Related papers: Unit sphere fibrations in Euclidean space

We consider unitary graphs attached to Z_d^n using an analogue of the Euclidean distance. These graphs are shown to be integral when n is odd or the dimension d is even.

We prove that all smooth sphere bundles that admit fiberwise 1/4-pinched metrics are induced bundles of vector bundles, so their structure groups reduce from the diffeomorphism group of the sphere to the orthogonal group. This result…

We show there exists a packing of identical spheres in $\mathbb{R}^d$ with density at least \[ (1-o(1))\frac{d \log d}{2^{d+1}}\, , \] as $d\to\infty$. This improves upon previous bounds for general $d$ by a factor of order $\log d$ and is…

We propose methods towards a systematic determination of d dimensional curved spaces where Euclidean field theories with rigid supersymmetry can be defined. The analysis is carried out from a group theory as well as from a supergravity…

A manifold $M$ is said to be a double disk bundle if it can be decomposed as a union of two disk bundles glued together by a diffeomorphism of their boundaries. We show that if $M^n$ is a closed simply connected $n$-manifold with $n$ even…

We study embeddings of the unit sphere of complex Hilbert spaces of dimension a power $2^n$ into the corresponding groups of non-singular linear transformations. For the case of $n=1$, the sphere $S_2$ of qubits is identified with…

We study Funk-type transforms on the unit sphere in R^n associated with cross-sections of the sphere by lower-dimensional planes passing through an arbitrary fixed point inside the sphere or outside. Our main concern is injectivity of the…

We prove the existence of optimal separators for intersection graphs of balls and spheres in any dimension $d$. One of our results is that if an intersection graph of $n$ spheres in $\mathbb{R}^d$ has $m$ edges, then it contains a balanced…

We show two sphere theorems for the Riemannian manifolds with scalar curvature bounded below and the non-collapsed $\mathrm{RCD}(n-1,n)$ spaces with mean distance close to $\frac{\pi}{2}$.

We provide sharp upper bounds for the number of symmetrizations required to transform a star shaped set in ${\mathbb R}^n$ arbitrarily close (in the Hausdorff metric) to the Euclidean ball.

Let $X$ be a compact real algebraic set of dimension $n$. We prove that every Euclidean continuous map from $X$ into the unit $n$-sphere can be approximated by regulous map. This strengthens and generalizes previously known results.

It is well-known that the Pachner graph of $n$-vertex triangulated $2$-spheres is connected, i.e., each pair of $n$-vertex triangulated $2$-spheres can be turned into each other by a sequence of edge flips for each $n\geq 4$. In this…

We show that the maximum number of unit distances or of diameters in a set of n points in d-dimensional Euclidean space is attained only by specific types of Lenz constructions, for all d >= 4 and n sufficiently large, depending on d. As a…

The Apollonius problem asks for a sphere tangent to $n+1$ given spheres or hyperplanes in $\mathbb{R}^n$. This problem has been widely studied for an isolated configuration of $n+1$ spheres. In this paper, we study relations among the…

In this survey, we remind some fibrations structure theorems (also called Milnor's fibrations) recently proved in the real and complex case, in the local and global settings. We give several Poincar\'e-Hopf type formulae which relates the…

For primes p>=3, Cohen, Moore, and Neisendorfer showed that the exponent of the p-torsion in the homotopy groups of S^2n+1 is p^n. This was obtained as a consequence of a thorough analysis of the homotopy theory of Moore spaces. Anick…

We consider the geometric join of a family of subsets of the Euclidean space. This is a construction frequently used in the (colorful) Carath\'eodory and Tverberg theorems, and their relatives. We conjecture that when the family has at…

In this paper, we prove that, for any integer $n\ge 2,$ there exists an $\epsilon_{n} \ge 0$ so that if $M$ is an n-dimensional complete manifold with sectional curvature $ K_{M}\ge 1$ and if $M$ has conjugate radius bigger than…

Nevo, Santos, and Wilson constructed $2^{\Omega(N^d)}$ combinatorially distinct simplicial $(2d-1)$-spheres with $N$ vertices. We prove that all spheres produced by one of their methods are shellable. Combining this with prior results of…

Let $\Lambda$ be any integral lattice in Euclidean space. It has been shown that for every integer $n>0$, there is a hypersphere that passes through exactly $n$ points of $\Lambda$. Using this result, we introduce new lattice invariants and…