We study a curvature-dependent variational problem for curves on an oriented surface S⊂ℝ^3 . The functional 𝒦_e=∫ (k_n-H) ds measures the signed deviation of the normal curvature k_n from the mean curvature H. Using Weierstrass’ symmetric form, we derive the Euler–Lagrange equation in curvature-line coordinates and reformulate it as a first-order differential equation for the angle between the tangent vector and a principal direction. This formulation is analogous to the tangent-angle equation for geodesics derived via Liouville’s formula. By reparametrizing the curves using conformal arclength, we express the equation’s coefficients in terms of the conformal principal curvatures. This allows us to prove that the family of extremals is invariant under spatial inversions. Additionally, for surfaces of revolution, we derive an explicit first integral. Finally, excluding the totally umbilic case of a sphere, we demonstrate that the existence of a non-principal loxodromic extremal forces the surface’s meridian to have constant Euclidean curvature. We also characterize Dupin cyclides as the unique surfaces where all (k_n-H) -extremals are loxodromes of the principal net. Furthermore, we establish that the circular cylinder and the sphere (as a degenerate case) are the only surfaces for which every Euclidean geodesic is also a (k_n-H) -extremal.
We give an estimate on magnetic focal values under an assumption that sectional curvatures are bounded from above.
The perimeter-analogue of the centroid of a triangle, when this is viewed as the intersection of the area-bisecting cevians, is the intersection of the perimeter-bisecting cevians, or the Nagel point. The length of the shortest part of the perimeter cut by a variable line through an interior point P of a triangle ranges over a closed interval [w, s], where s is the semiperimeter and w=w(P) is a real number between 0 and s. When P is the Nagel point, we compute the expression of w(P) in terms of the side lengths and show that the range of w(P)/(2s) over all triangles is the interval (0, 4/9], where 4/9 is attained by the equilateral triangle, and 0 is approached by triangles with sides in bi-ratio approaching 0 : 1 : 1. This result is a perimeter-analogue of the classical Winternitz theorem in a triangle, asserting that, in the case of area-ratio instead of perimeter-ratio and P is the centroid, in every triangle, w(P)/σ =4/9 , where σ is the area of the triangle. Hence, the range of w(P)/σ over all triangles is the singleton set {4/9} .
Abstract We present a novel proof for the fact that thick spherical buildings of types $$\mathsf {H_3}$$ H 3 and $$\mathsf {H_4}$$ H 4 do not exist. For that, we first provide an elementary axiom system for Lie incidence geometries associated with buildings of type $$\mathsf {H_3}$$ H 3 . This way, we can write our arguments purely in the language of point-line geometries, not needing the theory of buildings. Assuming the existence of thick spherical buildings of type $$\mathsf {H_3}$$ H 3 , we construct nontrivial root elations of generalized pentagons contained within them. This leads to a contradiction with Tits’s result on the nonexistence of Moufang generalized pentagons. Consequently, we obtain a new, direct, and geometric proof for the nonexistence of thick spherical buildings of types $$\mathsf {H_3}$$ H 3 and $$\mathsf {H_4}$$ H 4 , without invoking Tits’s extension theorem. Together with similar geometric constructions of the author for root elations of buildings of types $$\mathsf {B_n}$$ B n and $$\mathsf {C_n}$$ C n , and known constructions for buildings of type $$\mathsf {A_n}$$ A n , this yields an alternative, elementary proof for the fact that all thick irreducible spherical buildings of rank 3 have the Moufang property, not using Tits’s extension theorem.
Asking for syntactically simple axioms that are equivalent, with plane absolute geometry in the sense of Hilbert as background theory, to Aristotle’s axiom Ar (“The perpendiculars dropped from one side of an angle to the other grow without bound”) and to the Lotschnittaxiom L (“The perpendiculars raised on the sides of a right angle intersect”), we find that: (1) Ar can be expressed as a positive statement in terms of points, betweenness, and equidistance; (2) Ar cannot be expressed positively in terms of points and collinearity; (3) the simplest form of Ar, with respect to quantifier type, in terms of points and collinearity, is AAAAEA (a prenex statement in which four universal quantifiers are followed by an existential quantifier, which in turn is followed by a universal quantifier); (4) L can be expressed as a positive sentence, using only ∧ as logical connective, in terms of points and collinearity; (5) the simplest forms of L in terms of points and collinearity are prenex statements of quantifier types AAAAAAE and AAAAAEE.
