \(p \in U_i \cap U_j\), \((d \varphi_i \circ
+\varphi_j^{-1})_{\varphi_j(p)} : V_j \to V_i\) is a continuous linear
+isomorphism, so that we may assume that each connected component of \(M\) is
+modeled after a single Banach space \(V\). It is sometimes conviniant, however,
+to allow ourselves the more leniant notion of Banach manifold aforded 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
+Hausdorff is redundant. Although we are primarily interested in manifolds
+modeled after separable spaces, in the interest of afording 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
+
+\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 \(\id : U \to V\).
+\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 \(A\). 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 \(\mathscr{B}(V) = \mathscr{B}(V, V)\) under the
+ operator norm: \(\GL(V) = \mathscr{B}(V)^\times\).
+\end{example}
+
+\begin{example}
+ Given a complex Hilbert space \(H\), the space \(\operatorname{U}(H)\) of
+ unitary operators \(H \to H\) -- endowed with the topology of the operator
+ norm -- is a Banach manifold modeled after the closed subspace
+ \(\mathfrak{u}(H) \subset \mathscr{B}(H)\) of continuous skew-symmetric
+ operators \(H \to H\) \cite[p.~4]{unitary-group-strong-topology}.
+\end{example}
+
+This last two examples are examples of Banach Lie groups -- i.e. Banach
+manifolds endowed with a group structure whose group operations are smooth.
+