https://courses.pzhou.org/ 2026-04-17T10:01:23+00:00 https://courses.pzhou.org/ https://courses.pzhou.org/lib/tpl/dokuwiki/images/favicon.ico text/html 2026-02-21T14:41:14+00:00 Anonymous (anonymous@undisclosed.example.com) https://courses.pzhou.org/doku.php?id=math214:01-22&rev=1771684874&do=diff 2020-01-22, Wednesday Definition of topological manifold, examples, coordinate chart and smoothly compatible coordinate charts. Topological Manifold A topological manifold $M$ of dimension $n$ is a topological space such that * $M$ is a Hausdorff space, for every pair of distinct points $x, y \in M$, there are disjoint open subsets $U, V \subset M$ such that $x \in U, y \in V$. text/html 2026-02-21T14:41:13+00:00 Anonymous (anonymous@undisclosed.example.com) https://courses.pzhou.org/doku.php?id=math214:01-24&rev=1771684873&do=diff 2020-01-24, Friday $$ \gdef\In\subset $$ Smooth Structure Recall that an atlas for a topological manifold $M$ is a collection of coordinate charts $\{(U_\alpha, \varphi_\alpha)\}$ such that $M = \cup_\alpha U_\alpha$. And a smooth atlas is an atlas such that the transition functions between charts $$ g_{\alpha\beta}:= \varphi_\alpha \circ \varphi_\beta^{-1}: \varphi_\beta(U_\alpha \cap U_\beta) \to \varphi_\alpha(U_\alpha \cap U_\beta) $$ are diffeomorphism. text/html 2026-02-21T14:41:13+00:00 Anonymous (anonymous@undisclosed.example.com) https://courses.pzhou.org/doku.php?id=math214:01-27&rev=1771684873&do=diff 2020-01-27, Monday Today, we talk about the linear approximation of a smooth manifold at a point $p \in M$: the tangent space $T_p M$. There are several approaches to define the tangent space, one possible way is to define it in each coordinate patch, and then show the compatibilities in coordinate patch; another way is to characterize its action on a function by directional derivatives, and characterize the tangent space as such. text/html 2026-02-21T14:41:14+00:00 Anonymous (anonymous@undisclosed.example.com) https://courses.pzhou.org/doku.php?id=math214:01-29&rev=1771684874&do=diff 2020-01-29, Wednesday $$\gdef\d\partial, \gdef\t\tilde$$ We first introduce the notion of a tangent bundle. Next, we mention that a smooth map $f: M \to N$ induces a global differential $df: TM \to TN$. This finishes up Ch 3. Next, we introduce some more terminologies in Ch 4, Immersion and Submersion. text/html 2026-02-21T14:41:14+00:00 Anonymous (anonymous@undisclosed.example.com) https://courses.pzhou.org/doku.php?id=math214:01-31&rev=1771684874&do=diff 2020-01-31, Friday $$\gdef\d\partial$$ Today is our first encounter with a non-trivial theorem, the Sard theorem. Theorem(Sard)Given a smooth map $F: M \to N$, the set of critical values of $F$ is a measure zero set in $N$. We recall the definitions text/html 2026-02-21T14:41:14+00:00 Anonymous (anonymous@undisclosed.example.com) https://courses.pzhou.org/doku.php?id=math214:02-05&rev=1771684874&do=diff $$\gdef\Q{\mathbb Q} \gdef\P{\mathbb P} \gdef\RM{\backslash}$$ 2020-02-05 Wednesday We will do some basic definitions of submanifolds. Then we talk about Whitney Embedding Theorem Basic Terminology of Submanifolds (1) Let $M, N$ be two smooth manifolds. A text/html 2026-02-21T14:41:14+00:00 Anonymous (anonymous@undisclosed.example.com) https://courses.pzhou.org/doku.php?id=math214:02-07&rev=1771684874&do=diff 2020-02-07, Friday A vector field $X$ on a smooth manifold $M$, is an