We consider a non degenerate surface M in the Minkowski space and a regular curve on it with a well defined causality. Over such curve we construct, locally, a normal surface Σ as a ruled surface whose rules are non degenerate straight lines orthogonal to M. In the first part we consider immersed timelike surfaces. We prove that if the curve is a lightlike straight line in a timelike surface then it is a line of curvature. This is equivalent to the condition that Σ is a lightlike surface. As an application we deduce that if at every point of M pass two lightlike straight lines then M is umbilical. Moreover, we prove that through every point of M pass two lightlike geodesics with constant curvatures if and only if M is a parallel surface. Finally, we consider non degenerate immersed surfaces and we give another characterization of parallel surfaces: Through every point of M pass three different curves, of any well defined causality, such that their normal surfaces are either extremal or lightlike if and only if M is a parallel surface.
In this paper, envelopes of a family of parabolas in the plane are investigated. For a given family of parabolas, we obtain a necessary and sufficient condition concerning if the family of parabolas creates an envelope. Furthermore, we study the relation among an envelope of a family of parabolas in the plane and an envelope of the family of directrixes corresponding to the family of parabolas.
We investigate Cayley graphs of graph products of groups. We prove that if the vertex groups admit isomorphic Cayley graphs with respect to chosen symmetric generating sets, then the associated graph products have isomorphic Cayley graphs with respect to induced generating sets. Then we provide examples of non-isomorphic graph products of finite groups whose Cayley graphs are nevertheless isomorphic.
We investigate conformal Fedosov structures on (pseudo-) Riemannian manifolds of Roter type. Using the integrability condition for the conformal Fedosov equation ∇ω = θ⊗ω , we show that the Lee form is closed and can be gauged away, reducing the problem to the classical Fedosov case with a parallel symplectic form. By exploiting the Roter-type decomposition of the curvature tensor, we establish that any such manifold must have constant sectional curvature. In particular, conformal Fedosov structures do not exist on non-trivial Roter type manifolds. This result extends earlier non-existence theorems for symmetric, recurrent, and quasi-constant curvature manifolds to the full Roter type class, revealing the strong rigidity of conformally Fedosov geometry.
We have introduced lightcone framed surfaces and lightcone framed curves as mixed type surfaces and mixed type curves with singular points, respectively. A more general notion of singular surfaces, called generalised lightcone framed surfaces, is introduced in this paper. The notion of generalised lightcone framed surfaces includes not only the notion of lightcone framed surfaces, but also the notion of one-parameter families of lightcone framed curves at least locally. We investigate properties of generalised lightcone framed surfaces.
This paper introduces a new class of geometric mappings called pointwise almost h-conformal semi-slant Riemannian maps (pahcssR maps) between Riemannian manifolds and almost quaternionic Hermitian manifolds. These maps unify and extend existing concepts like Riemannian submersions, conformal maps, and semi-slant submersions to the quaternionic setting. We establish fundamental properties and provide necessary and sufficient conditions for these maps to be harmonic or totally geodesic. We also investigate the decomposition of the tangent bundle and analyze the geometric behavior of the associated distributions. Several illustrative examples are provided to support the theoretical framework. Finally, we discuss potential applications in mathematical physics, where quaternionic structures are prevalent in theories such as supergravity and string theory.
We present a novel proof for the fact that thick spherical buildings of types H3 and H4 do not exist. For that, we first provide an elementary axiom system for Lie incidence geometries associated with buildings of type H3. This way, we can write our arguments purely in the language of point-line geometries, not needing the theory of buildings. Assuming the existence of thick spherical buildings of type H3, we construct non-trivial root elations of generalized pentagons contained within them. This leads to a contradiction with Tits's result on the nonexistence of Moufang generalized pentagons. Consequently, we obtain a new, direct, and geometric proof for the nonexistence of thick spherical buildings of types H3 and H4, without invoking Tits's extension theorem. Together with similar geometric constructions of the author for root elations of buildings of types Bn and Cn, and known constructions for buildings of type An, this yields an alternative, elementary proof for the fact that all thick irreducible spherical buildings of rank 3 have the Moufang property, not using Tits's extension theorem.
Richelot, F., in the Journal für die reine und angewandte Mathematik, 250–267 (1830) developed the Fuss formulae for bicentric polygons of high order, circuminscribed between two circles. In particular, he demonstrated a doubling technique for deriving the formulae for 2n -sided polygons, from the formulae for n -sided polygons. While reviewing and implementing these results from 1830, we noticed that the formulae were not producing closed polygons. Upon further inspection, we uncovered the errant typos and fixed them.
