Invariant connections on Euclidean space

On the Invariant Theory of Weingarten Surfaces in Euclidean Space

We prove that any strongly regular Weingarten surface in Euclidean space carries locally geometric principal parameters. The basic theorem states that any strongly regular Weingarten surface is determined up to a motion by its structural functions and the normal curvature function satisfying a geometric differential equation. We apply these results to the special Weingarten surfaces: minimal su...

On the Quaternionic Curves in the Semi-Euclidean Space E_4_2

In this study, we investigate the semi-real quaternionic curves in the semi-Euclidean space E_4_2. Firstly, we introduce algebraic properties of semi-real quaternions. Then, we give some characterizations of semi-real quaternionic involute-evolute curves in the semi-Euclidean space E42 . Finally, we give an example illustrated with Mathematica Programme.

Invariant classification of orthogonally separable Hamiltonian systems in Euclidean space

The term “manifold of n dimensions” in this setting describes a set of n variables that independently take on the real values from −∞ to ∞ ([12], p 116). Motivated by this idea, one can assert that Euclidean geometry of E (Euclidean space) can be completely characterized by the invariants of the Euclidean group of transformations. As is well-known, this Lie group of (orientation-preserving) iso...

Linear Connections of Normal Space to a Variety in Euclidean Space

On invariant measures of the Euclidean algorithm

We study the ergodic properties of the additive Euclidean algorithm f defined in R+. A natural extension of f is obtained using the action of SL(2,Z) on a subset of SL(2,R). We prove that even though f is ergodic and has an infinite invariant measure equivalent to the Lebesgue measure, such a measure is not unique; (in fact there is a continuous family of such measures). While it is folklore th...