Related papers: Orthogonal separation of variables for spaces of c…
The classical Cartan's structural equations show in a compact way the relation between a connection and its curvature, and reveals their geometric interpretation in terms of moving frames. In order to study the mathematical properties of…
We discuss when static Killing vector fields are standard, that is, leading to a global orthogonal splitting of the spacetime. We prove that such an orthogonal splitting is unique whenever the natural space is compact. This is carried out…
The existence of orthogonal local coordinates is a generalization of the manifold being conformally flat. It is always possible to construct orthogonal coordinates on 2-manifolds, using geometric normal coordinates or isothermal…
We algebraically compute all possible sectional curvature values for canonical algebraic curvature tensors, and use this result to give a method for constructing general sectional curvature bounds. We use a well-known method to…
Orthogonal polynomials on quadratic curves in the plane are studied. These include orthogonal polynomials on ellipses, parabolas, hyperbolas, and two lines. For an integral with respect to an appropriate weight function defined on any…
Decomposing tensors into orthogonal factors is a well-known task in statistics, machine learning, and signal processing. We study orthogonal outer product decompositions where the factors in the summands in the decomposition are required to…
Integrable Killing tensors are used to classify orthogonal coordinates in which the classical Hamilton-Jacobi equation can be solved by a separation of variables. We completely solve the Nijenhuis integrability conditions for Killing…
We present a method for calculating the results of operation of differential operators operating on components of vector in generalized coordinates not restricted to orthogonal one. For this we use the relationships between covariant,…
We review the theory of orthogonal separation of variables on pseudo-Riemannian manifolds of constant non-zero curvature via concircular tensors and warped products. We then apply this theory simultaneously to both the three-dimensional…
We focus on studying, numerically, the scalar curvature tensor in a two-dimensional discrete space. The continuous metric of a two-sphere is transformed into that of a lattice using two possible slicings. In the first, we use two integers,…
We use ODEs and symmetry arguments to construct closed constant curvature space curves, first on cylinders, next on tori, at last with the Frenet-Serret equations.
In this article, we give a geometric description for any invertible operator on a finite dimensional inner--product space. With the aid of such a description, we are able to decompose any given conformal transformation as a product of…
We develop general expressions for the raising and lowering operators that belong to the orthogonal polynomials of hypergeometric type with discrete and continuous variable. We construct the creation and annihilation operators that…
A piecewise flat manifold is a triangulated manifold given a geometry by specifying edge lengths (lengths of 1-simplices) and specifying that all simplices are Euclidean. We consider the variation of angles of piecewise flat manifolds as…
Discrete orthogonal matrices have several applications in information technology, such as in coding and cryptography. It is often challenging to generate discrete orthogonal matrices. A common approach widely in use is to discretize…
The traditional study of plane and space algebraic curves by looking at their tangent vectors, curvatures and torsions provides geometric, but unfortunately not sufficient information about individual curves in order to be able to…
In the field of algebraic geometric codes (AG codes), the characterization of dual codes has long been a challenging problem which relies on differentials. In this paper, we provide some descriptions for certain differentials utilizing…
We describe a smooth structure, called Fr\"olicher space, on CW complexes and spaces of triangulations. This structure enables differential methods for e.g. minimization of functionnals. As an application, we exhibit how an optimized…
We establish several inequalities for manifolds with positive scalar curvature and, more generally, for the scalar curvature bounded from below, in the spirit of the classical bound on the distances between conjugates points in surfaces…
While every matrix admits a singular value decomposition, in which the terms are pairwise orthogonal in a strong sense, higher-order tensors typically do not admit such an orthogonal decomposition. Those that do have attracted attention…