We present a novel proof for the fact that thick spherical buildings of types 𝖧_3 and 𝖧_4 do not exist. For that, we first provide an elementary axiom system for Lie incidence geometries associated with buildings of type 𝖧_3 . This way, we can write our arguments purely in the language of point-line geometries, not needing the theory of buildings. Assuming the existence of thick spherical buildings of type 𝖧_3 , we construct nontrivial root elations of generalized pentagons contained within them. This leads to a contradiction with Tits’s result on the nonexistence of Moufang generalized pentagons. Consequently, we obtain a new, direct, and geometric proof for the nonexistence of thick spherical buildings of types 𝖧_3 and 𝖧_4 , without invoking Tits’s extension theorem. Together with similar geometric constructions of the author for root elations of buildings of types 𝖡_𝗇 and 𝖢_𝗇 , and known constructions for buildings of type 𝖠_𝗇 , this yields an alternative, elementary proof for the fact that all thick irreducible spherical buildings of rank 3 have the Moufang property, not using Tits’s extension theorem.
Grace’s inequality is a strengthening of the classical Euler inequality for arbitrary tetrahedra. In this paper, we present a simple and elementary proof of Grace’s inequality in the special case of a right tetrahedron.
The main problem considered in this paper is “how does a dual polar space Γ of rank 3 embed in a metasymplectic space Δ ?” The expected and generic answer is that Γ is isomorphic to a subgeometry of a point residual _Δ (p) and that it arises as a subgeometry of a trace geometry, that is, Γ⊆ p^⊥∩ q^⋈ , for two opposite points p and q, where q^⋈ is the set of points special to q. However, this is not always the case, and we describe some counterexamples, even classify them for certain classes of metasymplectic spaces Δ . These results complement the analogous results for the exceptional geometries of diameter at most 3 arising from groups of types 𝖤_6,𝖤_7,𝖤_8 recently treated by Cooperstein and the second author.
This paper presents a rigorous generalization of the classical Bäcklund transformation (BT) for space curves in Euclidean 3-space 𝔼^3 using the versatile quasi-frame (q-frame). The novelty lies in introducing the frame rotation angle, θ (s) , as a continuous geometric control parameter for designing BTs that satisfy fundamental geometric constraints, notably the constant separation distance r. We derive a coupled system of first-order ordinary differential equations (ODEs) governing the evolution of both the frame rotation θ (s) and the inclination angle γ (s) of the transformation. A key theoretical contribution is the demonstration that this class of BTs universally preserves the second quasi-curvature ( κ̃_2 ) and the quasi-torsion ( τ̃ ) of the q-frame, implying that the geometric action is concentrated solely on the first quasi-curvature ( κ _1 ). This invariance offers deep, intrinsic insight into the geometric nature of the transformation. Our formulation naturally encompasses the classical Frenet and Bishop frame BTs as special static cases. Numerical simulations starting from a circular helix illustrate the framework’s ability to generate complex, non-helical curves under specific dynamic geometric constraints, highlighting the utility of θ (s) in geometric control theory.
We investigate(pseudo-)Riemannian spaces whose curvature tensor possesses a specific structure called split curvature. First, we provide a direct and simplified proof that every space of split curvature is semisymmetric. This result refines previous approaches by eliminating unnecessary assumptions and clarifying inaccuracies in earlier classifications. We then propose a natural classification of these spaces according to the rank of the Ricci tensor, distinguishing three geometric types. The results of Sinyukov and Mikeš on geodesic mappings of semisymmetric spaces and on equidistant spaces are employed here. Furthermore, we analyze geodesic mappings of spaces of split curvature and demonstrate the existence of spaces that admit nontrivial geodesic mappings while having non-constant scalar curvature. This disproves a previously published theorem by Kiosak asserting that the scalar curvature must be constant in this context. By constructing explicit examples, we provide a more precise and transparent description of the geometric structure of spaces of split curvature, including their classification, mapping properties, and curvature characteristics. Special attention is given to the role of concircular vector fields in equidistant spaces, which yields new insights into their structure and mapping behavior.
We consider various well-known properties of Minkowski planes and clarify dependencies between them. In particular, we show that the existence of all circle symmetries implies that all reflections about generators exist.
As is well known from Fenchel’s theorem, the total absolute curvature of any closed space curve in Euclidean 3-space is always not less than 2π . By the fundamental theorem of space curves, a curve is determined by its curvature and torsion functions, and in the case of a closed curve, these functions must be periodic. However, it remains an open problem to decide which periodic curvature and torsion functions actually give rise to closed space curves; this is the so-called closed curve problem. In this paper, we propose a constructive method to obtain closed spherical curves by controlling only the values of the first integral of a periodic function, which then determines their periodic curvature and torsion. We further provide a solution to the closed curve problem in the special case of rotationally symmetric curves. In addition, our approach extends naturally to closed curves with singularities, including closed planar Legendre curves, closed spherical fronts and closed framed curves in Euclidean 3-space. For these cases, we show precise conditions that characterize when (2,3)-cusps occur on such curves.
