Merge branch 'master' of github.com:mseri/AoM

1-manifolds.tex

@@ -678,7 +678,7 @@ \section{Smooth maps and differentiability}\label{sec:smoothfn}
   Show that a map $F:M_1\to M_2$ is smooth if and only if either of the following conditions holds:
   \begin{enumerate}
     \item for every $p\in M_1$, there are smooth charts $(V_1,\phi_1)$, $p\in V_1$, and $(V_2,\phi_1)$, $F(p) \in V_2$, such that $F(V_1) \subseteq V_2$ and $\phi_2 \circ F \circ \phi_1^{-1}$ is smooth from $\phi_1(V_1)$ to $\phi_2(V_2)$;
-    \item $F$ is continuous and there exists two smooth atlases $\{(V^1_\alpha, \phi^1_\alpha)\}$ and $\{(V^2_\beta, \phi^2_\beta)\}$, respectively for $M_1$ and $M_2$, such that for each $\alpha$ and $\beta$, $\phi^2_\beta \circ F \circ (\phi^1_\alpha)^{-1}$ is a smooth maph from $\phi^1_\alpha(V^1_\alpha \cap F(V_\beta^2))$ to $\phi^2_\beta(V^2_\beta)$.
+    \item $F$ is continuous and there exists two smooth atlases $\{(V^1_\alpha, \phi^1_\alpha)\}$ and $\{(V^2_\beta, \phi^2_\beta)\}$, respectively for $M_1$ and $M_2$, such that for each $\alpha$ and $\beta$, $\phi^2_\beta \circ F \circ (\phi^1_\alpha)^{-1}$ is a smooth map from $\phi^1_\alpha(V^1_\alpha \cap F(V_\beta^2))$ to $\phi^2_\beta(V^2_\beta)$.
   \end{enumerate}
 \end{exercise}
 

2-tangentbdl.tex

@@ -837,7 +837,7 @@ \section{The tangent bundle}\label{sec:tangentbundle}
 
   \newthought{Step 3: $TM$ is a manifold.}
   With the procedure delineated above, a countable smooth atlas $\{(U_i, \varphi_i)\}$ of $M$ induces a countable atlas $\{(\pi^{-1}(U_i), \widetilde\varphi_i)\}$ of $TM$.
-  First of all, $\{(\pi^{-1}(U_i)\}$ provides a countable covering of $TM$.
+  First of all, $\{\pi^{-1}(U_i)\}$ provides a countable covering of $TM$.
   We need to show that the topology induced by those charts\footnote{Given a family of functions $\cF$ from the same set $X$ into (possibly different) topological spaces, the topology $\cT_{\cF}$ induced by the functions in $\cF$ is the smallest topology such that all the functions are continuous. It is possible to show that such a topology exists and it has as a basis the set $\{V\subset X \,\mid\, \exists n\in\N, f_i\in\cF, U_i \mbox{ open} : \bigcap_{i=1}^n f_i^{-1}(U_i) \}$.} is Hausdorff and second countable.
 
   Let $(p_1, v_1), (p_2, v_2) \in TM$ be different points: either $p_1\neq p_2$, or $p_1 = p_2$ and $v_1 \neq v_2$.

2b-submanifolds.tex

@@ -103,15 +103,15 @@ \section{Inverse function theorem}
   our choice of coordinates after the above observations implies that
   $\det \left( \frac{\partial Q^i}{\partial x^j} \right) \neq 0$ at $(x,y) = (0,0)$.
 
-  Since the gradient of $Q$ with respect to $z$ is regular, we are going to extend
+  Since the gradient of $Q$ with respect to $x$ is regular, we are going to extend
   the mapping with the identity on the rest of the coordinates to get a regular map
   on the whole neighbourhood.
   Let $\varphi : U \to \R^m$ be defined by $\varphi(x,y) = (Q(x,y), y)$. Then,
   \begin{equation}
     D\varphi(0,0) = 
     \begin{pmatrix}
       \frac{\partial Q^i}{\partial x^j}(0,0) & \frac{\partial Q^i}{\partial y^j}(0,0) \\
-      0 & \id_{\R^k}
+      0 & \id_{\R^{m-k}}
     \end{pmatrix}
   \end{equation}
   has nonvanishing determinant by hypothesis.
@@ -185,6 +185,12 @@ \section{Inverse function theorem}
   concluding the proof.
 \end{proof}
 
+\begin{exercise}
+  Formulate and prove a version of the Rank theorem for a map $F : M^m \to N^n$ of constant rank $k$,
+  where $M$ is a smooth manifold with boundary, $N$ is a smooth manifold without boundary
+  and $\ker dF_p \not\subseteq T_p\partial M$.
+\end{exercise}
+
 \section{Embeddings, submersions and immersions}
 
 Looking at the statement of the Rank Theorem, one can already see that there can be different possibilities depending on the relation between, $m$, $n$ and $k$. This warrants a definition.
@@ -261,7 +267,7 @@ \section{Embeddings, submersions and immersions}
 In the rest of this chapter we will try to give an answer to the following questions:
 \begin{itemize}
   \item if $F$ is an immersion, what can we say about its image $F(M)$ as a subset of $N$?
-  \item if $F$ is a submersion, what can we say about its levelsets $f^{-1}(q) \subset M$?
+  \item if $F$ is a submersion, what can we say about its levelsets $F^{-1}(q) \subset M$?
 \end{itemize}
 And what can we say about the corresponding tangent spaces?
 

aom.tex

@@ -275,7 +275,7 @@ \chapter*{Introduction}
 
 I am extremely grateful to Martijn Kluitenberg for his careful reading of the notes and his invaluable suggestions, comments and corrections, and to Bram Brongers\footnote{You can also have a look at \href{https://fse.studenttheses.ub.rug.nl/25344/}{his bachelor thesis} to learn more about some interesting advanced topics in differential geometry.} for his comments, corrections and the appendices that he contributed to these notes.
 
-Many thanks also to the following people for their comments and for reporting a number of misprints and corrections: Wojtek Anyszka, Bhavya Bhikha, Huub Bouwkamp, Daniel Cortlid, Harry Crane, Anna de Bruijn, Luuk de Ridder, Mollie Jagoe Brown, Wietze Koops, Henrieke Krijgsheld, Levi Moes, Nicol\'as Moro, Magnus Petz, Lisanne Sibma, Bo Tielman, Jesse van der Zeijden, Jordan van Ekelenburg, Hanneke van Harten, Martin Daan van IJcken, Marit van Straaten, Dave Verweg and Federico Zadra.
+Many thanks also to the following people for their comments and for reporting a number of misprints and corrections: Wojtek Anyszka, Bhavya Bhikha, Huub Bouwkamp, Daniel Cortlid, Harry Crane, Anna de Bruijn, Luuk de Ridder, Mollie Jagoe Brown, Wietze Koops, Henrieke Krijgsheld, Levi Moes, Nicol\'as Moro, Magnus Petz, Jorian Pruim, Lisanne Sibma, Bo Tielman, Jesse van der Zeijden, Jordan van Ekelenburg, Hanneke van Harten, Martin Daan van IJcken, Marit van Straaten, Dave Verweg and Federico Zadra.
 
 \mainmatter