assignment to each $p \in M$ an element $X(p) \in T_p M$. We say $X$ is smooth, if for any smooth function $f \in C^\infty(M)$, the derivative $X(f)$ is also a smooth function. text/html 2026-02-21T14:41:13+00:00 Anonymous (anonymous@undisclosed.example.com) https://courses.pzhou.org/doku.php?id=math214:02-10&rev=1771684873&do=diff 2020-02-10, Monday $$\gdef\wt\widetilde \gdef\RM\backslash$$ Whitney Approximation Theorem. Thm Suppose $M$ is a smooth manifold, $F: M \to \R^k$ is a continuous map. $\delta: M \to \R$ is a positive function. Then, we can find a smooth function $\wt F: M \to \R^k$, such that $|F(x) - \wt F(x)| < \delta(x)$ for all $x \in M$. Furthermore, if $F$ is already smooth on a closed set $A$, we can choose $\wt F = F$ on $A$. text/html 2026-02-21T14:41:14+00:00 Anonymous (anonymous@undisclosed.example.com) https://courses.pzhou.org/doku.php?id=math214:02-12&rev=1771684874&do=diff 2020-02-12, Wednesday Tubular Neighborhood Theorem Let $M \subset \R^n$ be a embedded submanifold. Let $NM$ be the normal bundle of $M$, defined by $$ NM= \{ (x, v) \in T\R^n \mid x \in M, v \perp w, \forall w \in T_x M \}. $$ Thm: $NM$ is an embedded $n$-dimensional submanifold of $T\R^n$. Let $M_0 =\{(x,v) \in NM| v = 0\} \subset NM$ be the zero section. text/html 2026-02-21T14:41:14+00:00 Anonymous (anonymous@undisclosed.example.com) https://courses.pzhou.org/doku.php?id=math214:02-14&rev=1771684874&do=diff 2020-02-14, Friday Vector Bundle Definition : A vector bundle is a quadruple $(E, \pi, M ,F)$, such that * $E, M$ are smooth manifolds * $\pi: E \to M$ is a surjective submersion. For each $U \subset M$, we set $E|_U := \pi^{-1}(U)$. * $F$ is a vector space of rank $n$ over $\R$. text/html 2026-02-21T14:41:14+00:00 Anonymous (anonymous@undisclosed.example.com) https://courses.pzhou.org/doku.php?id=math214:02-19&rev=1771684874&do=diff 2020-02-19, Wednesday Vector Bundle Definition : A vector bundle is a quadruple $(E, \pi, M ,F)$, such that * $E, M$ are smooth manifolds * $\pi: E \to M$ is a surjective submersion. For each $U \subset M$, we set $E|_U := \pi^{-1}(U)$. * text/html 2026-02-21T14:41:14+00:00 Anonymous (anonymous@undisclosed.example.com) https://courses.pzhou.org/doku.php?id=math214:02-21&rev=1771684874&do=diff 2020-02-21, Friday $$\gdef\ot\otimes$$ Tensor power of a vector space Let $V$ be a finite dimensional vector space. We denote the $k$-th tensor power of $V$ as $$ V^{\otimes k} = \underbrace{V\ot \cdots \ot V}_{\text{ $k$ times} } $$ Its elements are linear combinations of terms like $v_1 \otimes \cdots \ot v_k$, subject to the usual linearity relations. text/html 2026-02-21T14:41:14+00:00 Anonymous (anonymous@undisclosed.example.com) https://courses.pzhou.org/doku.php?id=math214:02-28&rev=1771684874&do=diff 2020-02-28, Friday It does not do much justice to give Lie derivative just one day, but it is nice to first meet it then slowly get famliar with it. $$\gdef\cD{\mathcal D} \gdef\cL{\mathcal L}$$ Let $M$ be a smooth manifold, $X$ be a vector field, $\Phi: \cD \to M$ is the flow, where $\cD \subset M \times \R$ is the open subset of the flow domain. For simplicity, assume $M$ is compact, and then $\cD = M \times \R$. text/html 2026-02-21T14:41:14+00:00 Anonymous (anonymous@undisclosed.example.com) https://courses.pzhou.org/doku.php?id=math214:03-04&rev=1771684874&do=diff 2020-03-04, Wednesday $$\gdef\T{\mathbb T}$$ Definitions