From e9071f7b3fd276a19e30ddbd9d6df36bb42af904 Mon Sep 17 00:00:00 2001 From: Marcello Seri Date: Thu, 30 Nov 2023 14:53:54 +0100 Subject: [PATCH 1/3] Update Signed-off-by: Marcello Seri --- 1-manifolds.tex | 2 +- aom.tex | 2 +- 2 files changed, 2 insertions(+), 2 deletions(-) diff --git a/1-manifolds.tex b/1-manifolds.tex index a8ea832..2770869 100644 --- a/1-manifolds.tex +++ b/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} diff --git a/aom.tex b/aom.tex index d99c3ed..335d8a3 100644 --- a/aom.tex +++ b/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 From 6172500a2b66d0b759dca89aa63b2466f3e352d8 Mon Sep 17 00:00:00 2001 From: Marcello Seri Date: Tue, 5 Dec 2023 17:24:53 +0100 Subject: [PATCH 2/3] Small typos fixed Signed-off-by: Marcello Seri --- 2b-submanifolds.tex | 12 +++++++++--- 1 file changed, 9 insertions(+), 3 deletions(-) diff --git a/2b-submanifolds.tex b/2b-submanifolds.tex index a279520..4d176cd 100644 --- a/2b-submanifolds.tex +++ b/2b-submanifolds.tex @@ -103,7 +103,7 @@ \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, @@ -111,7 +111,7 @@ \section{Inverse function theorem} 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? From 9b1384520f8b22c649aee8b5bcf54e9503b79066 Mon Sep 17 00:00:00 2001 From: Marcello Seri Date: Tue, 5 Dec 2023 17:34:30 +0100 Subject: [PATCH 3/3] Fix typo Signed-off-by: Marcello Seri --- 2-tangentbdl.tex | 2 +- 1 file changed, 1 insertion(+), 1 deletion(-) diff --git a/2-tangentbdl.tex b/2-tangentbdl.tex index f59894f..3f5e8c3 100644 --- a/2-tangentbdl.tex +++ b/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 $v_1 \neq v_2$.