Related papers: Some Elementary Aspects of 4-dimensional Geometry

This paper concentrates on the homogeneous (conformal) model of Euclidean space (Horosphere) with subspaces that intuitively correspond to Euclidean geometric objects in three dimensions. Mathematical details of the construction and…

We show that the sum over geometries in the Lorentzian 4-D state sum model for quantum GR in [1] includes terms which correspond to geometries on manifolds with conical singularities. Natural approximations suggest that they can be…

Noncommutative geometry has seen remarkable applications for high energy physics, viz. the geometrical interpretation of the Standard Model. The question whether it also allows for supersymmetric theories has so far not been answered in a…

Conditions, related to the so-called bending problem are considered for hypersurfaces of a pseudo-Euclidean space. Corresponding theorems are proved.

In this paper we introduce a new algebraic device, which enables us to treat the quaternions as though they were a commutative field. This is of interest both for its own sake, and because it can be applied to develop an "algebraic…

We investigate proper biharmonic hypersurfaces with at most three distinct principal curvatures in space forms. We obtain the full classification of proper biharmonic hypersurfaces in 4-dimensional space forms.

A Heron triangle is one that has all integer side lengths and integer area, which takes its name from Heron of Alexandria's area formula. From a more relaxed point of view, if rescaling is allowed, then one can define a Heron triangle to be…

In a recent paper, algebraic descriptions for all non-relativistic spins were derived by elementary means directly from the Lie algebra $\specialorthogonalliealgebra{3}$, and a connection between spin and the geometry of Euclidean…

This is the first comprehensive introduction to the authors' recent attempts toward a better understanding of the global concepts behind spinor representations of surfaces in 3-space. The important new aspect is a quaternionic-valued…

We seek for an alternative to the metric tensor $g_{\mu\nu}$ as a fundamental geometrical object in four-dimensional Riemannian manifolds. We suggest that the metric tensor $g_{\mu\nu}(P)$ at a given point $P$ of a manifold may be replaced…

The purpose of this paper is, first, to give an algorithm that enables to obtain the lines of curvature on parametric hypersurfaces in Euclidean 4-space, and then, to obtain the curvatures of such lines by using the extended Darboux frame…

Heron angle: both its sine and cosine are rational Heron triangle: all its sides and area are rational Heron Parallelogram: all its sides, diagonals and area are rational We give one-to-one (bijective) parametrizations for all three…

We show that the eigenschemes of $4 \times 4 \times 4$ symmetric tensors are parametrized by a linear subvariety of the Grassmannian $\operatorname{Gr}(3,\mathbb{P}^{14})$. We also study the decomposition of the eigenscheme into the…

We generalise the notions of supersymmetry and superspace by allowing generators and coordinates transforming according to more general Lorentz representations than the spinorial and vectorial ones of standard lore. This yields novel…

Following a previous proposition of quaternity spacetime for electronic orbitals in neon shell, this paper describes the geometrical course each electron takes as it oscillates harmonically within a certain quaternity space dimension and…

A comprehensive analysis of the morphology of the solution space for a special type of quadratic quaternion equation is presented. This equation, which arises in a surface construction problem, incorporates linear terms in a quaternion…

Several sets of quaternionic functions are described and studied with respect to hy-perholomorphy, addition and (non commutative) multiplication, on open sets of H, then Hamil-ton 4-manifolds analogous to Riemann surfaces, for H instead of…

A spherical quadrilateral is a bordered surface homeomorphic to a closed disk, with four distinguished boundary points called corners, equipped with a Riemannian metric of constant curvature 1, except at the corners, and such that the…

In this article we find connections between the values of Gauss hypergeometric functions and the dimension of the vector space of Hodge cycles of four dimensional cubic hypersurfaces. Since the Hodge conjecture is well-known for those…

It is known that the groups of Euclidean rotations in dimension 3 (isometries of $S^2$), general Lorentz transformations in dimension 4 (Hyperbolic isometries in dimension 3), and screw motions in dimension 3 can be represented by the…