The standard model of Mercator projection shows a cylinder wrapped around a spherical earth eg Wiki. Many sites describe the resulting square map like this: "...spherical Mercator …

Problem: Suppose that $f:S^n\to S^n$ is a mapping from the n-dimensional sphere ($n\geq 3$) into itself which maps circles into (instead of onto) circles. Can we say that f maps (n …

Consider the Euclidian space $E_n={\mathbb R}^n$, with standard scalar product $$x\cdot y=x_1y_1+\cdots+x_ny_n.$$ A closed convex cone $\Gamma\subset E_n$ defines an order by $y\ge …

Conjecture. If $n>1$ and $f$ is a mapping from $S^n$ to $S^n$ which maps circles into (instead of onto) circles, and whose range has n+3 distinct points any n+2 of which are in gen …

I know how to compute the content of orthoschemes in 3- and 4-dimensional spherical space from dihedral angles using Schlafli series computations. Can anyone direct me to a textboo …

What is known about isometric actions on $\mathbb S^n$ such that the quotient space is homeomorphic to $\mathbb S^n$? Comments. I am mostly interested in (maybe trivial) pro …

I am interested in finding a canonical general expression for the area of a spherical polygon in $\mathbb{S}^2$ knowing the side lengths of the polygon and a bound on the internal …

There is a well known pattern that turns up in nature involving the golden ratio $\phi = \frac{\sqrt{5}-1}{2}$. To get this "sunflower spiral" pattern, put the $k$th node at an …

using only a spherical ruler (to construct great lines) and a pair of compasses, how can you construct a regular dodecahedron on the surface of the sphere? thank you very much.

Let $X$ be the unit sphere in $\ell^2$, i.e. $X=\{x\in\ell^2: \|x\|=1\}$. Let the metric on $X$ be the geodesic metric, i.e. $d(x,y)=\cos^{-1}\langle x,y\rangle$. Call a set a ba …

Where would one put $n$ telescopes on the surface of the earth to see the whole sky as well as possible ? Use the cosine metric to define how well we can see in direction $x$: $ \q …

There is a well-known duality between compact symmetric spaces and symmetric spaces of noncompact type. Basically it goes as follows: If $G/K$ is a symmetric space of noncompact ty …

It is possible that on a sphere $S^n$ there is a natural Riemannian metric in $R^(n+1)$. But it is not always possible for pseudo Riemann metric since the sum of two symmetric matr …

I have been trying to figure this problem out for a while, and while I believe someone must have figured it out hundreds of years ago, I still can't quite get it. Suppose we have …

Does every homeomorphism of the unit sphere S^n, n=2, has diffeomorphic extension to the unit ball. I am indeed interesten about the reference of the following problem: I need a g …