Published by De Gruyter

Volume 21 Issue 2

Issue of Advances in Geometry

Contents
Journal Overview

Publicly Available April 16, 2021

Frontmatter

Page range: i-iii

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April 16, 2021

Complex geodesics in convex domains and ℂ-convexity of semitube domains

Sylwester Zając, Paweł Zapałowski

Page range: 149-162

Abstract

In this paper the complex geodesics of a convex domain in ℂ n are studied. One of the main results provides a certain necessary condition for a holomorphic map to be a complex geodesic for a convex domain in ℂ n . The established condition is of geometric nature and it allows to find a formula for every complex geodesic. The ℂ-convexity of semitube domains is also discussed.

February 5, 2021

Triharmonic Riemannian submersions from 3-dimensional space forms

Tomoya Miura, Shun Maeta

Page range: 163-168

Abstract

We show that any triharmonic Riemannian submersion from a 3-dimensional space form into a surface is harmonic. This is an affirmative partial answer to the submersion version of the generalized Chen conjecture. Moreover, a non-existence theorem for f -biharmonic Riemannian submersions is also presented.

September 11, 2019

Ricci almost solitons and contact geometry

Amalendu Ghosh

Page range: 169-178

Abstract

We prove that on a K -contact manifold, a Ricci almost soliton is a Ricci soliton if and only if the potential vector field V is a Jacobi field along the Reeb vector field ξ . Then we study contact metric as a Ricci almost soliton with parallel Ricci tensor. To this end, we consider Ricci almost solitons whose potential vector field is a contact vector field and prove some rigidity results.

April 16, 2021

Shapes of centrally symmetric octahedra with prescribed cone-deficits

Zili Wang

Page range: 179-185

Abstract

The space of Euclidean cone metrics on centrically symmetric octahedra with fixed cone angles θi < 2 π , with total surface area 1, has a natural hyperbolic metric, and is locally isometric to hyperbolic 3-space. The metric completion of the space is isometric to a hyperbolic ideal tetrahedron whose dihedral angles are half the cone-deficits 2 π − θi .

April 10, 2021

Equivariant Ulrich bundles on exceptional homogeneous varieties

Kyoung-Seog Lee, Kyeong-Dong Park

Page range: 187-205

Abstract

We prove that the only rational homogeneous varieties with Picard number 1 of the exceptional algebraic groups admitting irreducible equivariant Ulrich vector bundles are the Cayley plane E 6 / P 1 and the E 7 -adjoint variety E 7 / P 1 . From this result,we see that a general hyperplane section F 4 / P 4 of the Cayley plane also has an equivariant but non-irreducible Ulrich bundle.

April 13, 2021

Exotic Steiner chains in Miquelian Möbius planes of odd order

Norbert Hungerbühler, Gideon Villiger

Page range: 207-220

Abstract

In the Euclidean plane, two circles that intersect or are tangent clearly do not carry a finite Steiner chain of circles. We show that such exotic Steiner chains exist in finite Miquelian Möbius planes of odd order. We obtain explicit conditions in terms of the order of the plane and the capacitance of the two carrier circles for the existence, length, and number of Steiner chains.

April 16, 2021

Double cover K3 surfaces of Hirzebruch surfaces

Taro Hayashi

Page range: 221-225

Abstract

General K3 surfaces obtained as double covers of the n -th Hirzebruch surfaces with n = 0, 1, 4 are not double covers of other smooth surfaces. We give a criterion for such a K 3 surface to be a double covering of another smooth rational surface based on the branch locus of double covers and fibre spaces of Hirzebruch surfaces.

March 13, 2021

The dual cone of sums of non-negative circuit polynomials

Mareike Dressler, Helen Naumann, Thorsten Theobald

Page range: 227-236

Abstract

For a non-empty, finite subset A⊆ N0n $\mathcal{A} \subseteq \mathbb{N}_0^n$ let C sonc (𝒜) ∈ ℝ[ x 1 , . . . , x n ] be the cone of sums of non-negative circuit polynomials with support 𝒜. We derive a representation of the dual cone ( C sonc (𝒜)) ∗ and deduce an optimality criterion for sums of non-negative circuit polynomials in polynomial optimization.

March 11, 2021

Einstein tori and crooked surfaces

Jean-Philippe Burelle, Virginie Charette, Dominik Francoeur, William M. Goldman

Page range: 237-250

Abstract

In hyperbolic space, the angle of intersection and distance classify pairs of totally geodesic hyperplanes. A similar algebraic invariant classifies pairs of hyperplanes in the Einstein universe. In dimension 3, symplectic splittings of a 4-dimensional real symplectic vector space model Einstein hyperplanes and the invariant is a determinant. The classification contributes to a complete disjointness criterion for crooked surfaces in the 3-dimensional Einstein universe.