A Lie group $G$ is a smooth manifold that is also a group in the algebraic sense, such that the multiplication map $m: G \times G \to G$ and the inverse $i: G \to G$ are all smooth maps. Let $g \in G$, we define the left translation $L_g$ and right translation $R_g$ as maps $G \to G$ by $$ L_g(h) = gh, \quad R_g(h) = hg $$ text/html 2026-02-21T14:41:14+00:00 Anonymous (anonymous@undisclosed.example.com) https://courses.pzhou.org/doku.php?id=math214:04-01&rev=1771684874&do=diff $$ \gdef\vect{\text{Vect}} \gdef\lcal{\mathcal L} \gdef\End{\text{End}} \gdef\Hom{\text{Hom}}$$ 2020-04-01, Wednesday Today and Friday, we will follow Nicolascu's note 3.3, and discuss connection on vector bundle. Why we need something called 'connection'? text/html 2026-02-21T14:41:14+00:00 Anonymous (anonymous@undisclosed.example.com) https://courses.pzhou.org/doku.php?id=math214:04-03&rev=1771684874&do=diff 2020-04-03, Friday $$\gdef\End{\text{ End}}$$ Parallel Transport. Let $\gamma: [0,1] \to M$ be an embedded smooth curve. (If you worry about the boundary, think of an embedded curve $(-\epsilon, 1+\epsilon) \to M$.) Let $E \to M$ be a vector bundle, $\nabla$ be a connection. Our goal is to define the isomorphism $$P_\gamma: E_{\gamma(0)} \to E_{\gamma(1)}. $$ Suppose $u_0 \in E_{\gamma(0)}$, we want to find a section $u_t \in E_{\gamma(t)}$, such that $$ \nabla_{\dot \gamma(t)} u_t = 0. $$ … text/html 2026-02-21T14:41:14+00:00 Anonymous (anonymous@undisclosed.example.com) https://courses.pzhou.org/doku.php?id=math214:04-06&rev=1771684874&do=diff 2020-04-06, Monday $$\gdef\xto\xrightarrow \gdef\End{\z{ End}}$$ Summary of Connection and Curvature Let $E$ be a vector bundle over a smooth manifold. The connection $\nabla$ is given as such $$ \Omega^0(M, E) \xto{ \nabla} \Omega^1(M, E) \xto{ \nabla} \Omega^2(M, E) \cdots. $$ And the curvature is defined succinctly as $$ F_\nabla = \nabla^2 $$ which is a map $\Omega^k(M,E) \to \Omega^{k+2}(M,E)$ that commute with $C^\infty(M)$ action, hence $F_\nabla \in \Omega^2(M, \End(E)).$ text/html 2026-02-21T14:41:14+00:00 Anonymous (anonymous@undisclosed.example.com) https://courses.pzhou.org/doku.php?id=math214:04-15&rev=1771684874&do=diff 2020-04-15, Wednesday Cartan's Moving Frame Pick any Orthonormal Frame $X_\alpha$ of $TM$, and choose its dual frame $\theta^\alpha$ of $T^*M$. Introduce a collection of 1-forms using covariant derivatives $$ \nabla X_\alpha = \omega_\alpha^\beta X_\beta $$ text/html 2026-02-21T14:41:14+00:00 Anonymous (anonymous@undisclosed.example.com) https://courses.pzhou.org/doku.php?id=math214:final-solution&rev=1771684874&do=diff Final Solution $$\gdef\gfrak{\mathfrak g}$$ First, I have to admit that I stealed some problems from Prof Valentino Tosatti's final problems, with some modifications (problem 1,2,3). Here is his course website and the final with solution. I will write not below the most concise solution, but some remark and explanations along the way. text/html 2026-02-21T14:41:13+00:00 Anonymous (anonymous@undisclosed.example.com) https://courses.pzhou.org/doku.php?id=math214:final&rev=1771684873&do=diff Final $$\gdef\gfrak{\mathfrak g}$$ ---------- Due Date: May 10th (Sunday) 11:59PM. Submit online to gradescope. Policy : You can use textbook and your notes. There should be no discussion or collaborations, since this is suppose to be a exam on your own understanding. If you found some