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content/Math401/Extending_thesis/Math401_R2.md
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@@ -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 <text style="color: red;"> homeomorphic</text> 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