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partial updates, continue on Riemannian manifold

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2025-09-16 22:19:33 -05:00
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content/Math401/Extending_thesis/Math401_R2.md
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Not very edible for undergraduates.
## Crash course on Riemannian Geometry
> This section is designed for stupids like me skipping too much essential materials in the book.
### 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.
#### 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 $\mathbb{R}^m$.
Example is trivial that 1-manifold is a curve and 2-manifold is a surface.
#### Theorem of imbedded space
If $X$ is a compact $m$-manifold, then $X$ can be imbedded in $\mathbb{R}^n$ for some $n$.
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
## 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.

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content/Math401/Extending_thesis/Math401_S2.md
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# Math 401, Fall 2025: Thesis notes, S2, Majorana stellar representation of quantum states.
## Majorana representation of quantum states
## Majorana stellar representation of quantum states
> [!TIP]
>
@@ -26,6 +26,6 @@ Using stereographic projection of each root we can get a unordered collection of
> [!NOTE]
>
> TODO: Check more definition from different area of mathematics (algebraic geometry, complex analysis, etc.) of the Majorana representation of quantum states.
> TODO: Check more definition from different area of mathematics (algebraic geometry, complex analysis, etc.) of the Majorana stellar representation of quantum states.
>
> Read Chapter 5 and 6 of [Geometry of Quantum states](https://www.cambridge.org/core/books/geometry-of-quantum-states/46B62FE3F9DA6E0B4EDDAE653F61ED8C) for more details.