question that is unclear, please let me know via email. text/html 2026-02-21T14:41:14+00:00 Anonymous (anonymous@undisclosed.example.com) https://courses.pzhou.org/doku.php?id=math214:home&rev=1771684874&do=diff Math 214: Differentiable manifolds UC Berkeley, Spring 2020 Lecture: MWF 11-12, at 9 Lewis Hall Zoom Meeting: <https://berkeley.zoom.us/j/515298392> Zoom Office hour: Enter Office Wednesday 2-4pm, Thursday 1-2pm, at PMI number: 881-910-2324. Please also join the zoom channel, “Math 214”. If I am not in 'office', you can leave a message there. text/html 2026-02-21T14:41:13+00:00 Anonymous (anonymous@undisclosed.example.com) https://courses.pzhou.org/doku.php?id=math214:hw8&rev=1771684873&do=diff Homework 8 Ch 7: 13, Ch 8: 19,22,28,31 7.13 For each $n \geq 􏰆 1$, prove that $U(n)$ is a properly embedded $n^2$-dimensional Lie subgroup of $GL(n, \C)$. We follow the route of proof in Example 7.27 (page 166-167 of Lee). First, we construct a map from $U(n)$ to $M(n, \C)$, that maps $$\Phi: GL(n, \C) \to M(n, \C), \quad A \mapsto A^*A. $$ Then, $GL(n,\C)$ acts on the right of $M(n, \C)$ by conjugation, $(g, M) \mapsto g^{*} M g$. This is a right action of $U(n)$ on $M$. Then, by equiva… text/html 2026-02-21T14:41:14+00:00 Anonymous (anonymous@undisclosed.example.com) https://courses.pzhou.org/doku.php?id=math214:hw9-sol&rev=1771684874&do=diff HW 9 1. Show that if $a: G \times M \to M$ is proper, then $\theta= (\pi_M, a): G \times M \to M \times M $ is proper. But converse is not true. Take a compact set $K \In M \times M$, then $\theta^{-1}(K) \in a^{-1}( \pi_2(K))$, where $\pi_2: M \times M \to M$ is the projection to the second factor. Since $\pi_2$ is continuous, $\pi_2(K)$ is compact; since $a$ is proper, $a^{-1}(\pi_2(K))$ is compact. $\theta^{-1}(K)$ is closed subset of a compact set, hence is compact. text/html 2026-02-21T14:41:14+00:00 Anonymous (anonymous@undisclosed.example.com) https://courses.pzhou.org/doku.php?id=math214:hw10&rev=1771684874&do=diff Problem 4 $$ \gdef\End{\text{End}}$$ Let $M=\R^2$ and $L$ be the trivial line bundle on $M$. We identify sections of $L$ with smooth function on $M$. Let $$ \nabla = d + A$$, where $d$ is the trivial connection on $L$ and $A$ is the connection 1-form in $\Omega^1(M, \End(L)) = \Omega^1(M)$: $$ A = x dy - y d x $$ Let point $a=(1,0)$, $b=(-1,0)$, and $\gamma_\pm$ be path from $a$ to $b$, going along upper (or lower) semicircle: $$ \gamma_\pm: [0,\pi] \to \R^2, \quad t \mapsto (\cos t, \pm \s… text/html 2026-02-21T14:41:14+00:00 Anonymous (anonymous@undisclosed.example.com) https://courses.pzhou.org/doku.php?id=math214:hw11-hint&rev=1771684874&do=diff Hint 1. One can find a cone in $\R^3$ that is tangent to the sphere at the curve, or do the computation using the pullback metric tensor. The following might be helpful: * if $A$ is square matrix, to compute $\exp(A)$, we better do eigenvalue decomposition $A = U D U^{-1}$ where $D$ is diagonal,then $e^A = U e^D U^{-1}$ (proof by Taylor expansion). text/html 2026-02-21T14:41:14+00:00 Anonymous (anonymous@undisclosed.example.com) https://courses.pzhou.org/doku.php?id=math214:hw11&rev=1771684874&do=diff Homework 11 This week we studied curvature and connections, in particular the Levi-Cevita connection on the tangent bundle. It is important to do some calculation to see that our intuition agrees with the formula and calculationd. Many nice