Volume 17, Issue 4

We prove that there exist non-trivial (i.e. not Einstein) Ricci solitons on non-conformally flat four-dimensional Lorentzian Walker manifolds. Moreover, we show that only steady Ricci solitons may be gradient ones.

Let π be a subplane of PG(2, q 3 ) of order q that is exterior to ℓ ∞ . The exterior splash of π is the set of q 2 + q +1 points on ℓ ∞ that lie on the extended lines of π . Exterior splashes are projectively equivalent to scattered linear sets of rank 3, covers of the circle geometry CG (3, q ), and hyper-reguli of PG(5, q ). In this article we use the Bruck–Bose representation in PG(6, q ) to give a geometric characterisation of carrier conics of π in terms of the covers of the exterior splash of π . We also investigate properties of subplanes of order q with a common exterior splash, and study the intersection of two exterior splashes.

Let M be a complete connected C 2 -surface in ℝ 3 in general position, intersecting some plane along a clean figure-8 (a loop with total curvature zero) and such that all compact intersections with planes have central symmetry. We prove that M is a (geometric) cylinder over some central figure-8. On the way, we establish interesting facts about centrally symmetric loops in the plane; for instance, a clean loop with even rotation number 2 k can never be central unless it passes through its center exactly twice and k = 0.

Vieta’s classical formulae explicitly determine the coefficients of a polynomial p ∈ 𝔽[ x ] in terms of the roots of p , where 𝔽 is any commutative ring. In this paper, Vieta’s formulae are obtained for slice-regular polynomials over the noncommutative algebra of quaternions, by an argument which essentially relies on induction, without invoking quasideterminants or noncommutative symmetric functions.

We prove that sausages are the family of ‘extremal sets’ in relation to certain linear improvements of Minkowski’s first inequality when working with projection/sections assumptions. In particular they characterize the equality cases of the corresponding linear refinements of both the isoperimetric inequality and Urysohn’s inequality. We also characterize sausages by algebraic properties of the roots of Steiner polynomials, in which other functionals of convex bodies such as the inradius, the mean width or the diameter are involved.

New types of maximal symplectic partial spreads are constructed.

We study S -convex sets, which are the geometric objects obtained as the intersection of the usual convex sets in ℝ d with a proper subset S ⊂ ℝ d , and contribute new results about their S -Helly numbers. We extend prior work for S = ℝ d , ℤ d , and ℤ d − k × ℝ k , and give some sharp bounds for several new cases: low-dimensional situations, sets that have some algebraic structure, in particular when S is an arbitrary subgroup of ℝ d or when S is the difference between a lattice and some of its sublattices. By abstracting the ingredients of Lovász method we obtain colorful versions of many monochromatic Helly-type results, including several colorful versions of our own results.

Using the semidiameter (in connection to the mean radius and surface radius) of a convex closed hypersurface in ℝ n , n ≥ 2, as a sharp upper bound of the variational p -capacity radius where p ∈ (1, n ), this paper settles a restriction or variant of S.-T. Yau’s Problem 59 in [66] from the surface area to the variational p -capacity whose limit as p → 1 actually induces the surface area.

We show that a 30° circular sector of unit radius contains an isometric copy of every drapeable unit arc, and we describe the family of drapeable exit arcs of unit length in the sector. The conjecture that this sector is a cover for the family of all unit arcs remains unresolved.

Uniform Lie algebras are combinatorially defined two-step nilpotent Lie algebras which can be used to define Einstein solvmanifolds. These Einstein spaces often have nontrivial isotropy groups. We derive basic properties of uniform Lie algebras and we classify uniform Lie algebras with five or fewer generators. We define a type of directed colored graph called a uniformly colored graph and establish a correspondence between uniform Lie algebras and uniformly colored graphs. We present several methods of constructing infinite families of uniformly colored graphs and corresponding uniform Lie algebras.

Let 𝓒 be a finite family of distinct boxes in ℝ d , let S = ⋃ { C : C in 𝓒} and let G be the intersection graph of 𝓒. For each block of G , assume that the corresponding members of 𝓒 have a staircase convex union. Then S with the rectilinear metric is a median space. Moreover, if every two points of S see a common point via staircase paths in S , then S is staircase starshaped.