From 60173cbc64590111e71969f9dc9c0c8485554560 Mon Sep 17 00:00:00 2001 From: Trance-0 <60459821+Trance-0@users.noreply.github.com> Date: Wed, 17 Sep 2025 11:53:19 -0500 Subject: [PATCH] updates --- .../Math401/Extending_thesis/Math401_R2.md | 39 +++--- content/Math4201/Math4201_L10.md | 124 ++++++++++++++++++ 2 files changed, 144 insertions(+), 19 deletions(-) create mode 100644 content/Math4201/Math4201_L10.md diff --git a/content/Math401/Extending_thesis/Math401_R2.md b/content/Math401/Extending_thesis/Math401_R2.md index fd47f3c..efe7301 100644 --- a/content/Math401/Extending_thesis/Math401_R2.md +++ b/content/Math401/Extending_thesis/Math401_R2.md @@ -252,19 +252,25 @@ $$ Not very edible for undergraduates. -## Crash course on Riemannian Geometry +## Riemannian manifolds and geometry > This section is designed for stupids like me skipping too much essential materials in the book. +> This part might be extended to a separate note, let's check how far we can go from this part. +> +> References: +> +> - [Riemannian Geometry by John M. Lee](https://www.amazon.com/Introduction-Riemannian-Manifolds-Graduate-Mathematics/dp/3319917544?dib=eyJ2IjoiMSJ9.88u0uIXulwPpi3IjFn9EdOviJvyuse9V5K5wZxQEd6Rto5sCIowzEJSstE0JtQDW.QeajvjQEbsDmnEMfPzaKrfVR9F5BtWE8wFscYjCAR24&dib_tag=se&keywords=riemannian+manifold+by+john+m+lee&qid=1753238983&sr=8-1) + ### Manifold -Unexpectedly, a good definition of the manifold is defined in the topology I. - -Check section 36. This topic extends to a wonderful chapter 8 in the book where you can hardly understand chapter 2. +> Unexpectedly, a good definition of the manifold is defined in the topology I. +> +> Check section 36. This topic extends to a wonderful chapter 8 in the book where you can hardly understand chapter 2. #### Definition of m-manifold -An $m$-manifold is a Hausdorff space $X$ with a countable basis such that each point of $x$ of $X$ has a neighborhood homeomorphic to an open subset of $\mathbb{R}^m$. +An $m$-manifold is a [Hausdorff space](../../Math4201/Math4201_L9#hausdorff-space) $X$ with a countable basis such that each point of $x$ of $X$ has a neighborhood [homeomorphic](../../Math4201/Math4201_L10#definition-of-homeomorphism) to an open subset of $\mathbb{R}^m$. Example is trivial that 1-manifold is a curve and 2-manifold is a surface. @@ -274,17 +280,9 @@ If $X$ is a compact $m$-manifold, then $X$ can be imbedded in $\mathbb{R}^n$ for This theorem might save you from imagining abstract structures back to real dimension. Good news, at least you stay in some real numbers. -### Riemannian manifold +### Smooth manifold - - -## Crash course on Riemannian manifolds - -> This part might be extended to a separate note, let's check how far we can go from this part. -> -> References: -> -> - [Riemannian Geometry by John M. Lee](https://www.amazon.com/Introduction-Riemannian-Manifolds-Graduate-Mathematics/dp/3319917544?dib=eyJ2IjoiMSJ9.88u0uIXulwPpi3IjFn9EdOviJvyuse9V5K5wZxQEd6Rto5sCIowzEJSstE0JtQDW.QeajvjQEbsDmnEMfPzaKrfVR9F5BtWE8wFscYjCAR24&dib_tag=se&keywords=riemannian+manifold+by+john+m+lee&qid=1753238983&sr=8-1) +> This