chapter guide, fixed preface, minor fixes

Author: abogatskiy

geom_autistic_intro.tex

@@ -19,12 +19,18 @@
 \clearpage
 \printglossary[type=\acronymtype,title=Abbreviations,style=long,nonumberlist] % nonumberlist
 
-
 \clearpage
 \section*{Preface}
 \addcontentsline{toc}{section}{Preface}
+\markboth{Preface}{Preface}
+\input{parts/introduction}
+
+\clearpage
+\section*{Chapter guide}
+\addcontentsline{toc}{section}{Chapter guide}
+\markboth{Chapter guide}{Chapter guide}
+\input{parts/guide}
 
-\include{parts/introduction}
 
 \part{Basic Geometric Objects}
 \include{parts/topology}
@@ -56,7 +62,7 @@ \part{Classical Geometries \texorpdfstring{\ucmark}{}}\label{Part Classical Geom
 
 
 \clearpage
-\part{Homology and Cohomology}\label{Part II}
+\part{Homology and Cohomology}\label{Part Homology}
 \include{parts/homology}
 
 

parts/classical_geom.tex

@@ -1,6 +1,6 @@
 
 \clearpage
-\chapter{Riemannian Geometry \texorpdfstring{\ucmark}{}}
+\chapter{Riemannian Geometry \texorpdfstring{\ucmark}{}}\label{ch riemannian geom}
 
 \section{Riemannian manifolds}\label{sec: Riemannian mfds}
 
@@ -394,11 +394,8 @@ \section{Morse theory}
 
 
 
-
-
-
 \clearpage
-\chapter{Complex Geometry \texorpdfstring{\ucmark}{}}
+\chapter{Complex Geometry \texorpdfstring{\ucmark}{}}\label{ch complex geom}
 
 
 \section{Complex and Hermitian structures}
@@ -599,7 +596,7 @@ \chapter{Symplectic geometry \texorpdfstring{\ucmark}{}}
 
 
 
-\chapter{Applications in Lie Groups \texorpdfstring{\ucmark}{}}
+\chapter{Applications in Lie Groups \texorpdfstring{\ucmark}{}}\label{ch applications in Lie groups}
 
 
 \section{Geometry of Lie groups}

parts/diff_geom.tex

@@ -482,7 +482,7 @@ \section{Flat bundles}
 
 	We have shown that bundles over $M$ with a totally disconnected structure group $G$ are precisely classified by the conjugacy classes of homomorphisms $\Hom(\pi_1(M),G)$. The conjugacy class of such homomorphisms corresponding to a bundle $E\overset{\pi}{\to}M$ is called the \emph{characteristic class}\index{Characteristic class} of the bundle. Therefore, for totally disconnected structure groups, a single characteristic class determines the isomorphism class of the bundle.
 
-	The main goal for Parts \ref{Part II} and \ref{Part III} of these lectures will be to develop the machinery necessary for the full classification of \glspl{fb} with arbitrary topological structure groups, in particular by extending the theory of characteristic classes. The characteristic class we have just constructed (acting on one-dimensional cycles, i.e., closed loops) is only the first one in an infinite list of cohomology classes corresponding to a given bundle.
+	The main goal for \Part s~\ref{Part Homology} and \ref{Part III} of these lectures will be to develop the machinery necessary for the full classification of \glspl{fb} with arbitrary topological structure groups, in particular by extending the theory of characteristic classes. The characteristic class we have just constructed (acting on one-dimensional cycles, i.e., closed loops) is only the first one in an infinite list of cohomology classes corresponding to a given bundle.
 \end{example}
 
 
@@ -1242,7 +1242,7 @@ \section{Tangent bundle}
     \[\dd \varphi_b^i=\left[\left(\varphi_b\circ\varphi_a^{-1}\right)_{\ast \varphi_a(m)}\right]^i_j \dd \varphi_a^j,\quad \alpha^b_{i,m}=\left[\left(\varphi_a\circ\varphi_b^{-1}\right)_{\ast \varphi_b(m)}\right]^j_i \alpha^a_{j,m}.\label{eq 1.4.23 RS1}\]
 \end{rem}
 
-We will return to a more detailed study of the cotangent bundle in \Chap~\ref{sec: differential forms}.
+We will return to a more detailed study of the cotangent bundle in \Chap~\ref{ch: differential forms}.
 
