Volume 21, Issue 2

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.

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.

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.

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 .

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.

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.

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.

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.

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.

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.

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].

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 .

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.

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.