section is waiting for the completion of book Introduction to Smooth Manifolds by John M. Lee. ### Riemannian manifolds @@ -296,7 +294,7 @@ An example of Riemannian manifold is the sphere $\mathbb{C}P^n$. A Riemannian metric is a smooth assignment of an inner product to each tangent space $T_pM$ of the manifold. -An example of Riemannian metric is the Euclidean metric on $\mathbb{R}^n$. +An example of Riemannian metric is the Euclidean metric, the bilinear form of $d(p,q)=\|p-q\|_2$ on $\mathbb{R}^n$. ### Notion of Connection @@ -308,9 +306,12 @@ $$ D_VX=\lim_{h\to 0}\frac{X(p+h)-X(p)}{h} $$ -### Nabla notation and Levi-Civita connection +### Notion of Curvatures +> [!NOTE] +> +> Geometrically, the curvature of the manifold is radius of the tangent sphere of the manifold. -### Ricci curvature - +#### Nabla notation and Levi-Civita connection +#### Ricci curvature diff --git a/content/Math4201/Math4201_L10.md b/content/Math4201/Math4201_L10.md new file mode 100644 index 0000000..4d03fdd --- /dev/null +++ b/content/Math4201/Math4201_L10.md @@ -0,0 +1,124 @@ +# Math4201 Lecture 10 + +## Continuity + +### Continuous functions + +Let $X,Y$ be topological spaces and $f:X\to Y$. For any $x\in X$ and any open neighborhood $V$ of $f(x)$ in $Y$, $f^{-1}(V)$ contains an open neighborhood of $x$ in $X$. + + +#### Lemma for continuous functions + +Let $f:X\to Y$ be a function, then: + +1. $A\subseteq Y$: $f^{-1}(A^c) = (f^{-1}(A))^c$. +2. $\{A_\alpha\}_{\alpha\in I}\subseteq Y$: $f^{-1}(\bigcup_{\alpha\in I} A_\alpha) = \bigcup_{\alpha\in I} f^{-1}(A_\alpha)$. +3. $\{A_\alpha\}_{\alpha\in I}\subseteq Y$: $f^{-1}(\bigcap_{\alpha\in I} A_\alpha) = \bigcap_{\alpha\in I} f^{-1}(A_\alpha)$. + +
+Proof + +1. By definition of continuous functions, $\forall V$ open in $Y$, $f^{-1}(V)$ is open in $X$. + +2. It is sufficient to shoa that $x\in f^{-1}(\bigcup_{\alpha\in I} A_\alpha)$ if and only if $x\in \bigcup_{\alpha\in I} f^{-1}(A_\alpha)$. + +This condition holds if and only if $\exists \alpha\in I$ such that $f(x)\in A_\alpha$. + +Which is equivalent to $\exists \alpha\in I$ such that $x\in f^{-1}(A_\alpha)$. + +So $x\in f^{-1}(\bigcup_{\alpha\in I} A_\alpha)$ + +In particular, $f^{-1}(\bigcup_{\alpha\in I} A_\alpha) = \bigcup_{\alpha\in I} f^{-1}(A_\alpha)$. + +3. Similar to 2 but use forall. + +
+ +#### Properties of continuous functions + +A function $f:X\to Y$ is continuous if and only if: + +1. $f^{-1}(V)$ is open in $X$ for any open set $V\subset Y$. +2. $f$ is continuous at any point $x\in X$. +3. $f^{-1}(C)$ is closed in $X$ for any closed set $C\subset Y$. +4. Assume $\mathcal{B}$ is a basis for $Y$, then $f^{-1}(\mathcal{B})$ is open in $X$ for any $B\in \mathcal{B}$. +5. For any $A\subseteq X$, $f(\overline{A})\subseteq \overline{f(A)}$. + +