 Another important class of bundles, called jet bundles, are a generalization of the tangent and cotangent bundles. Recall that one way of defining tangent vectors was as directional derivatives acting on smooth functions. Similarly, covectors in $\T_p^\ast M$ are linear functionals on vectors, hence they can be identified with differentials $\dd_p f$ of smooth functions. The differential $\dd_p f$ can be identified with the linear term of the Taylor series for $f$ at $p$ in any local coordinate system. Thinking of functions $C^\infty(M)$ as sections of the trivial bundle $M\times\bbR$, we can alternatively describe covectors at $p\in M$ as equivalence classes of (local) sections of this bundle where two sections are considered equivalent if the order $1$ terms in their Taylor expansions at $p\in M$ in any local coordinate chart coincide. Extending this to higher derivatives naturally leads to the following construction.
 
@@ -2387,7 +2387,7 @@ \section{Vector fields as differential operators}
     This formula essentially states that $\mathbb{D}$ ``commutes with contractions''.
 \end{cor}
 
-This allows us to extend the definition of the Lie derivative to arbitrary tensor fields. This natural operator was first formalized by \'Elie Cartan around 1920, although he was primarily interested in derivatives of differential forms, which we will study in \S\ref{sec: differential forms}. It was named after Sophus Lie by others in the 1930's.
+This allows us to extend the definition of the Lie derivative to arbitrary tensor fields. This natural operator was first formalized by \'Elie Cartan around 1920, although he was primarily interested in derivatives of differential forms, which we will study in \Chap~\ref{ch: differential forms}. It was named after Sophus Lie by others in the 1930's.
 
 \begin{defn}[Lie derivatives of tensors]\index{Lie derivative!of a tensor field}
     If $X\in\fX(M)$ then the Lie derivative operator $\Lie_X$ on the tensor algebra $\Gamma^\infty(M)$ is defined as the unique tensor differential operator (established by Willmore's theorem) that coincides with the previously constructed Lie derivative on $C^\infty(M)$ and $\fX(M)$.
@@ -2853,7 +2853,7 @@ \section{Frobenius theorem I}\label{sec: frobenius i}
 
 
 \clearpage
-\chapter{Differential Forms and Integration}\label{sec: differential forms}
+\chapter{Differential Forms and Integration}\label{ch: differential forms}
 
 
 While symmetric algebras (or even commutative rings), and in particular symmetric tensors, can be identified with polynomials, the dual objects are antisymmetric tensors, which turn out to form an algebra of their own. Following the original invention of this algebra by Grassmann, it became important in the theory of differential equations, and eventually, with \'Elie Cartan, in geometry more broadly.

parts/group_actions.tex

@@ -2332,7 +2332,7 @@ \section{Yamabe's theorem}\label{sec: Yamabe's theorem}
 
 
 \clearpage
-\chapter{Group Actions on Manifolds}
+\chapter{Group Actions on Manifolds}\label{ch group actions}
 
 
 \section{Homogeneous spaces}\label{sec: homogeneous spaces}
@@ -4289,7 +4289,7 @@ \section{(*) Diffeomorphism groups}\label{sec: Diff groups}
 
 
 \clearpage 
-\chapter{Principal and Associated Bundles}
+\chapter{Principal and Associated Bundles}\label{ch pfb}
 
     Now that we have comprehensively reviewed the fundamentals of Lie groups and their actions on manifolds, we are finally ready to return to Klein's Erlangen program as described at the beginning of \Chap~\ref{chap: Lie theory ii}. We will do so using the more modern language of principal bundles.
 