April 16, 2021

Compact null hypersurfaces in Lorentzian manifolds

C. Atindogbé, M. Gutiérrez, R. Hounnonkpe

Page range: 251-263

Abstract

We show how the topological and geometric properties of the family of null hypersurfaces in a Lorentzian manifold are related with the properties of the ambient manifold itself. In particular, we focus in how the presence of global symmetries and curvature conditions restrict the existence of compact null hypersurfaces. We use these results to show the influence on the existence of compact totally umbilic null hypersurfaceswhich are not totally geodesic. Finally we describe the restrictions that they impose in causality theory.

April 16, 2021

Hyperbolic torsion polynomials of pretzel knots

Takayuki Morifuji, Anh T. Tran

Page range: 265-272

Abstract

In this paper, we explicitly calculate the highest degree term of the hyperbolic torsion polynomial of an infinite family of pretzel knots. This gives supporting evidence for a conjecture of Dunfield, Friedl and Jackson that the hyperbolic torsion polynomial determines the genus and fiberedness of a hyperbolic knot. The verification of the genus part of the conjecture for this family of knots also follows from the work of Agol and Dunfield [1] or Porti [19].

September 22, 2020

Geodesic orbit Randers metrics on spheres

Shaoxiang Zhang, Zaili Yan

Page range: 273-280

Abstract

Studying geodesic orbit Randers metrics on spheres, we obtain a complete classification of such metrics. Our method relies upon the classification of geodesic orbit Riemannian metrics on the spheres S n in [17] and the navigation data in Finsler geometry. We also construct some explicit U( n + 1)-invariant metrics on S 2 n +1 and Sp( n + 1)U(1)-invariant metrics on S 4 n +3 .

March 11, 2021

Rational conic fibrations of sectional genus two

Antonio Lanteri, Raquel Mallavibarrena

Page range: 281-292

Abstract

Polarized rational surfaces ( X , L) of sectional genus two ruled in conics are studied. When they are not minimal, they are described as the blow-up of 𝔽 1 at some points lying on distinct fibers. Ampleness and very ampleness of L are studied in terms of their location. When L is very ample and there is a line contained in X and transverse to the fibers, the conic fibrations ( X , L) are classified and a related property concerned with the inflectional locus is discussed.

March 11, 2021

Hodge numbers of hypersurfaces in ℙ⁴ with ordinary triple points

Sławomir Cynk

Page range: 293-298

Abstract

We give a formula for the Hodge numbers of a three-dimensional hypersurface in a weighted projective space with only ordinary triple points as singularities.

About this journal

Objective
Advances in Geometry is a mathematical journal for the publication of original research articles of excellent quality in the area of geometry. Geometry is a field of long standing-tradition and eminent importance. The study of space and spatial patterns is a major mathematical activity; geometric ideas and geometric language permeate all of mathematics.

Topics

Finite geometry
Incidence geometry
Tits buildings
Combinatorial aspects of geometry and geometric combinatorics
Geometric group theory
Geometric aspects of Lie groups and of algebraic groups
Algebraic geometry
Topological geometry
Differential geometry
Convex sets and polytopes
Discrete geometry

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Volume 21 Issue 2

Issue of Advances in Geometry

Frontmatter

Complex geodesics in convex domains and ℂ-convexity of semitube domains

Abstract

Triharmonic Riemannian submersions from 3-dimensional space forms

Abstract

Ricci almost solitons and contact geometry

Abstract

Shapes of centrally symmetric octahedra with prescribed cone-deficits

Abstract

Equivariant Ulrich bundles on exceptional homogeneous varieties

Abstract

Exotic Steiner chains in Miquelian Möbius planes of odd order

Abstract

Double cover K3 surfaces of Hirzebruch surfaces

Abstract

The dual cone of sums of non-negative circuit polynomials

Abstract

Einstein tori and crooked surfaces

Abstract

Compact null hypersurfaces in Lorentzian manifolds

Abstract

Hyperbolic torsion polynomials of pretzel knots

Abstract

Geodesic orbit Randers metrics on spheres

Abstract

Rational conic fibrations of sectional genus two

Abstract

Hodge numbers of hypersurfaces in ℙ4 with ordinary triple points

Abstract

Hodge numbers of hypersurfaces in ℙ⁴ with ordinary triple points