Abstract We study global maxima of distance functions on most Alexandrov surfaces with curvature bounded below, where most is used in the sense of Baire categories.

Abstract We investigate slant surfaces in the almost Hermitian manifold 𝕊3 × ℝ, considering the position of the Reeb vector field ξ of the Sasakian structure on 𝕊3 with respect to the surfaces. We examine two cases: ξ normal or tangent to the surfaces. In the first case, we prove that every surface is totally real. In the second case, we characterize and locally describe complex surfaces. Finally, we completely classify non-complex slant surfaces, giving explicit examples.

Abstract We present a way of computing the degree of the secant (resp. tangent) variety of a smooth projective surface, under the assumption that the divisor giving the embedding in the projective space is 3-very ample. This method exploits the link between these varieties and the Hilbert scheme 0-dimensional subschemes of length 2 of the surface.

Abstract In the first part, we define and investigate new classes of almost 3-contact metric manifolds, with two guiding ideas in mind: first, what geometric objects are best suited for capturing the key properties of almost 3-contact metric manifolds, and second, the new classes should admit ‘good’ metric connections with skew torsion. In particular, we introduce the Reeb commutator function and the Reeb Killing function, we define the new classes of canonical almost 3-contact metric manifolds and of 3-(α, δ)-Sasaki manifolds (including as special cases 3-Sasaki manifolds, quaternionic Heisenberg groups, and many others) and prove that the latter are hypernormal, thus generalizing a seminal result of Kashiwada. We study their behaviour under a new class of deformations, called 𝓗-homothetic deformations, and prove that they admit an underlying quaternionic contact structure, from which we deduce the Ricci curvature. For example, a 3-(α, δ)-Sasaki manifold is Einstein either if α = δ (the 3-α-Sasaki case) or if δ = (2n + 3)α, where dim M = 4n + 3. In the second part we find these adapted connections. We start with a very general notion of φ-compatible connections, where φ denotes any element of the associated sphere of almost contact structures, and make them unique by a certain extra condition, thus yielding the notion of canonical connection (they exist exactly on canonical manifolds, hence the name). For 3-(α, δ)-Sasaki manifolds, we compute the torsion of this connection explicitly and we prove that it is parallel, we describe the holonomy, the ∇-Ricci curvature, and we show that the metric cone is a HKT-manifold. In dimension 7, we construct a cocalibrated G 2-structure inducing the canonical connection and we prove the existence of four generalized Killing spinors.

Abstract We exploit an elementary specialization technique to study rational curves on Fano varieties of index one less than their dimension, known as del Pezzo manifolds. First, we study the splitting type of the normal bundles of the rational curves. Second, we prove a simple formula relating the number of rational curves passing through a suitable number of points in the case of threefolds and the analogous invariants for del Pezzo surfaces.

Abstract In this short note we prove a sharp lower bound for the second moment of a lattice Voronoi cell in terms of the respective covering radius. This gives an affirmative answer to a conjecture by Haviv, Lyubashevsky and Regev. We also characterize those lattice Voronoi cells for which this lower bound is attained.

Abstract We classify the rank two commutative semifields which are 8-dimensional over their center 𝔽 q . This is done using computational methods utilizing the connection to linear sets in PG(2, q 4). We then apply our methods to complete the classification of rank two commutative semifields which are 10-dimensional over 𝔽3. The implications of these results are detailed for other geometric structures such as semifield flocks, ovoids of parabolic quadrics, and eggs.

Abstract After nearly 50 years of research the Chern conjecture for isoparametric hypersurfaces in spheres is still an unsolved and important problem. Here we give a partial result for CMC hypersurfaces with constant Gauß–Kronecker curvature, mainly using a result given in [3] by Otsuki.

Abstract We define generalized null 2-type submanifolds in the m-dimensional Euclidean space 𝔼 m . Generalized null 2-type submanifolds are a generalization of null 2-type submanifolds defined by B.-Y. Chen satisfying the condition Δ H = f H + gC for some smooth functions f, g and a constant vector C in 𝔼 m , where Δ and H denote the Laplace operator and the mean curvature vector of a submanifold, respectively. We study developable surfaces in 𝔼3 and investigate developable surfaces of generalized null 2-type surfaces. As a result, all cylindrical surfaces are proved to be of generalized null 2-type. Also, we show that planes are the only tangent developable surfaces which are of generalized null 2-type. Finally, we characterize generalized null 2-type conical surfaces.