parts/guide.tex

@@ -0,0 +1,18 @@
+The following tracks provide shorter routes through the book focused more on specific topics. Some of them are significantly longer than others, and some are not yet finished (particularly the last three). A special warning is in order for \Chap s \ref{ch: general topology} and \ref{ch manifolds}: both of them are currently not written in as comprehensive or complete form as the other completed chapters, and many of the proofs are omitted. Thus, it is generally recommended to have some basic familiarity with point-set topology and basic properties of smooth manifolds and smooth maps before reading.
+
+\begin{itemize}
+    \item \textbf{Basic differential geometry:} \Chap s \ref{ch category I}-\ref{ch: differential forms}.
+    \item \textbf{Lie theory:} requires basic differential geometry; \Chap s \ref{ch Lie I}, \ref{chap: Lie theory ii}-\ref{sec: Lie theory iii}, \ref{ch applications in Lie groups}, \Part~\ref{part Lie Theory}.
+    \item \textbf{Bundles and connections:} requires basic differential geometry; \Chap s \ref{chap: Fiber bundles}-\ref{ch: differential forms}, \ref{ch group actions}-\ref{ch pfb}, \ref{ch: connections}-\ref{ch affine structures}.
+    \item \textbf{Cartan geometry:} requires bundles and includes connections; \Part s \ref{Part Structured Geom I}-\ref{Part Structured Geom II}. This includes several relatively independent parts:
+        \begin{itemize}
+            \item Symmetries in differential equations: \Chap~\ref{ch: diff eq and sym}.
+            \item Equivalence problems: \S\ref{sec: jets and cartan distr}, \Chap~\ref{ch equivalence problems}.
+            \item Exterior differential systems: \Chap s \ref{chap: EDSs}.
+            \item Theory of surfaces in Euclidean space: \Chap s \ref{chap: EDSs}-\ref{chap curves and surfaces}
+            \item Klein and Cartan geometries: \Chap~\ref{chap: Klein geom}, \Part~\ref{Part Structured Geom II}.
+        \end{itemize}
+    \item \textbf{Homology and cohomology:} particular \chap s require basic differential geometry; \Chap s \ref{ch category I}-\ref{ch: general topology}, \ref{ch homotopy theory}, \Part~\ref{Part Homology}.
+    \item \textbf{Classical geometries:} requires bundles and connections; \Part~\ref{Part Classical Geom}.
+    \item \textbf{Gauge theory:} requires bundles and connections; \Chap s \ref{ch riemannian geom}-\ref{ch complex geom}, \Part~\ref{Part III}.
+\end{itemize}

parts/introduction.tex

@@ -1,20 +1,27 @@
 
 These notes started when I, along with my dear friend Umang Mehta, taught a series of seminars to our fellow physics grad students at UChicago in 2018/19. My desire to do so was based, in part, on the fact that another friend of mine, Sam Pramodh, taught me many of these topics in a similar fashion two years earlier.
 
-In 2024 these notes started getting greatly expanded as part of my effort to continue deepening my understanding of these subjects, as well as to continue teaching my friends. The subtitle \emph{"an autistic introduction"} refers to the only style of teaching that I find fully satisfying: \emph{bottom up}. In short, this means that I try to apply the following principle as extensively as reasonably possible:
+In 2024 these notes started getting greatly expanded as part of my effort to continue deepening my understanding of these subjects, as well as to continue teaching my friends. The subtitle \emph{``an autistic introduction''} refers to the only style of teaching that I find fully satisfying: \emph{bottom up}. In short, this means that I try to apply the following principle as extensively as reasonably possible:
 \begin{quote}
     No sophisticated construction is introduced until it is motivated enough to feel necessary.
 \end{quote}
 Teaching in this way inherently forces one to understand the objects being introduced much deeper, always keeping one's eyes on the ultimate goal. It also helps one retain the sense of discovery associated with any good learning experience: instead of being ``passed down from the top of a mountain'', definitions and theorems come up as the most natural objects and questions that one should consider. 
 
-This is why, even though most of the material in these notes is copied from external sources, it is heavily refactored and synthesized to provide a bottom up narrative that, to a maximum reasonable extent, moves from general to specific, adding new structure only when it becomes necessary. The ``prize'' on which we keep our eyes in this entire process is the classification of fiber bundles and geometric structures on them, which eventually forces us to learn about homology and cohomology, to which an separate \partt\ of the book is dedicated. 
+This is why, even though most of the material in these notes is copied from external sources, it is heavily refactored and synthesized to provide a bottom up narrative that, to a maximum reasonable extent, moves from general to specific, adding new structure only when it becomes necessary. The ``prize'' on which we keep our eyes in this entire process is the classification of fiber bundles and geometric structures on them, which eventually forces us to learn about homology and cohomology, to which an entire separate \partt\ of the book is dedicated. 
 
-Nevertheless, there is no such thing as a ``complete understanding'' of any subject. One of the wonderful unique properties of mathematics is that one can \emph{always} dig deeper and find ever more ``fundamental'' objects through which familiar results can be unified and reformulated. In other words, \emph{things are never understood in themselves, but only in relation to each other}. As such, one of the main goals of this book is not to speed to the finish line of, say, gauge theory. The goal is to give relatively comprehensive introductions to a variety of geometric tools, exploring their connections to each other, looking at each one from multiple sides, and applying them to as many familiar and simple examples as possible. This is in contrast with too much of academic mathematical literature, which is overloaded with hard theory and neglects giving the reader a simple example that illustrates the approach.
+Nevertheless, there is no such thing as a ``complete understanding'' of any subject. One of the wonderful unique properties of mathematics is that one can \emph{always} dig deeper and find ever more ``fundamental'' objects through which familiar results can be unified and reformulated. In other words, \emph{things are never understood in themselves, but only in relation to each other}. As such, one of the main goals of this book is not to speed to the finish line of, say, gauge theory. The goal is to give relatively comprehensive introductions to a variety of geometric tools, exploring their connections to each other, looking at each one from multiple sides, and applying them to as many familiar and simple examples as possible. This is in contrast with so much of academic mathematical literature which is overloaded with hard theory and neglects giving the reader a simple example that illustrates the approach.
 
