Recently, in Innamorati and Zuanni (J. Geom 111:45, 2020. https://doi.org/10.1007/s00022-020-00557-0) the authors give a characterization of a Baer cone of mathrm {PG}(3, q^2), q a prime power, as a subset of points of the projective space intersected by any line in at least one point and by every plane in q^2+1, q^2+q+1 or q^3+q^2+1 points. In this paper, we show that a similar characterization holds even without assuming that the order of the projective space is a square, and weakening the assumptions on the three intersection numbers with respect to the planes.

It is known that minimal surfaces in Euclidean space can be represented in terms of holomorphic functions. For example, we have the well-known Weierstrass representation, where part of the holomorphic data is chosen to be the stereographic projection of the normal of the corresponding surface, and also the Bj\"orling representation, where it is prescribed a curve on the surface and the unit normal on this curve. In this work, we are interested in the holomorphic representation of minimal surfaces in simply isotropic space, a three-dimensional space equipped with a rank 2 metric of index zero. Since the isotropic metric is degenerate, a surface normal cannot be unequivocally defined based on metric properties only, which leads to distinct definitions of an isotropic normal. As a consequence, this may also lead to distinct forms of a Weierstrass and of a Bj\"orling representation. Here, we show how to represent simply isotropic minimal surfaces in accordance with the choice of an isotropic surface normal.

In this paper some new geometric characterizations are given of the following algebraic properties of the underlying field K of an Euclidean plane $$E: -1$$ is a square in K, the group of squares is of index 2 in $$K^*$$ , K is Pythagorean, and K is Euclidean. In the first part ordinary incidence conditions of lines and circles in E are used for this purpose, while in the second these properties are characterized by comparing four betweenness relations introduced in E in a natural way.

In the three dimensional Riemannian space forms, we introduce a natural moving frame to define associate curve of a curve. Using the notion of associate curve we give a new necessary and sufficient condition of which a Frenet curve is a Mannheim curve or Mannheim partner curve in the three dimensional Euclidean space. Then we generalize these conclusions to the curves which lie on the three dimensional Riemannian sphere and the curves which lie in the three dimensional hyperbolic space. We also give the geometric characterizations of these curves. Our methods can be easily used to reveal the properties of the curves on space forms.

Object of study in the present paper are slant and Legendre null curves in 3-dimensional Sasaki-like almost contact B-metric manifolds. For the examined curves we express the general Frenet frame and the Frenet frame for which the original parameter is distinguished, as well as the corresponding curvatures, in terms of the structure on the manifold. We prove that the curvatures of a framed null slant and Legendre curve are constants if and only if a specific function for the considered manifolds is a constant. We find a necessary and sufficient condition a slant null curve to be a generalized helix and a Legendre null curve to be a null cubic. For some of investigated curves we show that they are non-null slant or Legendre curves with respect to the associated B-metric on the manifold. We give examples of the examined curves. Some of them are constructed in a 3-dimensional Lie group as Sasaki-like manifold and their matrix representation is obtained.

Given a locally finite set X subseteq {{mathbb {R}}}^d and an integer k ge 0, we consider the function {mathbf{w}_{k}^{}} :{mathrm{Del}_{k}{({X})}} rightarrow {{mathbb {R}}} on the dual of the order-k Voronoi tessellation, whose sublevel sets generalize the notion of alpha shapes from order-1 to order-k (Edelsbrunner et al. in IEEE Trans Inf Theory IT-29:551–559, 1983; Krasnoshchekov and Polishchuk in Inf Process Lett 114:76–83, 2014). While this function is not necessarily generalized discrete Morse, in the sense of Forman (Adv Math 134:90–145, 1998) and Freij (Discrete Math 309:3821–3829, 2009), we prove that it satisfies similar properties so that its increments can be meaningfully classified into critical and non-critical steps. This result extends to the case of weighted points and sheds light on k-fold covers with balls in Euclidean space.

For a given curve $${\mathcal {C}}$$ and a given angle $$\theta $$ , the $$\theta $$ -isoptic curve of $${\mathcal {C}}$$ is the geometric locus of points through which passes a pair of tangents to $${\mathcal {C}}$$ making an angle equal to $$\theta $$ . If the curve $${\mathcal {C}}$$ is smooth and convex, isoptics exist for any angle, and through every point exterior to the curve, there is exactly one pair of tangents. The isoptics of conics are well known. In this paper, we explore the inner isoptics of ellipses, i.e. the envelopes of the lines joining the points of contact of the ellipse with the tangents through points on a given isoptic. If $$\theta =90^{\circ }$$ , the isoptic is called orthoptic and the corresponding inner isoptic is called the inner orthoptic. We show that the inner orthoptic of an ellipse is an ellipse, but in general the inner isoptics are more complicated.

In the present paper, we derive optimal inequalities involving generalized normalized $$\delta $$ -Casorati curvatures for slant submanifolds in a golden Riemannian space form. We obtain these inequalities by analysing a suitable constrained extrememum problem on submanifold.

In this paper, we deal with a parabolic stable spacelike hypersurface immersed in a generalized Robertson–Walker spacetime. We consider the support type function as the product between the warping function and the hyperbolic cosine of the hyperbolic angle of the normal vector field with the timelike parallel direction $$\partial _t$$ . We provide some conditions under which the support type function must be constant or, particularly, when the hypersurface must be a slice, yielding a Calabi–Bernstein type result.