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Results tagged with mg.metric-geometry
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user 7311
Euclidean, hyperbolic, discrete, convex, coarse geometry, metric spaces, comparisons in Riemannian geometry, symmetric spaces.
32
votes
5
answers
6k
views
What is a good method to find random points on the n-sphere when n is large?
As part of a more complex algorithm, I need a fast method to find random points of the n-sphere, $S^n$, starting with a RNG (random number generator). A simple way to do this (in low dimensions at lea …
- 15.3k
27
votes
$C^1$ isometric embedding of flat torus into $\mathbb{R}^3$
On the other hand, if you are willing to settle for conformally flat, there is a beautiful theory of these. (The idea is to consider flat embeddings in the three-sphere, and then "project them into $R …
- 15.3k
18
votes
1
answer
859
views
What sort of models did Bolyai and Lobachevsky use to demonstrate the consistency of their m...
As is well-known, in the 1820s both Bolyai and Lobachevsky showed, at long last, the independence of the Parallel Postulate from the rest of the axioms of Euclidean geometry by developing what we now …
- 15.3k
16
votes
Shortest-path Distances Determining the Metric?
There is an old paper of mine called "On the Differentiability of Isometries" in which I show that if you know a Riemannian manifold $M$ only as a metric space, i.e., you just know its point set and t …
- 15.3k
6
votes
Intuitive proof that the first $(n-2)$ coordinates on a sphere are uniform in a ball
I find myself very confused by all this, and I suspect I must be missing something very important, and I am hoping someone (Greg?) can set me straight.
Let's just consider the classic case $n=3$, so …
- 15.3k
5
votes
Easy proof of the fact that isotropic spaces are Euclidean
There is a classic paper by Jordan and von Neumann where they prove results that allows this question is settled in an elementary way.
On Inner Products in Linear, Metric Spaces Author(s): P. Jordan …
- 15.3k
2
votes
History of the triangle inequality
Here is a suggestion, to get the idea across in an informal way---it is what I always tell the students when I introduce the triangle inequality: I tell them that its essential content, and the way it …
- 15.3k