Volume 17, Issue 3

We study left-invariant almost paracontact metric structures on arbitrary three-dimensional Lorentzian Lie groups. We obtain a complete classification and description under a natural assumption, which includes relevant classes as normal and almost para-cosymplectic structures, and we investigate geometric properties of these structures.

Plane curves with the same endpoints are homotopic; an analogous claim for plane curves with the same endpoints and bounded curvature still remains open. We find necessary and sufficient conditions for two plane curves with bounded curvature to be deformed into each other by a continuous one-parameter family of curves also having bounded curvature. We conclude that the space of these curves has either one or two connected components, depending on the distance between the endpoints. The classification theorem presented here answers a question raised in 1961 by L. E. Dubins.

It is known that manifolds with G 2 structures have almost contact metric structures; see [3; 16]. In this manuscript, we deform a parallel G 2 structure by a parallel vector field and investigate the properties of the almost contact metric structure obtained by the deformation of the G 2 structure.

The k th projection function υ k ( K , ·) of a convex body K ⊂ ℝ d , d ≥ 3, is a function on the Grassmannian G ( d , k ) which measures the k -dimensional volume of the projection of K onto members of G ( d , k ). For k = 1 and k = d − 1, simple formulas for the projection functions exist. In particular, υ d -1 ( K , ·) can be written as a spherical integral with respect to the surface area measure of K . Here, we generalize this result and prove two integral representations for υ k ( K , ·), k = 1,..., d − 1, over flag manifolds. Whereas the first representation generalizes a result of Ambartzumian (1987), but uses a flag measure which is not continuous in K , the second representation is related to a recent flag formula for mixed volumes by Hug, Rataj and Weil (2013) and depends continuously on K .

We prove a Bonnet theorem for isometric immersions of submanifolds into the products of an arbitrary number of simply connected real space forms. Then we prove the existence of associate families of minimal surfaces in such products. Finally, in the case of 𝕊 2 × 𝕊 2 , we give a complex version of the main theorem in terms of the two canonical complex structures of 𝕊 2 × 𝕊 2 .

A hemisystem on the Hermitian surface ℋ(3, q 2 ), q ≥ 7 odd, admitting a subgroup of PΩ - (4, q ) of order q 2 ( q +1) is constructed. Also, a new family of Cameron-Liebler line classes of PG(3, q ), q ≥ 5 odd, with parameter ( q 2 + 1)/2 is provided.

We introduce successive radii in generalized Minkowski spaces (that is, with respect to gauges) and study some first properties. This is done via formulating some kind of minimal containment problems, where intersections or Minkowski sums of the latter set and affine flats of a certain dimension are incorporated. Since this is strongly related to minimax location problems, we also look at ball intersections and ball hulls.

In this paper, we establish two sharp inequalities for the normalized δ -Casorati curvatures of submanifolds in a Kenmotsu space form, tangent to the structure vector field of the ambient space.Moreover, we show that in both cases the equality at all points characterizes the totally geodesic submanifolds.

Let H be a hypersurface in ℝ n and let π be an orthogonal projection in ℝ n restricted to H . We say that H satisfies the Archimedean projection property corresponding to π if there exists a constant C such that Vol( π –1 ( U )) = C ·Vol( U ) for every measurable U in the range of π . It iswell-known that the ( n –1)-dimensional sphere, as a hypersurface in ℝ n , satisfies the Archimedean projection property corresponding to any codimension 2 orthogonal projection in ℝ n , the range of any such projection being an ( n – 2)-dimensional ball. Here we construct new hypersurfaces that satisfy Archimedean projection properties. Our construction works for any projection codimension k with 2 ≤ k ≤ n – 1, and it allows us to specify a wide variety of desired projection ranges Ω n – k ⊂ ℝ n – k . Letting Ω n – k be the ( n – k )-dimensional ball for each k , it produces a new family of smooth, compact hypersurfaces in ℝ n satisfying codimension k Archimedean projection properties that includes, in the special case k = 2, the ( n – 1)-dimensional spheres.

Let π : X → Y be a morphism of projective varieties and π ∗ : N k ( X ) → N k ( Y ) the pushforward map of numerical cycle classes. We show that when the Chow groups of points of the fibers are as simple as they can be, then the kernel of π ∗ is spanned by k -cycles contracted by π .

We construct anew family of flat Minkowski planes that admit the simple group PSL 2 (ℝ) as a group of automorphisms. These planes are obtained from the classical flat Minkowski plane by ‘bending’ circles at a distinguished circle.