-Another side effect of this approach is that different sections of this book are essentially comprehensive introductions into subjects that are usually taught separately, without too much cross-talk, so that it is relatively easy to start reading ``from the middle''. For example, the theory of Lie groups is rarely included in differential geometry courses, but its historical development was actually inseparably connected with the creation of modern geometry. The word ``geometry'' itself, at the end of the 19th century, usually referred to a certain set of properties shared by what we now call homogeneous spaces of Lie groups. Even though I am not a proponent of the strict chronological approach to teaching (which is all too common in physics), the general historic progression of the development of geometric ideas was ``correct'' in that Lie groups provide the simplest \emph{models} for most geometric structures. Thus, a deep understanding of the structure and geometry of Lie groups makes the development of the theory of connections much more natural and easier to digest, not to mention the endless examples easily computable ``by hand'' that are provided by Lie groups.
+Another side effect of this approach is that different \chap\ s of this book are essentially comprehensive introductions into subjects that are usually taught separately, without too much cross-talk, so that it is relatively easy to start reading ``from the middle''. For example, the theory of Lie groups is rarely included in differential geometry courses, but its historical development was actually inseparably connected with the creation of modern geometry. The word ``geometry'' itself, at the end of the 19th century, usually referred to a certain set of properties shared by what we now call homogeneous spaces of Lie groups and largely excluded what we now call ``Riemannian geometry''. Even though I am not a proponent of the strict chronological approach to teaching (which is all too common in physics), the general historic progression of the development of geometric ideas was ``correct'' in that Lie groups provide the simplest \emph{models} for most geometric structures. Thus, a deep understanding of the structure and geometry of Lie groups makes the development of the theory of connections much more natural and easier to digest, not to mention the endless examples easily computable ``by hand'' that are provided by Lie groups.
+
+One important thematic difference that makes this book unique among ``introductory'' texts on geometry is its focus on Cartan geometry. In fact, \Chap s \ref{ch: diff eq and sym} through \ref{chap: EDSs}, especially \Chap~\ref{ch equivalence problems}, are in many ways the culmination of the book's narrative. If I had to pick a single \chap\ which I am most proud of and which I would highly recommend to anyone already familiar with much of standard differential geometry, it would be \Chap~\ref{ch equivalence problems}. Equivalence problems provide a unified concept of all differential geometry and answer the question of \emph{what geometry is about}. Particular geometric structures, after all, matter only up to (appropriately defined class of) isomorphisms. The isomorphism-invariant data, called \emph{differential invariants}, are the ``physically measurable quantities'' that classify a geometric structure. It is exactly in this context that notoriously complicated concepts such as torsion and curvature naturally arise -- indeed, Gauss, Riemann and Christoffel first discovered them exactly by considering equivalence problems. This allows one to view the standard theory of connections as presented in \Chap~\ref{ch: connections} not as an imposition of \emph{additional} structure, but as an essential tool in the study of more fundamental geometric structures. \Chap~\ref{ch: cartan geom} then unifies all of these approaches by examining Cartan geometries via the common language of principal bundles and connections.
+
+Having said that, the book does not have to be read in the sequence in which it is written. For example, the reader can easily skip the entirety of \Part~\ref{Part Structured Geom I}, thus jumping from principal bundles immediately to connections, to approximate a more standard university telling of differential geometry. A chapter guide is provided to aid in the navigation.
 
 I hope that someone will find this material helpful. It is certainly a passion project for me, so I am eager to keep adding to it and correcting errors (which are, alas, undoubtedly numerous). Lastly, keep in mind that the contents are being updated regularly, many later sections are incomplete (marked by \ucmark\ in their title), and the last part is completely empty. Please share any typos you find via the GitHub page for this book:
 \begin{center}
     \url{https://github.com/abogatskiy/Geometry-Autistic-Intro}.
 \end{center}
-If you feel like something needs to be added to the book, you are welcome to message me with a request. Moreover, if you feel inspired to contribute to the book yourself, then you can fork the GitHub repo! Thank you for reading!
+If you feel like something needs to be added to the book, you are welcome to message me with a request. Moreover, if you feel inspired to contribute to the book yourself, then you can fork the GitHub repo! 
+
+Thank you for reading!
+