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date: 2022-01-15 21:49
modification date: Wednesday 23rd February 2022 18:09:21
title: riemannian manifold
aliases: [riemannian manifold]

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	- #todo/untagged
- Refs:
	- #todo/add-references
- Links:
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# riemannian manifold

# Notes

- Creating an interesting Riemannian manifold: let $G\in\Lie\Grp^\ss$ with $K\leq G$ its maximal compact subgroup.
	The Riemannian structure comes from an invariant metric on $G$, and so $G\to \Isom(G/K)$ via left translation and $G\actson G/K$.
	Take $\Gamma \leq G$  and form the double quotient $\dcoset{\Gamma}{G}{K}$. These are some of the most celebrated manifolds in mathematics.
	- Take $G=\SL_2(\RR)$ to get [uniformization](uniformization.md): every $\Sigma_g$ for $g\geq 2$ can be described this way.
		- $\SL_2(\RR)/\SO_2 \cong \HH$
	- Take $G = \SL_2(\CC)$ to get [hyperbolic 3-manifolds](Unsorted/hyperbolic%203-manifold.md).
	- For [songruence subgroups](Unsorted/songruence%20subgroups.md), this is the subject of [Langlands](Unsorted/Langlands.md) and