An isometry is a map between two metric spaces which preserves all distances: for all . (After typing a bunch of such formulas, one tends to prefer shorter notation: , with the metric inferred from contexts.) It’s a popular exercise to prove that every isometry from a compact metric space into itself is surjective. 355 more words

## Tags » Geodesics

#### Sofia - A vantage point

When you start talking about moving to Sofia, you will feel a lot of *pushback*. It’s funny because I’ve lived a lot of places, and Sofia is the first city where I’ve felt pushback. 559 more words

#### Poem for October...

**October**O hushed October morning mild,

Thy leaves have ripened to the fall;

Tomorrow’s wind, if it be wild,

Should waste them all. 108 more words

#### Dark Matter & Energy - Part 2.3

### Fate of the Universe

### Part 2.3 – Determining Density (Omega Ω)

## Introduction

In the last post we saw that the fate of universe, whether it will a ‘Big Crunch or a ‘Big Chill’ or a ‘Big Rip’ depends on the density of matter in the universe, represented by greek letter Omega Ω. 589 more words

#### TUNE OF THE DAY: UNBOUND BY CATHEDRALS (GEODESICS REMIX)

Geodesics, an Austin, TX based project, is one of the many bedroom producer outfits popping up in today’s music world, but this remix of Cathedral’s “Unbound” is good enough for us to take note of this relatively unknown artist. 51 more words

#### Four-point CAT(0) condition

The definition of a metric space guarantees that if we pick three points from it, we can draw a triangle in the Euclidean plane, with vertices labeled so that the side lengths are equal to the corresponding distances in the metric space. 533 more words

#### Three hyperbolic metrics

Up to a constant factor, there is just one conformally invariant Riemannian metric on the disk . Indeed, on the tangent space at the metric must be a multiple of the Euclidean one, due to rotational invariance. 514 more words