Contact path geometries are curved geometric structures on a contact man-ifold comprising smooth families of paths modeled on the family of all isotropic lines in the projectivization of a symplectic vector space. Locally such a structure is equivalent to the graphs in the space of independent and depedent variables of the family of solutions of a system of an odd number of second order ODE's s...

The Willmore energy for Frenet curves in quaternionic projective space HP is the generalization of the Willmore functional for immersions into S. Critical points of the Willmore energy are called Willmore curves in HP. Using a Bäcklund transformation on Willmore curves, we generalize Bryant’s result on Willmore spheres in 3–space: a Willmore sphere in HP has integer Willmore energy, and is give...

The spectral action functional, considered as a model of gravity coupled to matter, provides, in its non-perturbative form, a slow-roll potential for inflation, whose form and corresponding slowroll parameters can be sensitive to the underlying cosmic topology. We explicitly compute the non-perturbative spectral action for some of the main candidates for cosmic topologies, namely the quaternion...

We construct a continuous family of new homogeneous Einstein spaces with negative Ricci curvature, obtained by deforming from the quaternionic hyperbolic space of real dimension 12. We give an explicit description of this family, which is made up of Einstein solvmanifolds which share the same algebraic structure (eigenvalue type) as the rank one symmetric space HH. This deformation includes a c...

Jørgensen’s inequality gives a necessary condition for a non-elementary two generator group of isometries of real hyperbolic 2-space to be discrete. We give analogues of Jørgensen’s inequality for non-elementary groups of isometries of quaternionic hyperbolic n-space generated by two elements, one of which is loxodromic. Mathematics Subject Classifications (2000): 20H10, 22E40, 57S30.

The properties of spinors and vectors in (2 + 2) space split quaternions are studied. Quaternionic representation rotations naturally separates two SO(2,1) subgroups the full group symmetry norms quaternions, SO(2,2). One them represents symmetries three-dimensional Minkowski space-time. Then, second subgroup, generated by additional time-like coordinate from basis can be viewed as internal mod...

The original aim of this paper is to construct a graph associated to a vector space. By inspiration of the classical definition for the Cayley graph related to a group we define Cayley graph of a vector space. The vector space Cayley graph ${rm Cay(mathcal{V},S)}$ is a graph with the vertex set the whole vectors of the vector space $mathcal{V}$ and two vectors $v_1,v_2$ join by an edge whenever...

Quaternionic polynomials are generated by quaternionic variables and the quaternionic product. This paper proposes the generating ideal of quaternionic polynomials in tensor algebra, finds the Gröbner base of the ideal in the case of pure imaginary quaternionic variables, and describes the normal forms of such quaternionic polynomials explicitly.