We discuss neutral harmonic maps from Riemann surfaces into Lie groups, that is, maps which are harmonic with respect to the so called neutral connection, a natural bi-invariant torsion free affine connection. In particular we study neutral harmonic maps from oriented surfaces into the solvable Lie model of the complex hyperbolic plane ℂℍ^2(-1) . We use a representation formula of Weierstrass type to construct some elementary examples of neutral harmonic maps from the plane into the complex hyperbolic plane ℂℍ^2(-1) .
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.