things happens for Lie groups text/html 2026-02-21T14:41:14+00:00 Anonymous (anonymous@undisclosed.example.com) https://courses.pzhou.org/doku.php?id=math214:hw12&rev=1771684874&do=diff Homework 12 1. (Geodesics are length extremizing). Recall the following facts * Consider the space $\R^2$. If we equip $\R^2$ with the flat metric, then it is well-known that every of two points has a unique geodesic connecting them. * Consider the sphere $S^2$, if we equip $S^2$ with the usual round metric, then every pair of two points (not antipodal pairs) has exactly two geodesics connecting them. text/html 2026-02-21T14:41:14+00:00 Anonymous (anonymous@undisclosed.example.com) https://courses.pzhou.org/doku.php?id=math214:hw13&rev=1771684874&do=diff Homework 13 This is our last homework. In this week, we discussed the variational approach of geodesics, mentioning the first and second variation formula, and then Jacobi field equation. 1. (2pt) Jacobi Equation in a nice coordinate. Let $\gamma: [0,1] \to M$ be a geodesic. Let $e_1, \cdots, e_n$ be parallel, orthonormal tangent vectors along $\gamma$, i.e. $e_i(t) \in T_{\gamma(t)} M$ and $\nabla_{\dot \gamma(t)} e_i(t)=0$, and $\la e_i(0), e_j(0) \ra = \delta_{ij}$ (enforced at one time,… text/html 2026-02-21T14:41:14+00:00 Anonymous (anonymous@undisclosed.example.com) https://courses.pzhou.org/doku.php?id=math214:hwsol&rev=1771684874&do=diff Students Homework Solutions If you would also like to share your solution, please let me know and I can upload your as well. From Mason Thanks to Mason Haberle who kindly shared with us his solutions. * * * * * * From Yulong text/html 2026-02-21T14:41:14+00:00 Anonymous (anonymous@undisclosed.example.com) https://courses.pzhou.org/doku.php?id=math214:latex_template&rev=1771684874&do=diff Output is like [this] Source file is like the following. Just go to Overleaf, create an account, create and empty project, and copy the following into it. \documentclass{article} \title{HW7 for 214} \author{John Doe} \usepackage{mathrsfs, amssymb,amsmath,amsthm,amsfonts, mathtools, tikz-cd, tikz} \def\R{\mathbb R} \def\C{\mathbb C} \def\acal{\mathcal A} \def\gfrak{\mathfrak g} \newtheorem{theo}{Theorem}[section] \newtheorem{lemma}[theo]{Lemma} \newtheorem{problem}[theo]{Problem} \newth… text/html 2026-02-21T14:41:14+00:00 Anonymous (anonymous@undisclosed.example.com) https://courses.pzhou.org/doku.php?id=math214:note&rev=1771684874&do=diff 2020-01-24, Friday $$ \gdef\In\subset $$ Smooth Structure Recall that an atlas for a topological manifold $M$ is a collection of coordinate charts $\{(U_\alpha, \varphi_\alpha)\}$ such that $M = \cup_\alpha U_\alpha$. And a smooth atlas is an atlas such that the transition functions between charts $$ g_{\alpha\beta}:= \varphi_\alpha \circ \varphi_\beta^{-1}: \varphi_\beta(U_\alpha \cap U_\beta) \to \varphi_\alpha(U_\alpha \cap U_\beta) $$ are diffeomorphism. text/html 2026-02-21T14:41:14+00:00 Anonymous (anonymous@undisclosed.example.com) https://courses.pzhou.org/doku.php?id=math214:start&rev=1771684874&do=diff Math 214: Differentiable manifolds UC Berkeley, Spring 2020 Lecture: MWF 11-12, at 9 Lewis Hall Zoom Meeting: <https://berkeley.zoom.us/j/515298392> Zoom Office hour: Enter Office Wednesday 2-4pm, Thursday 1-2pm, at PMI number: 881-910-2324. Please also join the zoom channel, “Math 214”. If I am not in 'office', you can leave a message there.