MathOverflow

We know that spectral rigidity fails in general (Gordon–Webb–Wolpert, 1992). However, consider the following highly symmetric family of 9‑dimensional closed manifolds:
$$M(a)=\mathbb{S}^3(a)\times\mathbb{S}^3\!\left(\frac{a}{\sqrt{150}}\right)\times\mathbb{S}^3\!\left(\frac{a}{\sqrt{150k_0}}\right),\qquad a>0,\;k_0>1\text{ fixed}.$$...

I would like to gain a better understanding of Günther's trick---the one used by Matthias Günther in his proof of Nash's embedding theorem.I’ve heard rumors that many systems of PDEs can be solved using the same trick.
If you know of such examples, please share them....

Many of you know dynamic geometry exercises in Euclidea; if not, here is one example.It lets you do a geometric construction and sends a message once you achieve the result.
I am looking for a way to produce exercises for students in a similar form.At the moment I use C.a.R. which uses Java plugins.But as you may know, all major browsers stopped supporting Java plugins in 2014 or so.So I have to make my students install...

Is there a standard/good reference text that does category of metric spaces?
Say, it seems that by looking at this category, one can recover everything about a particular metric space up to scaling --- is it written somewhere, or does it follow from something standard?...