Update

Signed-off-by: Marcello Seri <[email protected]>

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}
 

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