# Tagged Questions

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### What is a manifold with cusp?

What I am primarily currently learning about is hyperbolic geometry and methods to find hyperbolic structures on triangulated manifolds. I see phrases such as 'cusp ends' and 'manifold with one cusp' ...
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### Fundamental solution to Laplace equation on arbitrary Riemann surfaces

So, I've seen in a few places this method of calculating the heat kernel on a manifold given the kernel of its universal cover, through a so-called 'tiling method' as in section five of this paper ...
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### JSJ-decomposition of a non-hyperbolic 3-manifold

Suppose $M$ is a $3$-manifold. Then you can split it over spheres. This is the "prime decomposition" and is unique. You can then split the components of this decomposition along tori. If you leave the ...
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### geometrically finite hyperbolic surface of infinite volume

I am starting to read some papers involving analysis on hyperbolic manifolds. In these the notion of a "geometrically finite hyperbolic surface of infinite volume" is mentioned frequently and I am ...
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### Embedding manifolds of constant curvature in manifolds of other curvatures

I know that there is no complete surface embedded in $\mathbb{R}^3$ of constant curvature -$k$ for any $k$. But you can clearly embed the hyperbolic plane (curvature -1) into hyperbolic 3-space ...
Let $\mathbb{H}$ be the Poincare upper half-plane, seen as a Riemannian manifold with the metric $$\frac{dx^2+dy^2}{y^2}.$$ Moreover, we consider the action of $\text{SL}_2(\mathbb{R})$ on ...
### Universal cover of complete hyperbolic surfaces and torsion-free, discrete groups of isometries of $\mathbb{H}^2$
I'm taking a course this semester, and in it we proved that any complete hyperbolic surface is universally covered by $\mathbb{H}^2$. The text, found at ...