+Proof + +**Showing $1\iff 3$**: + +> Use the lemma for continuous functions (1) + +**Showing $1\iff 4$**: + +$1 \implies 4$: + +Because any $B\in \mathcal{B}$ is open in $Y$, so $f^{-1}(B)$ is open in $X$. + +$4 \implies 1$: + +Let $V\subset Y$ be an open set. Then there are basis elements $\{B_\alpha\}_{\alpha\in I}$ such that $V=\bigcup_{\alpha\in I} B_\alpha$. + +So $f^{-1}(V) = f^{-1}(\bigcup_{\alpha\in I} B_\alpha) = \bigcup_{\alpha\in I} f^{-1}(B_\alpha)$ (by lemma (2)) is a union of open sets, so $f^{-1}(V)$ is open in $X$. + +**Showing $1\implies 5$**: + +Take $A\subseteq X$ and $x\in \overline{A}$. It suffices to show $f(x)$ is an element of the closure of $f(A)$. This is equivalent to say that any open neighborhood $V$ of $f(x)$ intersects $f(A)$ has a non-trivial intersection with $f(A)$. + +For any such $V$, 1 implies that $f^{-1}(V)$ is open in $X$. Moreover, $x\in f^{-1}(V)$ because $f(x)\in V$. + +This means that $f^{-1}(V)$ is an open neighborhood of $x$. Since $x\in \overline{A}$, we have $f^{-1}(V)\cap A\neq \emptyset$ and contains a point $x'\in X$. + +So $x'\in f^{-1}(V)\cap A$, this implies that $f(x')\in V$ and $f(x')\in f(A)$, so $f(x')\in V\cap f(A)$. + +> [!NOTE] +> +> This verifies our claim. Proof of $5\implies 1$ is similar and left as an exercise. + +
+ +
+Example of property 5 + +Let $X=(0,1)\cup (1,2)$ and $Y=\mathbb{R}$ equipped with the subspace topology induced by the standard topology on $\mathbb{R}$. + +Let $f:X\to Y$ be the inclusion map, $f(x)=x$ for all $x\in X$. This is continuous. + +Let $A=(0,1)\cup (1,2)$. Then $\overline{A}=A$. So $f(\overline{A})=f(A)=(0,1)\cup (1,2)$. + +However, $\overline{f(A)}=\overline{(0,1)\cup (1,2)}=[0,2]$. + +So $f(\overline{A})\subsetneq \overline{f(A)}$. + +
+ +#### Definition of homeomorphism + +A **homeomorphism** $f:X\to Y$ is a continuous map of topological spaces that is a bijection and $f^{-1}:Y\to X$ is also continuous. + +
+Example of homeomorphism + +Let $X=\mathbb{R}$ and $Y=\mathbb{R}+$ with standard topology. + +$f:\mathbb{R}\to \mathbb{R}^+$ be defined by $f(x)=e^x$ is continuous and bijective. + +$f^{-1}:\mathbb{R}^+\to \mathbb{R}$ be defined by $f^{-1}(y)=\ln(y)$ is continuous and homeomorphism. + +
+ +### Epsilon delta definition of continuity + +Let $f:\mathbb{R}\to \mathbb{R}$ be a continuous function where we use the standard topology on $\mathbb{R}$. + +Then [property 4](#properties-of-continuous-functions) implies that for any open interval $(a,b)\in \mathbb{R}$, $f^{-1}((a,b))$ is open in $\mathbb{R}$. + +Now take an arbitrary $x\in \mathbb{R}$ and $\epsilon > 0$. In particular $f^{-1}((f(x)-\epsilon, f(x)+\epsilon))$ is an open set containing $x$. + +In particular, there is an open interval (by the standard topology on $\mathbb{R}$) $(c,d)$ such that $x\in (c,d)\subseteq f^{-1}((f(x)-\epsilon, f(x)+\epsilon))$. + +Let $\delta = \min\{x-c, d-x\}$. Then $(x-\delta, x+\delta)\subseteq (c,d)\subseteq f^{-1}((f(x)-\epsilon, f(x)+\epsilon))$. + +This says that if $|y-x| < \delta$, then $|f(y)-f(x)| < \epsilon$. \ No newline at end of file